D7xC4:1D4 - GroupNames

G = D₇×C₄⋊₁D₄ order 448 = 2⁶·7

Direct product of D₇ and C₄⋊₁D₄

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₇×C₄⋊₁D₄, C₄²⋊₃₄D₁₄, C₄⋊₁(D₄×D₇), (C₄×D₇)⋊₇D₄, C₂₈⋊₂(C₂×D₄), (C₂×D₄)⋊₂₅D₁₄, Dic₇⋊₁(C₂×D₄), C₂₈⋊D₄⋊₂₅C₂, C₂₈⋊₄D₄⋊₁₆C₂, (C₄×C₂₈)⋊₂₅C₂², D₁₄.₆₁(C₂×D₄), (D₇×C₄²)⋊₁₂C₂, (C₂×D₂₈)⋊₃₀C₂², (D₄×C₁₄)⋊₁₇C₂², C₁₄.₉₂(C₂²×D₄), (C₂×C₂₈).₅₀₇C₂³, (C₂×C₁₄).₂₅₈C₂⁴, (C₄×Dic₇)⋊₆₅C₂², C₂³.₆₄(C₂²×D₇), (C₂²×C₁₄).₇₂C₂³, (C₂³×D₇).₇₁C₂², C₂².₂₇₉(C₂³×D₇), (C₂×Dic₇).₂₆₈C₂³, (C₂²×D₇).₂₅₉C₂³, (C₂×D₄×D₇)⋊₁₈C₂, C₇⋊₂(C₂×C₄⋊₁D₄), C₂.₆₅(C₂×D₄×D₇), (C₇×C₄⋊₁D₄)⋊₅C₂, (C₂×C₇⋊D₄)⋊₂₅C₂², (C₂×C₄×D₇).₂₅₀C₂², (C₂×C₄).₅₉₆(C₂²×D₇), SmallGroup(448,1167)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — D₇×C₄⋊₁D₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×D₇ — C₂×C₄×D₇ — C₂×D₄×D₇ — D₇×C₄⋊₁D₄

Lower central

C₇ — C₂×C₁₄ — D₇×C₄⋊₁D₄

Upper central

C₁ — C₂² — C₄⋊₁D₄

Generators and relations for D₇×C₄⋊₁D₄
G = < a,b,c,d,e | a⁷=b²=c⁴=d⁴=e²=1, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c^-1, ede=d^-1 >

Subgroups: 2988 in 498 conjugacy classes, 131 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, D₇, D₇, C₁₄, C₁₄, C₄², C₄², C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, Dic₇, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₄², C₄⋊₁D₄, C₄⋊₁D₄, C₂²×D₄, C₄×D₇, D₂₈, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₇×D₄, C₂²×D₇, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂×C₄⋊₁D₄, C₄×Dic₇, C₄×C₂₈, C₂×C₄×D₇, C₂×D₂₈, D₄×D₇, C₂×C₇⋊D₄, D₄×C₁₄, C₂³×D₇, D₇×C₄², C₂₈⋊₄D₄, C₂₈⋊D₄, C₇×C₄⋊₁D₄, C₂×D₄×D₇, D₇×C₄⋊₁D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₂⁴, D₁₄, C₄⋊₁D₄, C₂²×D₄, C₂²×D₇, C₂×C₄⋊₁D₄, D₄×D₇, C₂³×D₇, C₂×D₄×D₇, D₇×C₄⋊₁D₄

Smallest permutation representation of D₇×C₄⋊₁D₄
►On 112 points

Generators in S₁₁₂

(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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)(64 66)(67 70)(68 69)(71 73)(74 77)(75 76)(78 80)(81 84)(82 83)(85 87)(88 91)(89 90)(92 94)(95 98)(96 97)(99 101)(102 105)(103 104)(106 108)(109 112)(110 111)

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

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

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

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

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

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

70 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	2L	2M	2N	2O	4A	···	4F	4G	···	4L	7A	7B	7C	14A	···	14I	14J	···	14U	28A	···	28R
order	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	4	···	4	4	···	4	7	7	7	14	···	14	14	···	14	28	···	28
size	1	1	1	1	4	4	4	4	7	7	7	7	28	28	28	28	2	···	2	14	···	14	2	2	2	2	···	2	8	···	8	4	···	4

70 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	4
type	+	+	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	D₇	D₁₄	D₁₄	D₄×D₇
kernel	D₇×C₄⋊₁D₄	D₇×C₄²	C₂₈⋊₄D₄	C₂₈⋊D₄	C₇×C₄⋊₁D₄	C₂×D₄×D₇	C₄×D₇	C₄⋊₁D₄	C₄²	C₂×D₄	C₄
# reps	1	1	1	6	1	6	12	3	3	18	18

Matrix representation of D₇×C₄⋊₁D₄ ►in GL₆(𝔽₂₉)

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	28	1
0	0	0	0	9	19

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	28	0
0	0	0	0	9	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	2	0	0
0	0	28	28	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	28	0	0	0	0
1	0	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

0	28	0	0	0	0
28	0	0	0	0	0
0	0	1	2	0	0
0	0	0	28	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,9,0,0,0,0,1,19],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,9,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,28,0,0,0,0,2,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,2,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D₇×C₄⋊₁D₄ in GAP, Magma, Sage, TeX

D_7\times C_4\rtimes_1D_4

% in TeX

G:=Group("D7xC4:1D4");

// GroupNames label

G:=SmallGroup(448,1167);

// by ID

G=gap.SmallGroup(448,1167);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,570,185,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^2=c^4=d^4=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;

// generators/relations

G = D7×C4⋊1D4 order 448 = 26·7

Direct product of D7 and C4⋊1D4

G = D₇×C₄⋊₁D₄ order 448 = 2⁶·7

Direct product of D₇ and C₄⋊₁D₄