C2^3.689C2^4 - GroupNames

G = C₂³.₆₈₉C₂⁴ order 128 = 2⁷

406^th central stem extension by C₂³ of C₂⁴

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂³.₆₈₉C₂⁴, C₂².₄₆₂2₊¹⁺⁴, C₂².₃₅₂2_-¹⁺⁴, C₄²⋊₈C₄.₄₉C₂, (C₂×C₄²).₇₁₆C₂², (C₂²×C₄).₆₀₁C₂³, (C₂²×Q₈).₂₂₁C₂², C₂³.₈₄C₂³.₈C₂, C₂³.₈₁C₂³.₄₂C₂, C₂.C₄².₃₉₃C₂², C₂³.₇₈C₂³.₂₄C₂, C₂³.₆₃C₂³.₅₂C₂, C₂³.₈₃C₂³.₃₉C₂, C₂³.₆₇C₂³.₅₈C₂, C₂.₁₂₀(C₂².₄₅C₂⁴), C₂.₆₀(C₂².₃₅C₂⁴), C₂.₅₀(C₂².₅₇C₂⁴), C₂.₇₂(C₂².₅₀C₂⁴), C₂.₁₀₇(C₂².₃₃C₂⁴), (C₂×C₄).₂₃₁(C₄○D₄), (C₂×C₄⋊C₄).₄₉₉C₂², C₂².₅₅₀(C₂×C₄○D₄), SmallGroup(128,1521)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂³ — C₂³.₆₈₉C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂³.₆₃C₂³ — C₂³.₆₈₉C₂⁴

Lower central

C₁ — C₂³ — C₂³.₆₈₉C₂⁴

Upper central

C₁ — C₂³ — C₂³.₆₈₉C₂⁴

Jennings

C₁ — C₂³ — C₂³.₆₈₉C₂⁴

Generators and relations for C₂³.₆₈₉C₂⁴
G = < a,b,c,d,e,f,g | a²=b²=c²=1, d²=abc, e²=g²=ba=ab, f²=a, ac=ca, ede^-1=ad=da, geg^-1=ae=ea, af=fa, ag=ga, bc=cb, fdf^-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef^-1=ce=ec, cf=fc, cg=gc, gdg^-1=abd, fg=gf >

Subgroups: 308 in 176 conjugacy classes, 88 normal (22 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₄², C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂.C₄², C₂.C₄², C₂×C₄², C₂×C₄², C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²×Q₈, C₄²⋊₈C₄, C₂³.₆₃C₂³, C₂³.₆₇C₂³, C₂³.₇₈C₂³, C₂³.₈₁C₂³, C₂³.₈₃C₂³, C₂³.₈₄C₂³, C₂³.₆₈₉C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂².₃₃C₂⁴, C₂².₃₅C₂⁴, C₂².₄₅C₂⁴, C₂².₅₀C₂⁴, C₂².₅₇C₂⁴, C₂³.₆₈₉C₂⁴

Smallest permutation representation of C₂³.₆₈₉C₂⁴
►Regular action on 128 points

Generators in S₁₂₈

(1 103)(2 104)(3 101)(4 102)(5 69)(6 70)(7 71)(8 72)(9 75)(10 76)(11 73)(12 74)(13 77)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 42)(38 43)(39 44)(40 41)(45 106)(46 107)(47 108)(48 105)(49 112)(50 109)(51 110)(52 111)(53 114)(54 115)(55 116)(56 113)(57 120)(58 117)(59 118)(60 119)(61 122)(62 123)(63 124)(64 121)(65 128)(66 125)(67 126)(68 127)

(1 12)(2 9)(3 10)(4 11)(5 44)(6 41)(7 42)(8 43)(13 47)(14 48)(15 45)(16 46)(17 51)(18 52)(19 49)(20 50)(21 55)(22 56)(23 53)(24 54)(25 59)(26 60)(27 57)(28 58)(29 63)(30 64)(31 61)(32 62)(33 67)(34 68)(35 65)(36 66)(37 71)(38 72)(39 69)(40 70)(73 102)(74 103)(75 104)(76 101)(77 108)(78 105)(79 106)(80 107)(81 110)(82 111)(83 112)(84 109)(85 116)(86 113)(87 114)(88 115)(89 118)(90 119)(91 120)(92 117)(93 124)(94 121)(95 122)(96 123)(97 126)(98 127)(99 128)(100 125)

(1 76)(2 73)(3 74)(4 75)(5 37)(6 38)(7 39)(8 40)(9 102)(10 103)(11 104)(12 101)(13 106)(14 107)(15 108)(16 105)(17 112)(18 109)(19 110)(20 111)(21 114)(22 115)(23 116)(24 113)(25 120)(26 117)(27 118)(28 119)(29 122)(30 123)(31 124)(32 121)(33 128)(34 125)(35 126)(36 127)(41 72)(42 69)(43 70)(44 71)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)(65 66 67 68)(69 70 71 72)(73 74 75 76)(77 78 79 80)(81 82 83 84)(85 86 87 88)(89 90 91 92)(93 94 95 96)(97 98 99 100)(101 102 103 104)(105 106 107 108)(109 110 111 112)(113 114 115 116)(117 118 119 120)(121 122 123 124)(125 126 127 128)

(1 90 74 60)(2 27 75 120)(3 92 76 58)(4 25 73 118)(5 114 39 23)(6 54 40 88)(7 116 37 21)(8 56 38 86)(9 57 104 91)(10 117 101 28)(11 59 102 89)(12 119 103 26)(13 124 108 29)(14 64 105 94)(15 122 106 31)(16 62 107 96)(17 67 110 97)(18 127 111 34)(19 65 112 99)(20 125 109 36)(22 72 113 43)(24 70 115 41)(30 78 121 48)(32 80 123 46)(33 81 126 51)(35 83 128 49)(42 85 71 55)(44 87 69 53)(45 95 79 61)(47 93 77 63)(50 100 84 66)(52 98 82 68)

(1 111 103 52)(2 83 104 19)(3 109 101 50)(4 81 102 17)(5 29 69 93)(6 64 70 121)(7 31 71 95)(8 62 72 123)(9 112 75 49)(10 84 76 20)(11 110 73 51)(12 82 74 18)(13 85 77 21)(14 113 78 56)(15 87 79 23)(16 115 80 54)(22 48 86 105)(24 46 88 107)(25 35 89 99)(26 66 90 125)(27 33 91 97)(28 68 92 127)(30 40 94 41)(32 38 96 43)(34 117 98 58)(36 119 100 60)(37 122 42 61)(39 124 44 63)(45 114 106 53)(47 116 108 55)(57 67 120 126)(59 65 118 128)

(1 77 74 47)(2 48 75 78)(3 79 76 45)(4 46 73 80)(5 66 39 100)(6 97 40 67)(7 68 37 98)(8 99 38 65)(9 14 104 105)(10 106 101 15)(11 16 102 107)(12 108 103 13)(17 24 110 115)(18 116 111 21)(19 22 112 113)(20 114 109 23)(25 96 118 62)(26 63 119 93)(27 94 120 64)(28 61 117 95)(29 90 124 60)(30 57 121 91)(31 92 122 58)(32 59 123 89)(33 41 126 70)(34 71 127 42)(35 43 128 72)(36 69 125 44)(49 56 83 86)(50 87 84 53)(51 54 81 88)(52 85 82 55)

G:=sub<Sym(128)| (1,103)(2,104)(3,101)(4,102)(5,69)(6,70)(7,71)(8,72)(9,75)(10,76)(11,73)(12,74)(13,77)(14,78)(15,79)(16,80)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,42)(38,43)(39,44)(40,41)(45,106)(46,107)(47,108)(48,105)(49,112)(50,109)(51,110)(52,111)(53,114)(54,115)(55,116)(56,113)(57,120)(58,117)(59,118)(60,119)(61,122)(62,123)(63,124)(64,121)(65,128)(66,125)(67,126)(68,127), (1,12)(2,9)(3,10)(4,11)(5,44)(6,41)(7,42)(8,43)(13,47)(14,48)(15,45)(16,46)(17,51)(18,52)(19,49)(20,50)(21,55)(22,56)(23,53)(24,54)(25,59)(26,60)(27,57)(28,58)(29,63)(30,64)(31,61)(32,62)(33,67)(34,68)(35,65)(36,66)(37,71)(38,72)(39,69)(40,70)(73,102)(74,103)(75,104)(76,101)(77,108)(78,105)(79,106)(80,107)(81,110)(82,111)(83,112)(84,109)(85,116)(86,113)(87,114)(88,115)(89,118)(90,119)(91,120)(92,117)(93,124)(94,121)(95,122)(96,123)(97,126)(98,127)(99,128)(100,125), (1,76)(2,73)(3,74)(4,75)(5,37)(6,38)(7,39)(8,40)(9,102)(10,103)(11,104)(12,101)(13,106)(14,107)(15,108)(16,105)(17,112)(18,109)(19,110)(20,111)(21,114)(22,115)(23,116)(24,113)(25,120)(26,117)(27,118)(28,119)(29,122)(30,123)(31,124)(32,121)(33,128)(34,125)(35,126)(36,127)(41,72)(42,69)(43,70)(44,71)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96)(97,98,99,100)(101,102,103,104)(105,106,107,108)(109,110,111,112)(113,114,115,116)(117,118,119,120)(121,122,123,124)(125,126,127,128), (1,90,74,60)(2,27,75,120)(3,92,76,58)(4,25,73,118)(5,114,39,23)(6,54,40,88)(7,116,37,21)(8,56,38,86)(9,57,104,91)(10,117,101,28)(11,59,102,89)(12,119,103,26)(13,124,108,29)(14,64,105,94)(15,122,106,31)(16,62,107,96)(17,67,110,97)(18,127,111,34)(19,65,112,99)(20,125,109,36)(22,72,113,43)(24,70,115,41)(30,78,121,48)(32,80,123,46)(33,81,126,51)(35,83,128,49)(42,85,71,55)(44,87,69,53)(45,95,79,61)(47,93,77,63)(50,100,84,66)(52,98,82,68), (1,111,103,52)(2,83,104,19)(3,109,101,50)(4,81,102,17)(5,29,69,93)(6,64,70,121)(7,31,71,95)(8,62,72,123)(9,112,75,49)(10,84,76,20)(11,110,73,51)(12,82,74,18)(13,85,77,21)(14,113,78,56)(15,87,79,23)(16,115,80,54)(22,48,86,105)(24,46,88,107)(25,35,89,99)(26,66,90,125)(27,33,91,97)(28,68,92,127)(30,40,94,41)(32,38,96,43)(34,117,98,58)(36,119,100,60)(37,122,42,61)(39,124,44,63)(45,114,106,53)(47,116,108,55)(57,67,120,126)(59,65,118,128), (1,77,74,47)(2,48,75,78)(3,79,76,45)(4,46,73,80)(5,66,39,100)(6,97,40,67)(7,68,37,98)(8,99,38,65)(9,14,104,105)(10,106,101,15)(11,16,102,107)(12,108,103,13)(17,24,110,115)(18,116,111,21)(19,22,112,113)(20,114,109,23)(25,96,118,62)(26,63,119,93)(27,94,120,64)(28,61,117,95)(29,90,124,60)(30,57,121,91)(31,92,122,58)(32,59,123,89)(33,41,126,70)(34,71,127,42)(35,43,128,72)(36,69,125,44)(49,56,83,86)(50,87,84,53)(51,54,81,88)(52,85,82,55)>;

G:=Group( (1,103)(2,104)(3,101)(4,102)(5,69)(6,70)(7,71)(8,72)(9,75)(10,76)(11,73)(12,74)(13,77)(14,78)(15,79)(16,80)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,42)(38,43)(39,44)(40,41)(45,106)(46,107)(47,108)(48,105)(49,112)(50,109)(51,110)(52,111)(53,114)(54,115)(55,116)(56,113)(57,120)(58,117)(59,118)(60,119)(61,122)(62,123)(63,124)(64,121)(65,128)(66,125)(67,126)(68,127), (1,12)(2,9)(3,10)(4,11)(5,44)(6,41)(7,42)(8,43)(13,47)(14,48)(15,45)(16,46)(17,51)(18,52)(19,49)(20,50)(21,55)(22,56)(23,53)(24,54)(25,59)(26,60)(27,57)(28,58)(29,63)(30,64)(31,61)(32,62)(33,67)(34,68)(35,65)(36,66)(37,71)(38,72)(39,69)(40,70)(73,102)(74,103)(75,104)(76,101)(77,108)(78,105)(79,106)(80,107)(81,110)(82,111)(83,112)(84,109)(85,116)(86,113)(87,114)(88,115)(89,118)(90,119)(91,120)(92,117)(93,124)(94,121)(95,122)(96,123)(97,126)(98,127)(99,128)(100,125), (1,76)(2,73)(3,74)(4,75)(5,37)(6,38)(7,39)(8,40)(9,102)(10,103)(11,104)(12,101)(13,106)(14,107)(15,108)(16,105)(17,112)(18,109)(19,110)(20,111)(21,114)(22,115)(23,116)(24,113)(25,120)(26,117)(27,118)(28,119)(29,122)(30,123)(31,124)(32,121)(33,128)(34,125)(35,126)(36,127)(41,72)(42,69)(43,70)(44,71)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96)(97,98,99,100)(101,102,103,104)(105,106,107,108)(109,110,111,112)(113,114,115,116)(117,118,119,120)(121,122,123,124)(125,126,127,128), (1,90,74,60)(2,27,75,120)(3,92,76,58)(4,25,73,118)(5,114,39,23)(6,54,40,88)(7,116,37,21)(8,56,38,86)(9,57,104,91)(10,117,101,28)(11,59,102,89)(12,119,103,26)(13,124,108,29)(14,64,105,94)(15,122,106,31)(16,62,107,96)(17,67,110,97)(18,127,111,34)(19,65,112,99)(20,125,109,36)(22,72,113,43)(24,70,115,41)(30,78,121,48)(32,80,123,46)(33,81,126,51)(35,83,128,49)(42,85,71,55)(44,87,69,53)(45,95,79,61)(47,93,77,63)(50,100,84,66)(52,98,82,68), (1,111,103,52)(2,83,104,19)(3,109,101,50)(4,81,102,17)(5,29,69,93)(6,64,70,121)(7,31,71,95)(8,62,72,123)(9,112,75,49)(10,84,76,20)(11,110,73,51)(12,82,74,18)(13,85,77,21)(14,113,78,56)(15,87,79,23)(16,115,80,54)(22,48,86,105)(24,46,88,107)(25,35,89,99)(26,66,90,125)(27,33,91,97)(28,68,92,127)(30,40,94,41)(32,38,96,43)(34,117,98,58)(36,119,100,60)(37,122,42,61)(39,124,44,63)(45,114,106,53)(47,116,108,55)(57,67,120,126)(59,65,118,128), (1,77,74,47)(2,48,75,78)(3,79,76,45)(4,46,73,80)(5,66,39,100)(6,97,40,67)(7,68,37,98)(8,99,38,65)(9,14,104,105)(10,106,101,15)(11,16,102,107)(12,108,103,13)(17,24,110,115)(18,116,111,21)(19,22,112,113)(20,114,109,23)(25,96,118,62)(26,63,119,93)(27,94,120,64)(28,61,117,95)(29,90,124,60)(30,57,121,91)(31,92,122,58)(32,59,123,89)(33,41,126,70)(34,71,127,42)(35,43,128,72)(36,69,125,44)(49,56,83,86)(50,87,84,53)(51,54,81,88)(52,85,82,55) );

G=PermutationGroup([[(1,103),(2,104),(3,101),(4,102),(5,69),(6,70),(7,71),(8,72),(9,75),(10,76),(11,73),(12,74),(13,77),(14,78),(15,79),(16,80),(17,81),(18,82),(19,83),(20,84),(21,85),(22,86),(23,87),(24,88),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(33,97),(34,98),(35,99),(36,100),(37,42),(38,43),(39,44),(40,41),(45,106),(46,107),(47,108),(48,105),(49,112),(50,109),(51,110),(52,111),(53,114),(54,115),(55,116),(56,113),(57,120),(58,117),(59,118),(60,119),(61,122),(62,123),(63,124),(64,121),(65,128),(66,125),(67,126),(68,127)], [(1,12),(2,9),(3,10),(4,11),(5,44),(6,41),(7,42),(8,43),(13,47),(14,48),(15,45),(16,46),(17,51),(18,52),(19,49),(20,50),(21,55),(22,56),(23,53),(24,54),(25,59),(26,60),(27,57),(28,58),(29,63),(30,64),(31,61),(32,62),(33,67),(34,68),(35,65),(36,66),(37,71),(38,72),(39,69),(40,70),(73,102),(74,103),(75,104),(76,101),(77,108),(78,105),(79,106),(80,107),(81,110),(82,111),(83,112),(84,109),(85,116),(86,113),(87,114),(88,115),(89,118),(90,119),(91,120),(92,117),(93,124),(94,121),(95,122),(96,123),(97,126),(98,127),(99,128),(100,125)], [(1,76),(2,73),(3,74),(4,75),(5,37),(6,38),(7,39),(8,40),(9,102),(10,103),(11,104),(12,101),(13,106),(14,107),(15,108),(16,105),(17,112),(18,109),(19,110),(20,111),(21,114),(22,115),(23,116),(24,113),(25,120),(26,117),(27,118),(28,119),(29,122),(30,123),(31,124),(32,121),(33,128),(34,125),(35,126),(36,127),(41,72),(42,69),(43,70),(44,71),(45,77),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,85),(54,86),(55,87),(56,88),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,99),(68,100)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64),(65,66,67,68),(69,70,71,72),(73,74,75,76),(77,78,79,80),(81,82,83,84),(85,86,87,88),(89,90,91,92),(93,94,95,96),(97,98,99,100),(101,102,103,104),(105,106,107,108),(109,110,111,112),(113,114,115,116),(117,118,119,120),(121,122,123,124),(125,126,127,128)], [(1,90,74,60),(2,27,75,120),(3,92,76,58),(4,25,73,118),(5,114,39,23),(6,54,40,88),(7,116,37,21),(8,56,38,86),(9,57,104,91),(10,117,101,28),(11,59,102,89),(12,119,103,26),(13,124,108,29),(14,64,105,94),(15,122,106,31),(16,62,107,96),(17,67,110,97),(18,127,111,34),(19,65,112,99),(20,125,109,36),(22,72,113,43),(24,70,115,41),(30,78,121,48),(32,80,123,46),(33,81,126,51),(35,83,128,49),(42,85,71,55),(44,87,69,53),(45,95,79,61),(47,93,77,63),(50,100,84,66),(52,98,82,68)], [(1,111,103,52),(2,83,104,19),(3,109,101,50),(4,81,102,17),(5,29,69,93),(6,64,70,121),(7,31,71,95),(8,62,72,123),(9,112,75,49),(10,84,76,20),(11,110,73,51),(12,82,74,18),(13,85,77,21),(14,113,78,56),(15,87,79,23),(16,115,80,54),(22,48,86,105),(24,46,88,107),(25,35,89,99),(26,66,90,125),(27,33,91,97),(28,68,92,127),(30,40,94,41),(32,38,96,43),(34,117,98,58),(36,119,100,60),(37,122,42,61),(39,124,44,63),(45,114,106,53),(47,116,108,55),(57,67,120,126),(59,65,118,128)], [(1,77,74,47),(2,48,75,78),(3,79,76,45),(4,46,73,80),(5,66,39,100),(6,97,40,67),(7,68,37,98),(8,99,38,65),(9,14,104,105),(10,106,101,15),(11,16,102,107),(12,108,103,13),(17,24,110,115),(18,116,111,21),(19,22,112,113),(20,114,109,23),(25,96,118,62),(26,63,119,93),(27,94,120,64),(28,61,117,95),(29,90,124,60),(30,57,121,91),(31,92,122,58),(32,59,123,89),(33,41,126,70),(34,71,127,42),(35,43,128,72),(36,69,125,44),(49,56,83,86),(50,87,84,53),(51,54,81,88),(52,85,82,55)]])

32 conjugacy classes

class	1	2A	···	2G	4A	···	4R	4S	···	4X
order	1	2	···	2	4	···	4	4	···	4
size	1	1	···	1	4	···	4	8	···	8

32 irreducible representations

dim	1	1	1	1	1	1	1	1	2	4	4
type	+	+	+	+	+	+	+	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄○D₄	2₊¹⁺⁴	2_-¹⁺⁴
kernel	C₂³.₆₈₉C₂⁴	C₄²⋊₈C₄	C₂³.₆₃C₂³	C₂³.₆₇C₂³	C₂³.₇₈C₂³	C₂³.₈₁C₂³	C₂³.₈₃C₂³	C₂³.₈₄C₂³	C₂×C₄	C₂²	C₂²
# reps	1	1	6	2	1	1	2	2	12	1	3

Matrix representation of C₂³.₆₈₉C₂⁴ ►in GL₆(𝔽₅)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	4

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	1	0
0	0	0	0	0	1

4	0	0	0	0	0
0	4	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

2	0	0	0	0	0
0	2	0	0	0	0
0	0	3	3	0	0
0	0	0	2	0	0
0	0	0	0	0	3
0	0	0	0	3	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	2	0
0	0	0	0	0	3

1	0	0	0	0	0
0	4	0	0	0	0
0	0	1	0	0	0
0	0	3	4	0	0
0	0	0	0	2	0
0	0	0	0	0	2

1	0	0	0	0	0
0	1	0	0	0	0
0	0	3	0	0	0
0	0	4	2	0	0
0	0	0	0	0	1
0	0	0	0	4	0

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,3,2,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,3,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,4,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;

C₂³.₆₈₉C₂⁴ in GAP, Magma, Sage, TeX

C_2^3._{689}C_2^4

% in TeX

G:=Group("C2^3.689C2^4");

// GroupNames label

G:=SmallGroup(128,1521);

// by ID

G=gap.SmallGroup(128,1521);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,344,758,723,100,1571,346,192]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=a*b*c,e^2=g^2=b*a=a*b,f^2=a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,g*d*g^-1=a*b*d,f*g=g*f>;

// generators/relations

G = C23.689C24 order 128 = 27

406th central stem extension by C23 of C24

G = C₂³.₆₈₉C₂⁴ order 128 = 2⁷

406^th central stem extension by C₂³ of C₂⁴