Copied to
clipboard

## G = C2×He3⋊3C8order 432 = 24·33

### Direct product of C2 and He3⋊3C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×He3⋊3C8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×He3 — He3⋊3C8 — C2×He3⋊3C8
 Lower central C32 — C2×He3⋊3C8
 Upper central C1 — C2×C4

Generators and relations for C2×He33C8
G = < a,b,c,d,e | a2=b3=c3=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=b-1, cd=dc, ece-1=c-1, de=ed >

Subgroups: 257 in 93 conjugacy classes, 46 normal (32 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4 [×2], C22, C6, C6 [×2], C6 [×9], C8 [×2], C2×C4, C32 [×2], C32, C12 [×2], C12 [×6], C2×C6, C2×C6 [×3], C2×C8, C3×C6 [×2], C3×C6 [×4], C3×C6 [×3], C3⋊C8 [×4], C24 [×2], C2×C12, C2×C12 [×3], He3, C3×C12 [×4], C3×C12 [×2], C62 [×2], C62, C2×C3⋊C8 [×2], C2×C24, C2×He3, C2×He3 [×2], C3×C3⋊C8 [×2], C324C8 [×2], C6×C12 [×2], C6×C12, C4×He3 [×2], C22×He3, C6×C3⋊C8, C2×C324C8, He33C8 [×2], C2×C4×He3, C2×He33C8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, Dic3 [×2], C12 [×2], D6, C2×C6, C2×C8, C3×S3, C3⋊C8 [×2], C24 [×2], C2×Dic3, C2×C12, C3×Dic3 [×2], S3×C6, C2×C3⋊C8, C2×C24, C32⋊C6, C3×C3⋊C8 [×2], C6×Dic3, C32⋊C12 [×2], C2×C32⋊C6, C6×C3⋊C8, He33C8 [×2], C2×C32⋊C12, C2×He33C8

Smallest permutation representation of C2×He33C8
On 144 points
Generators in S144
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 49)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 65)(17 77)(18 78)(19 79)(20 80)(21 73)(22 74)(23 75)(24 76)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 132)(34 133)(35 134)(36 135)(37 136)(38 129)(39 130)(40 131)(41 126)(42 127)(43 128)(44 121)(45 122)(46 123)(47 124)(48 125)(57 108)(58 109)(59 110)(60 111)(61 112)(62 105)(63 106)(64 107)(89 138)(90 139)(91 140)(92 141)(93 142)(94 143)(95 144)(96 137)(97 120)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)
(1 75 143)(2 144 76)(3 77 137)(4 138 78)(5 79 139)(6 140 80)(7 73 141)(8 142 74)(9 119 83)(10 84 120)(11 113 85)(12 86 114)(13 115 87)(14 88 116)(15 117 81)(16 82 118)(17 96 52)(18 53 89)(19 90 54)(20 55 91)(21 92 56)(22 49 93)(23 94 50)(24 51 95)(25 72 102)(26 103 65)(27 66 104)(28 97 67)(29 68 98)(30 99 69)(31 70 100)(32 101 71)
(1 75 143)(2 144 76)(3 77 137)(4 138 78)(5 79 139)(6 140 80)(7 73 141)(8 142 74)(9 83 119)(10 120 84)(11 85 113)(12 114 86)(13 87 115)(14 116 88)(15 81 117)(16 118 82)(17 96 52)(18 53 89)(19 90 54)(20 55 91)(21 92 56)(22 49 93)(23 94 50)(24 51 95)(25 102 72)(26 65 103)(27 104 66)(28 67 97)(29 98 68)(30 69 99)(31 100 70)(32 71 101)(33 46 105)(34 106 47)(35 48 107)(36 108 41)(37 42 109)(38 110 43)(39 44 111)(40 112 45)(57 126 135)(58 136 127)(59 128 129)(60 130 121)(61 122 131)(62 132 123)(63 124 133)(64 134 125)
(1 48 15)(2 41 16)(3 42 9)(4 43 10)(5 44 11)(6 45 12)(7 46 13)(8 47 14)(17 58 27)(18 59 28)(19 60 29)(20 61 30)(21 62 31)(22 63 32)(23 64 25)(24 57 26)(33 115 141)(34 116 142)(35 117 143)(36 118 144)(37 119 137)(38 120 138)(39 113 139)(40 114 140)(49 124 71)(50 125 72)(51 126 65)(52 127 66)(53 128 67)(54 121 68)(55 122 69)(56 123 70)(73 105 87)(74 106 88)(75 107 81)(76 108 82)(77 109 83)(78 110 84)(79 111 85)(80 112 86)(89 129 97)(90 130 98)(91 131 99)(92 132 100)(93 133 101)(94 134 102)(95 135 103)(96 136 104)
(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)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 6A 6B 6C 6D ··· 6I 6J ··· 6R 8A ··· 8H 12A 12B 12C 12D 12E ··· 12L 12M ··· 12X 24A ··· 24P order 1 2 2 2 3 3 3 3 3 3 4 4 4 4 6 6 6 6 ··· 6 6 ··· 6 8 ··· 8 12 12 12 12 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 2 3 3 6 6 6 1 1 1 1 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9 2 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 6 6 6 6 6 type + + + + - + - + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 S3 Dic3 D6 Dic3 C3×S3 C3⋊C8 C3×Dic3 S3×C6 C3×Dic3 C3×C3⋊C8 C32⋊C6 C32⋊C12 C2×C32⋊C6 C32⋊C12 He3⋊3C8 kernel C2×He3⋊3C8 He3⋊3C8 C2×C4×He3 C2×C32⋊4C8 C4×He3 C22×He3 C32⋊4C8 C6×C12 C2×He3 C3×C12 C62 C3×C6 C6×C12 C3×C12 C3×C12 C62 C2×C12 C3×C6 C12 C12 C2×C6 C6 C2×C4 C4 C4 C22 C2 # reps 1 2 1 2 2 2 4 2 8 4 4 16 1 1 1 1 2 4 2 2 2 8 1 1 1 1 4

Matrix representation of C2×He33C8 in GL8(𝔽73)

 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72
,
 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 0 22 0 0 0 0 0 0 22 0 0 0 0 0 0 0 0 0 55 20 0 0 0 0 0 0 2 18 0 0 0 0 0 0 0 0 55 20 0 0 0 0 0 0 2 18 0 0 0 0 0 0 0 0 55 20 0 0 0 0 0 0 2 18

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,55,2,0,0,0,0,0,0,20,18,0,0,0,0,0,0,0,0,55,2,0,0,0,0,0,0,20,18,0,0,0,0,0,0,0,0,55,2,0,0,0,0,0,0,20,18] >;

C2×He33C8 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_3C_8
% in TeX

G:=Group("C2xHe3:3C8");
// GroupNames label

G:=SmallGroup(432,136);
// by ID

G=gap.SmallGroup(432,136);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,80,4037,2035,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=b^-1,c*d=d*c,e*c*e^-1=c^-1,d*e=e*d>;
// generators/relations

׿
×
𝔽