metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊2Dic7, (Q8×C14)⋊1C4, C14.17C4≀C2, (C2×Q8)⋊1Dic7, (C2×C14).3Q16, C22⋊Q8.1D7, (C2×C28).230D4, (C2×C14).11SD16, (C22×C14).45D4, (C22×C4).60D14, C22.5(Q8⋊D7), C14.21(C23⋊C4), C2.3(Q8⋊Dic7), C7⋊3(C23.31D4), C23.49(C7⋊D4), C2.6(C23⋊Dic7), C22.2(C7⋊Q16), C14.11(Q8⋊C4), C2.5(D4⋊2Dic7), C28.55D4.16C2, C14.C42.35C2, (C22×C28).372C22, C22.38(C23.D7), (C7×C4⋊C4)⋊2C4, (C2×C4).8(C2×Dic7), (C2×C28).168(C2×C4), (C7×C22⋊Q8).10C2, (C2×C4).164(C7⋊D4), (C2×C14).96(C22⋊C4), SmallGroup(448,95)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊Dic7
G = < a,b,c,d | a4=b4=c14=1, d2=c7, bab-1=a-1, ac=ca, dad-1=ab2, cbc-1=a2b-1, dbd-1=ab, dcd-1=c-1 >
Subgroups: 348 in 80 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C2.C42, C22⋊C8, C22⋊Q8, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×C14, C23.31D4, C2×C7⋊C8, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C22×Dic7, C22×C28, Q8×C14, C28.55D4, C14.C42, C7×C22⋊Q8, C4⋊C4⋊Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, SD16, Q16, Dic7, D14, C23⋊C4, Q8⋊C4, C4≀C2, C2×Dic7, C7⋊D4, C23.31D4, Q8⋊D7, C7⋊Q16, C23.D7, C23⋊Dic7, Q8⋊Dic7, D4⋊2Dic7, C4⋊C4⋊Dic7
►On 112 points
Generators in S112
(1 49 13 53)(2 43 14 54)(3 44 8 55)(4 45 9 56)(5 46 10 50)(6 47 11 51)(7 48 12 52)(15 29 22 36)(16 30 23 37)(17 31 24 38)(18 32 25 39)(19 33 26 40)(20 34 27 41)(21 35 28 42)(57 87 100 82)(58 88 101 83)(59 89 102 84)(60 90 103 71)(61 91 104 72)(62 92 105 73)(63 93 106 74)(64 94 107 75)(65 95 108 76)(66 96 109 77)(67 97 110 78)(68 98 111 79)(69 85 112 80)(70 86 99 81)
(1 87 27 75)(2 95 28 83)(3 89 22 77)(4 97 23 71)(5 91 24 79)(6 85 25 73)(7 93 26 81)(8 84 15 96)(9 78 16 90)(10 72 17 98)(11 80 18 92)(12 74 19 86)(13 82 20 94)(14 76 21 88)(29 66 55 102)(30 60 56 110)(31 68 50 104)(32 62 51 112)(33 70 52 106)(34 64 53 100)(35 58 54 108)(36 109 44 59)(37 103 45 67)(38 111 46 61)(39 105 47 69)(40 99 48 63)(41 107 49 57)(42 101 43 65)
(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)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(7 14)(15 25)(16 24)(17 23)(18 22)(19 28)(20 27)(21 26)(29 47)(30 46)(31 45)(32 44)(33 43)(34 49)(35 48)(36 51)(37 50)(38 56)(39 55)(40 54)(41 53)(42 52)(57 82 64 75)(58 81 65 74)(59 80 66 73)(60 79 67 72)(61 78 68 71)(62 77 69 84)(63 76 70 83)(85 109 92 102)(86 108 93 101)(87 107 94 100)(88 106 95 99)(89 105 96 112)(90 104 97 111)(91 103 98 110)
G:=sub<Sym(112)| (1,49,13,53)(2,43,14,54)(3,44,8,55)(4,45,9,56)(5,46,10,50)(6,47,11,51)(7,48,12,52)(15,29,22,36)(16,30,23,37)(17,31,24,38)(18,32,25,39)(19,33,26,40)(20,34,27,41)(21,35,28,42)(57,87,100,82)(58,88,101,83)(59,89,102,84)(60,90,103,71)(61,91,104,72)(62,92,105,73)(63,93,106,74)(64,94,107,75)(65,95,108,76)(66,96,109,77)(67,97,110,78)(68,98,111,79)(69,85,112,80)(70,86,99,81), (1,87,27,75)(2,95,28,83)(3,89,22,77)(4,97,23,71)(5,91,24,79)(6,85,25,73)(7,93,26,81)(8,84,15,96)(9,78,16,90)(10,72,17,98)(11,80,18,92)(12,74,19,86)(13,82,20,94)(14,76,21,88)(29,66,55,102)(30,60,56,110)(31,68,50,104)(32,62,51,112)(33,70,52,106)(34,64,53,100)(35,58,54,108)(36,109,44,59)(37,103,45,67)(38,111,46,61)(39,105,47,69)(40,99,48,63)(41,107,49,57)(42,101,43,65), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(7,14)(15,25)(16,24)(17,23)(18,22)(19,28)(20,27)(21,26)(29,47)(30,46)(31,45)(32,44)(33,43)(34,49)(35,48)(36,51)(37,50)(38,56)(39,55)(40,54)(41,53)(42,52)(57,82,64,75)(58,81,65,74)(59,80,66,73)(60,79,67,72)(61,78,68,71)(62,77,69,84)(63,76,70,83)(85,109,92,102)(86,108,93,101)(87,107,94,100)(88,106,95,99)(89,105,96,112)(90,104,97,111)(91,103,98,110)>;
G:=Group( (1,49,13,53)(2,43,14,54)(3,44,8,55)(4,45,9,56)(5,46,10,50)(6,47,11,51)(7,48,12,52)(15,29,22,36)(16,30,23,37)(17,31,24,38)(18,32,25,39)(19,33,26,40)(20,34,27,41)(21,35,28,42)(57,87,100,82)(58,88,101,83)(59,89,102,84)(60,90,103,71)(61,91,104,72)(62,92,105,73)(63,93,106,74)(64,94,107,75)(65,95,108,76)(66,96,109,77)(67,97,110,78)(68,98,111,79)(69,85,112,80)(70,86,99,81), (1,87,27,75)(2,95,28,83)(3,89,22,77)(4,97,23,71)(5,91,24,79)(6,85,25,73)(7,93,26,81)(8,84,15,96)(9,78,16,90)(10,72,17,98)(11,80,18,92)(12,74,19,86)(13,82,20,94)(14,76,21,88)(29,66,55,102)(30,60,56,110)(31,68,50,104)(32,62,51,112)(33,70,52,106)(34,64,53,100)(35,58,54,108)(36,109,44,59)(37,103,45,67)(38,111,46,61)(39,105,47,69)(40,99,48,63)(41,107,49,57)(42,101,43,65), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(7,14)(15,25)(16,24)(17,23)(18,22)(19,28)(20,27)(21,26)(29,47)(30,46)(31,45)(32,44)(33,43)(34,49)(35,48)(36,51)(37,50)(38,56)(39,55)(40,54)(41,53)(42,52)(57,82,64,75)(58,81,65,74)(59,80,66,73)(60,79,67,72)(61,78,68,71)(62,77,69,84)(63,76,70,83)(85,109,92,102)(86,108,93,101)(87,107,94,100)(88,106,95,99)(89,105,96,112)(90,104,97,111)(91,103,98,110) );
G=PermutationGroup([[(1,49,13,53),(2,43,14,54),(3,44,8,55),(4,45,9,56),(5,46,10,50),(6,47,11,51),(7,48,12,52),(15,29,22,36),(16,30,23,37),(17,31,24,38),(18,32,25,39),(19,33,26,40),(20,34,27,41),(21,35,28,42),(57,87,100,82),(58,88,101,83),(59,89,102,84),(60,90,103,71),(61,91,104,72),(62,92,105,73),(63,93,106,74),(64,94,107,75),(65,95,108,76),(66,96,109,77),(67,97,110,78),(68,98,111,79),(69,85,112,80),(70,86,99,81)], [(1,87,27,75),(2,95,28,83),(3,89,22,77),(4,97,23,71),(5,91,24,79),(6,85,25,73),(7,93,26,81),(8,84,15,96),(9,78,16,90),(10,72,17,98),(11,80,18,92),(12,74,19,86),(13,82,20,94),(14,76,21,88),(29,66,55,102),(30,60,56,110),(31,68,50,104),(32,62,51,112),(33,70,52,106),(34,64,53,100),(35,58,54,108),(36,109,44,59),(37,103,45,67),(38,111,46,61),(39,105,47,69),(40,99,48,63),(41,107,49,57),(42,101,43,65)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(7,14),(15,25),(16,24),(17,23),(18,22),(19,28),(20,27),(21,26),(29,47),(30,46),(31,45),(32,44),(33,43),(34,49),(35,48),(36,51),(37,50),(38,56),(39,55),(40,54),(41,53),(42,52),(57,82,64,75),(58,81,65,74),(59,80,66,73),(60,79,67,72),(61,78,68,71),(62,77,69,84),(63,76,70,83),(85,109,92,102),(86,108,93,101),(87,107,94,100),(88,106,95,99),(89,105,96,112),(90,104,97,111),(91,103,98,110)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | - | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D7 | SD16 | Q16 | Dic7 | D14 | Dic7 | C4≀C2 | C7⋊D4 | C7⋊D4 | C23⋊C4 | Q8⋊D7 | C7⋊Q16 | C23⋊Dic7 | D4⋊2Dic7 |
| kernel | C4⋊C4⋊Dic7 | C28.55D4 | C14.C42 | C7×C22⋊Q8 | C7×C4⋊C4 | Q8×C14 | C2×C28 | C22×C14 | C22⋊Q8 | C2×C14 | C2×C14 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C2×C4 | C23 | C14 | C22 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 2 | 2 | 3 | 3 | 3 | 4 | 6 | 6 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of C4⋊C4⋊Dic7 ►in GL6(𝔽113)
| 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 111 |
| 0 | 0 | 0 | 0 | 0 | 98 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 58 | 0 | 0 |
| 0 | 0 | 5 | 84 | 0 | 0 |
| 0 | 0 | 0 | 0 | 69 | 87 |
| 0 | 0 | 0 | 0 | 18 | 44 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 102 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 16 | 0 | 0 |
| 0 | 0 | 98 | 76 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 98 | 1 |
G:=sub<GL(6,GF(113))| [98,0,0,0,0,0,0,15,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,15,0,0,0,0,0,111,98],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,29,5,0,0,0,0,58,84,0,0,0,0,0,0,69,18,0,0,0,0,87,44],[1,0,0,0,0,0,0,112,0,0,0,0,0,0,103,1,0,0,0,0,102,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,98,0,0,0,0,0,0,37,98,0,0,0,0,16,76,0,0,0,0,0,0,112,98,0,0,0,0,0,1] >;
C4⋊C4⋊Dic7 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes {\rm Dic}_7 % in TeX
G:=Group("C4:C4:Dic7"); // GroupNames label
G:=SmallGroup(448,95);
// by ID
G=gap.SmallGroup(448,95);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,232,219,1571,570,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=c^7,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations