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## G = C2×C62⋊5C4order 288 = 25·32

### Direct product of C2 and C62⋊5C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C62⋊5C4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C2×C62⋊5C4
 Lower central C32 — C3×C6 — C2×C62⋊5C4
 Upper central C1 — C23 — C24

Generators and relations for C2×C625C4
G = < a,b,c,d | a2=b6=c6=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c3, dcd-1=c-1 >

Subgroups: 996 in 396 conjugacy classes, 173 normal (11 characteristic)
C1, C2, C2 [×6], C2 [×4], C3 [×4], C4 [×4], C22, C22 [×10], C22 [×12], C6 [×28], C6 [×16], C2×C4 [×8], C23, C23 [×6], C23 [×4], C32, Dic3 [×16], C2×C6 [×44], C2×C6 [×48], C22⋊C4 [×4], C22×C4 [×2], C24, C3×C6, C3×C6 [×6], C3×C6 [×4], C2×Dic3 [×32], C22×C6 [×28], C22×C6 [×16], C2×C22⋊C4, C3⋊Dic3 [×4], C62, C62 [×10], C62 [×12], C6.D4 [×16], C22×Dic3 [×8], C23×C6 [×4], C2×C3⋊Dic3 [×4], C2×C3⋊Dic3 [×4], C2×C62, C2×C62 [×6], C2×C62 [×4], C2×C6.D4 [×4], C625C4 [×4], C22×C3⋊Dic3 [×2], C22×C62, C2×C625C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], D4 [×4], C23, Dic3 [×16], D6 [×12], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C3⋊S3, C2×Dic3 [×24], C3⋊D4 [×16], C22×S3 [×4], C2×C22⋊C4, C3⋊Dic3 [×4], C2×C3⋊S3 [×3], C6.D4 [×16], C22×Dic3 [×4], C2×C3⋊D4 [×8], C2×C3⋊Dic3 [×6], C327D4 [×4], C22×C3⋊S3, C2×C6.D4 [×4], C625C4 [×4], C22×C3⋊Dic3, C2×C327D4 [×2], C2×C625C4

Smallest permutation representation of C2×C625C4
On 144 points
Generators in S144
(1 57)(2 58)(3 59)(4 60)(5 55)(6 56)(7 10)(8 11)(9 12)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 87)(20 88)(21 89)(22 90)(23 85)(24 86)(25 53)(26 54)(27 49)(28 50)(29 51)(30 52)(31 72)(32 67)(33 68)(34 69)(35 70)(36 71)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(73 76)(74 77)(75 78)(79 82)(80 83)(81 84)(91 94)(92 95)(93 96)(97 100)(98 101)(99 102)(103 106)(104 107)(105 108)(109 117)(110 118)(111 119)(112 120)(113 115)(114 116)(121 124)(122 125)(123 126)(127 130)(128 131)(129 132)(133 136)(134 137)(135 138)(139 142)(140 143)(141 144)
(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)(127 128 129 130 131 132)(133 134 135 136 137 138)(139 140 141 142 143 144)
(1 50 35 60 25 67)(2 51 36 55 26 68)(3 52 31 56 27 69)(4 53 32 57 28 70)(5 54 33 58 29 71)(6 49 34 59 30 72)(7 108 134 40 47 130)(8 103 135 41 48 131)(9 104 136 42 43 132)(10 105 137 37 44 127)(11 106 138 38 45 128)(12 107 133 39 46 129)(13 24 111 64 89 116)(14 19 112 65 90 117)(15 20 113 66 85 118)(16 21 114 61 86 119)(17 22 109 62 87 120)(18 23 110 63 88 115)(73 126 142 101 80 94)(74 121 143 102 81 95)(75 122 144 97 82 96)(76 123 139 98 83 91)(77 124 140 99 84 92)(78 125 141 100 79 93)
(1 129 18 81)(2 138 13 126)(3 127 14 79)(4 136 15 124)(5 131 16 83)(6 134 17 122)(7 109 96 34)(8 119 91 71)(9 113 92 32)(10 117 93 69)(11 111 94 36)(12 115 95 67)(19 100 52 44)(20 77 53 104)(21 98 54 48)(22 75 49 108)(23 102 50 46)(24 73 51 106)(25 107 88 74)(26 45 89 101)(27 105 90 78)(28 43 85 99)(29 103 86 76)(30 47 87 97)(31 37 112 141)(33 41 114 139)(35 39 110 143)(38 116 142 68)(40 120 144 72)(42 118 140 70)(55 128 64 80)(56 137 65 125)(57 132 66 84)(58 135 61 123)(59 130 62 82)(60 133 63 121)

G:=sub<Sym(144)| (1,57)(2,58)(3,59)(4,60)(5,55)(6,56)(7,10)(8,11)(9,12)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,87)(20,88)(21,89)(22,90)(23,85)(24,86)(25,53)(26,54)(27,49)(28,50)(29,51)(30,52)(31,72)(32,67)(33,68)(34,69)(35,70)(36,71)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(73,76)(74,77)(75,78)(79,82)(80,83)(81,84)(91,94)(92,95)(93,96)(97,100)(98,101)(99,102)(103,106)(104,107)(105,108)(109,117)(110,118)(111,119)(112,120)(113,115)(114,116)(121,124)(122,125)(123,126)(127,130)(128,131)(129,132)(133,136)(134,137)(135,138)(139,142)(140,143)(141,144), (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)(127,128,129,130,131,132)(133,134,135,136,137,138)(139,140,141,142,143,144), (1,50,35,60,25,67)(2,51,36,55,26,68)(3,52,31,56,27,69)(4,53,32,57,28,70)(5,54,33,58,29,71)(6,49,34,59,30,72)(7,108,134,40,47,130)(8,103,135,41,48,131)(9,104,136,42,43,132)(10,105,137,37,44,127)(11,106,138,38,45,128)(12,107,133,39,46,129)(13,24,111,64,89,116)(14,19,112,65,90,117)(15,20,113,66,85,118)(16,21,114,61,86,119)(17,22,109,62,87,120)(18,23,110,63,88,115)(73,126,142,101,80,94)(74,121,143,102,81,95)(75,122,144,97,82,96)(76,123,139,98,83,91)(77,124,140,99,84,92)(78,125,141,100,79,93), (1,129,18,81)(2,138,13,126)(3,127,14,79)(4,136,15,124)(5,131,16,83)(6,134,17,122)(7,109,96,34)(8,119,91,71)(9,113,92,32)(10,117,93,69)(11,111,94,36)(12,115,95,67)(19,100,52,44)(20,77,53,104)(21,98,54,48)(22,75,49,108)(23,102,50,46)(24,73,51,106)(25,107,88,74)(26,45,89,101)(27,105,90,78)(28,43,85,99)(29,103,86,76)(30,47,87,97)(31,37,112,141)(33,41,114,139)(35,39,110,143)(38,116,142,68)(40,120,144,72)(42,118,140,70)(55,128,64,80)(56,137,65,125)(57,132,66,84)(58,135,61,123)(59,130,62,82)(60,133,63,121)>;

G:=Group( (1,57)(2,58)(3,59)(4,60)(5,55)(6,56)(7,10)(8,11)(9,12)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,87)(20,88)(21,89)(22,90)(23,85)(24,86)(25,53)(26,54)(27,49)(28,50)(29,51)(30,52)(31,72)(32,67)(33,68)(34,69)(35,70)(36,71)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(73,76)(74,77)(75,78)(79,82)(80,83)(81,84)(91,94)(92,95)(93,96)(97,100)(98,101)(99,102)(103,106)(104,107)(105,108)(109,117)(110,118)(111,119)(112,120)(113,115)(114,116)(121,124)(122,125)(123,126)(127,130)(128,131)(129,132)(133,136)(134,137)(135,138)(139,142)(140,143)(141,144), (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)(127,128,129,130,131,132)(133,134,135,136,137,138)(139,140,141,142,143,144), (1,50,35,60,25,67)(2,51,36,55,26,68)(3,52,31,56,27,69)(4,53,32,57,28,70)(5,54,33,58,29,71)(6,49,34,59,30,72)(7,108,134,40,47,130)(8,103,135,41,48,131)(9,104,136,42,43,132)(10,105,137,37,44,127)(11,106,138,38,45,128)(12,107,133,39,46,129)(13,24,111,64,89,116)(14,19,112,65,90,117)(15,20,113,66,85,118)(16,21,114,61,86,119)(17,22,109,62,87,120)(18,23,110,63,88,115)(73,126,142,101,80,94)(74,121,143,102,81,95)(75,122,144,97,82,96)(76,123,139,98,83,91)(77,124,140,99,84,92)(78,125,141,100,79,93), (1,129,18,81)(2,138,13,126)(3,127,14,79)(4,136,15,124)(5,131,16,83)(6,134,17,122)(7,109,96,34)(8,119,91,71)(9,113,92,32)(10,117,93,69)(11,111,94,36)(12,115,95,67)(19,100,52,44)(20,77,53,104)(21,98,54,48)(22,75,49,108)(23,102,50,46)(24,73,51,106)(25,107,88,74)(26,45,89,101)(27,105,90,78)(28,43,85,99)(29,103,86,76)(30,47,87,97)(31,37,112,141)(33,41,114,139)(35,39,110,143)(38,116,142,68)(40,120,144,72)(42,118,140,70)(55,128,64,80)(56,137,65,125)(57,132,66,84)(58,135,61,123)(59,130,62,82)(60,133,63,121) );

G=PermutationGroup([(1,57),(2,58),(3,59),(4,60),(5,55),(6,56),(7,10),(8,11),(9,12),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,87),(20,88),(21,89),(22,90),(23,85),(24,86),(25,53),(26,54),(27,49),(28,50),(29,51),(30,52),(31,72),(32,67),(33,68),(34,69),(35,70),(36,71),(37,40),(38,41),(39,42),(43,46),(44,47),(45,48),(73,76),(74,77),(75,78),(79,82),(80,83),(81,84),(91,94),(92,95),(93,96),(97,100),(98,101),(99,102),(103,106),(104,107),(105,108),(109,117),(110,118),(111,119),(112,120),(113,115),(114,116),(121,124),(122,125),(123,126),(127,130),(128,131),(129,132),(133,136),(134,137),(135,138),(139,142),(140,143),(141,144)], [(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),(127,128,129,130,131,132),(133,134,135,136,137,138),(139,140,141,142,143,144)], [(1,50,35,60,25,67),(2,51,36,55,26,68),(3,52,31,56,27,69),(4,53,32,57,28,70),(5,54,33,58,29,71),(6,49,34,59,30,72),(7,108,134,40,47,130),(8,103,135,41,48,131),(9,104,136,42,43,132),(10,105,137,37,44,127),(11,106,138,38,45,128),(12,107,133,39,46,129),(13,24,111,64,89,116),(14,19,112,65,90,117),(15,20,113,66,85,118),(16,21,114,61,86,119),(17,22,109,62,87,120),(18,23,110,63,88,115),(73,126,142,101,80,94),(74,121,143,102,81,95),(75,122,144,97,82,96),(76,123,139,98,83,91),(77,124,140,99,84,92),(78,125,141,100,79,93)], [(1,129,18,81),(2,138,13,126),(3,127,14,79),(4,136,15,124),(5,131,16,83),(6,134,17,122),(7,109,96,34),(8,119,91,71),(9,113,92,32),(10,117,93,69),(11,111,94,36),(12,115,95,67),(19,100,52,44),(20,77,53,104),(21,98,54,48),(22,75,49,108),(23,102,50,46),(24,73,51,106),(25,107,88,74),(26,45,89,101),(27,105,90,78),(28,43,85,99),(29,103,86,76),(30,47,87,97),(31,37,112,141),(33,41,114,139),(35,39,110,143),(38,116,142,68),(40,120,144,72),(42,118,140,70),(55,128,64,80),(56,137,65,125),(57,132,66,84),(58,135,61,123),(59,130,62,82),(60,133,63,121)])

84 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3A 3B 3C 3D 4A ··· 4H 6A ··· 6BH order 1 2 ··· 2 2 2 2 2 3 3 3 3 4 ··· 4 6 ··· 6 size 1 1 ··· 1 2 2 2 2 2 2 2 2 18 ··· 18 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 type + + + + + + - + image C1 C2 C2 C2 C4 S3 D4 Dic3 D6 C3⋊D4 kernel C2×C62⋊5C4 C62⋊5C4 C22×C3⋊Dic3 C22×C62 C2×C62 C23×C6 C62 C22×C6 C22×C6 C2×C6 # reps 1 4 2 1 8 4 4 16 12 32

Matrix representation of C2×C625C4 in GL7(𝔽13)

 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 9 8 0 0 0 0 0 0 3
,
 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 12 11 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 0 0 0 11 11 0 0 0 0 0 8 2

G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,9,0,0,0,0,0,0,8,3],[1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,8,0,0,0,0,0,11,2] >;

C2×C625C4 in GAP, Magma, Sage, TeX

C_2\times C_6^2\rtimes_5C_4
% in TeX

G:=Group("C2xC6^2:5C4");
// GroupNames label

G:=SmallGroup(288,809);
// by ID

G=gap.SmallGroup(288,809);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,422,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^6=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^3,d*c*d^-1=c^-1>;
// generators/relations

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