Copied to
clipboard

## G = C2×He3⋊4C8order 432 = 24·33

### Direct product of C2 and He3⋊4C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C2×He3⋊4C8
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊4C8 — C2×He3⋊4C8
 Lower central He3 — C2×He3⋊4C8
 Upper central C1 — C2×C12

Generators and relations for C2×He34C8
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, ce=ec, ede-1=d-1 >

Subgroups: 281 in 121 conjugacy classes, 59 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C32, C12, C12, C2×C6, C2×C6, C2×C8, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, He3, C3×C12, C62, C2×C3⋊C8, C2×C24, C2×He3, C2×He3, C3×C3⋊C8, C6×C12, C4×He3, C22×He3, C6×C3⋊C8, He34C8, C2×C4×He3, C2×He34C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, C3⋊S3, C3⋊C8, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C2×C3⋊C8, He3⋊C2, C324C8, C2×C3⋊Dic3, He33C4, C2×He3⋊C2, C2×C324C8, He34C8, C2×He33C4, C2×He34C8

Smallest permutation representation of C2×He34C8
On 144 points
Generators in S144
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 89)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 105)(24 106)(25 123)(26 124)(27 125)(28 126)(29 127)(30 128)(31 121)(32 122)(41 88)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 101)(50 102)(51 103)(52 104)(53 97)(54 98)(55 99)(56 100)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 137)(65 118)(66 119)(67 120)(68 113)(69 114)(70 115)(71 116)(72 117)(73 136)(74 129)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)
(1 141 44)(2 45 142)(3 143 46)(4 47 144)(5 137 48)(6 41 138)(7 139 42)(8 43 140)(9 110 51)(10 52 111)(11 112 53)(12 54 105)(13 106 55)(14 56 107)(15 108 49)(16 50 109)(17 33 100)(18 101 34)(19 35 102)(20 103 36)(21 37 104)(22 97 38)(23 39 98)(24 99 40)(25 120 129)(26 130 113)(27 114 131)(28 132 115)(29 116 133)(30 134 117)(31 118 135)(32 136 119)(57 95 88)(58 81 96)(59 89 82)(60 83 90)(61 91 84)(62 85 92)(63 93 86)(64 87 94)(65 80 121)(66 122 73)(67 74 123)(68 124 75)(69 76 125)(70 126 77)(71 78 127)(72 128 79)
(1 106 121)(2 107 122)(3 108 123)(4 109 124)(5 110 125)(6 111 126)(7 112 127)(8 105 128)(9 76 48)(10 77 41)(11 78 42)(12 79 43)(13 80 44)(14 73 45)(15 74 46)(16 75 47)(17 32 91)(18 25 92)(19 26 93)(20 27 94)(21 28 95)(22 29 96)(23 30 89)(24 31 90)(33 136 84)(34 129 85)(35 130 86)(36 131 87)(37 132 88)(38 133 81)(39 134 82)(40 135 83)(49 67 143)(50 68 144)(51 69 137)(52 70 138)(53 71 139)(54 72 140)(55 65 141)(56 66 142)(57 104 115)(58 97 116)(59 98 117)(60 99 118)(61 100 119)(62 101 120)(63 102 113)(64 103 114)
(1 141 13)(2 14 142)(3 143 15)(4 16 144)(5 137 9)(6 10 138)(7 139 11)(8 12 140)(17 136 100)(18 101 129)(19 130 102)(20 103 131)(21 132 104)(22 97 133)(23 134 98)(24 99 135)(25 120 85)(26 86 113)(27 114 87)(28 88 115)(29 116 81)(30 82 117)(31 118 83)(32 84 119)(33 61 91)(34 92 62)(35 63 93)(36 94 64)(37 57 95)(38 96 58)(39 59 89)(40 90 60)(41 70 126)(42 127 71)(43 72 128)(44 121 65)(45 66 122)(46 123 67)(47 68 124)(48 125 69)(49 74 108)(50 109 75)(51 76 110)(52 111 77)(53 78 112)(54 105 79)(55 80 106)(56 107 73)
(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)| (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,89)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,105)(24,106)(25,123)(26,124)(27,125)(28,126)(29,127)(30,128)(31,121)(32,122)(41,88)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,101)(50,102)(51,103)(52,104)(53,97)(54,98)(55,99)(56,100)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,137)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(73,136)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135), (1,141,44)(2,45,142)(3,143,46)(4,47,144)(5,137,48)(6,41,138)(7,139,42)(8,43,140)(9,110,51)(10,52,111)(11,112,53)(12,54,105)(13,106,55)(14,56,107)(15,108,49)(16,50,109)(17,33,100)(18,101,34)(19,35,102)(20,103,36)(21,37,104)(22,97,38)(23,39,98)(24,99,40)(25,120,129)(26,130,113)(27,114,131)(28,132,115)(29,116,133)(30,134,117)(31,118,135)(32,136,119)(57,95,88)(58,81,96)(59,89,82)(60,83,90)(61,91,84)(62,85,92)(63,93,86)(64,87,94)(65,80,121)(66,122,73)(67,74,123)(68,124,75)(69,76,125)(70,126,77)(71,78,127)(72,128,79), (1,106,121)(2,107,122)(3,108,123)(4,109,124)(5,110,125)(6,111,126)(7,112,127)(8,105,128)(9,76,48)(10,77,41)(11,78,42)(12,79,43)(13,80,44)(14,73,45)(15,74,46)(16,75,47)(17,32,91)(18,25,92)(19,26,93)(20,27,94)(21,28,95)(22,29,96)(23,30,89)(24,31,90)(33,136,84)(34,129,85)(35,130,86)(36,131,87)(37,132,88)(38,133,81)(39,134,82)(40,135,83)(49,67,143)(50,68,144)(51,69,137)(52,70,138)(53,71,139)(54,72,140)(55,65,141)(56,66,142)(57,104,115)(58,97,116)(59,98,117)(60,99,118)(61,100,119)(62,101,120)(63,102,113)(64,103,114), (1,141,13)(2,14,142)(3,143,15)(4,16,144)(5,137,9)(6,10,138)(7,139,11)(8,12,140)(17,136,100)(18,101,129)(19,130,102)(20,103,131)(21,132,104)(22,97,133)(23,134,98)(24,99,135)(25,120,85)(26,86,113)(27,114,87)(28,88,115)(29,116,81)(30,82,117)(31,118,83)(32,84,119)(33,61,91)(34,92,62)(35,63,93)(36,94,64)(37,57,95)(38,96,58)(39,59,89)(40,90,60)(41,70,126)(42,127,71)(43,72,128)(44,121,65)(45,66,122)(46,123,67)(47,68,124)(48,125,69)(49,74,108)(50,109,75)(51,76,110)(52,111,77)(53,78,112)(54,105,79)(55,80,106)(56,107,73), (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( (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,89)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,105)(24,106)(25,123)(26,124)(27,125)(28,126)(29,127)(30,128)(31,121)(32,122)(41,88)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,101)(50,102)(51,103)(52,104)(53,97)(54,98)(55,99)(56,100)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,137)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(73,136)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135), (1,141,44)(2,45,142)(3,143,46)(4,47,144)(5,137,48)(6,41,138)(7,139,42)(8,43,140)(9,110,51)(10,52,111)(11,112,53)(12,54,105)(13,106,55)(14,56,107)(15,108,49)(16,50,109)(17,33,100)(18,101,34)(19,35,102)(20,103,36)(21,37,104)(22,97,38)(23,39,98)(24,99,40)(25,120,129)(26,130,113)(27,114,131)(28,132,115)(29,116,133)(30,134,117)(31,118,135)(32,136,119)(57,95,88)(58,81,96)(59,89,82)(60,83,90)(61,91,84)(62,85,92)(63,93,86)(64,87,94)(65,80,121)(66,122,73)(67,74,123)(68,124,75)(69,76,125)(70,126,77)(71,78,127)(72,128,79), (1,106,121)(2,107,122)(3,108,123)(4,109,124)(5,110,125)(6,111,126)(7,112,127)(8,105,128)(9,76,48)(10,77,41)(11,78,42)(12,79,43)(13,80,44)(14,73,45)(15,74,46)(16,75,47)(17,32,91)(18,25,92)(19,26,93)(20,27,94)(21,28,95)(22,29,96)(23,30,89)(24,31,90)(33,136,84)(34,129,85)(35,130,86)(36,131,87)(37,132,88)(38,133,81)(39,134,82)(40,135,83)(49,67,143)(50,68,144)(51,69,137)(52,70,138)(53,71,139)(54,72,140)(55,65,141)(56,66,142)(57,104,115)(58,97,116)(59,98,117)(60,99,118)(61,100,119)(62,101,120)(63,102,113)(64,103,114), (1,141,13)(2,14,142)(3,143,15)(4,16,144)(5,137,9)(6,10,138)(7,139,11)(8,12,140)(17,136,100)(18,101,129)(19,130,102)(20,103,131)(21,132,104)(22,97,133)(23,134,98)(24,99,135)(25,120,85)(26,86,113)(27,114,87)(28,88,115)(29,116,81)(30,82,117)(31,118,83)(32,84,119)(33,61,91)(34,92,62)(35,63,93)(36,94,64)(37,57,95)(38,96,58)(39,59,89)(40,90,60)(41,70,126)(42,127,71)(43,72,128)(44,121,65)(45,66,122)(46,123,67)(47,68,124)(48,125,69)(49,74,108)(50,109,75)(51,76,110)(52,111,77)(53,78,112)(54,105,79)(55,80,106)(56,107,73), (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([[(1,90),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,89),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,107),(18,108),(19,109),(20,110),(21,111),(22,112),(23,105),(24,106),(25,123),(26,124),(27,125),(28,126),(29,127),(30,128),(31,121),(32,122),(41,88),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,101),(50,102),(51,103),(52,104),(53,97),(54,98),(55,99),(56,100),(57,138),(58,139),(59,140),(60,141),(61,142),(62,143),(63,144),(64,137),(65,118),(66,119),(67,120),(68,113),(69,114),(70,115),(71,116),(72,117),(73,136),(74,129),(75,130),(76,131),(77,132),(78,133),(79,134),(80,135)], [(1,141,44),(2,45,142),(3,143,46),(4,47,144),(5,137,48),(6,41,138),(7,139,42),(8,43,140),(9,110,51),(10,52,111),(11,112,53),(12,54,105),(13,106,55),(14,56,107),(15,108,49),(16,50,109),(17,33,100),(18,101,34),(19,35,102),(20,103,36),(21,37,104),(22,97,38),(23,39,98),(24,99,40),(25,120,129),(26,130,113),(27,114,131),(28,132,115),(29,116,133),(30,134,117),(31,118,135),(32,136,119),(57,95,88),(58,81,96),(59,89,82),(60,83,90),(61,91,84),(62,85,92),(63,93,86),(64,87,94),(65,80,121),(66,122,73),(67,74,123),(68,124,75),(69,76,125),(70,126,77),(71,78,127),(72,128,79)], [(1,106,121),(2,107,122),(3,108,123),(4,109,124),(5,110,125),(6,111,126),(7,112,127),(8,105,128),(9,76,48),(10,77,41),(11,78,42),(12,79,43),(13,80,44),(14,73,45),(15,74,46),(16,75,47),(17,32,91),(18,25,92),(19,26,93),(20,27,94),(21,28,95),(22,29,96),(23,30,89),(24,31,90),(33,136,84),(34,129,85),(35,130,86),(36,131,87),(37,132,88),(38,133,81),(39,134,82),(40,135,83),(49,67,143),(50,68,144),(51,69,137),(52,70,138),(53,71,139),(54,72,140),(55,65,141),(56,66,142),(57,104,115),(58,97,116),(59,98,117),(60,99,118),(61,100,119),(62,101,120),(63,102,113),(64,103,114)], [(1,141,13),(2,14,142),(3,143,15),(4,16,144),(5,137,9),(6,10,138),(7,139,11),(8,12,140),(17,136,100),(18,101,129),(19,130,102),(20,103,131),(21,132,104),(22,97,133),(23,134,98),(24,99,135),(25,120,85),(26,86,113),(27,114,87),(28,88,115),(29,116,81),(30,82,117),(31,118,83),(32,84,119),(33,61,91),(34,92,62),(35,63,93),(36,94,64),(37,57,95),(38,96,58),(39,59,89),(40,90,60),(41,70,126),(42,127,71),(43,72,128),(44,121,65),(45,66,122),(46,123,67),(47,68,124),(48,125,69),(49,74,108),(50,109,75),(51,76,110),(52,111,77),(53,78,112),(54,105,79),(55,80,106),(56,107,73)], [(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 ··· 6F 6G ··· 6R 8A ··· 8H 12A ··· 12H 12I ··· 12X 24A ··· 24P order 1 2 2 2 3 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 1 1 6 6 6 6 1 1 1 1 1 ··· 1 6 ··· 6 9 ··· 9 1 ··· 1 6 ··· 6 9 ··· 9

80 irreducible representations

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

Matrix representation of C2×He34C8 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 0 72 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 64 0 0 0 0 0 64 0 0 0 0 0 64
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 64 0 0 1 0 0
,
 0 22 0 0 0 22 0 0 0 0 0 0 0 0 51 0 0 0 51 0 0 0 51 0 0

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[0,1,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,64,0],[0,22,0,0,0,22,0,0,0,0,0,0,0,0,51,0,0,0,51,0,0,0,51,0,0] >;

C2×He34C8 in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_4C_8
% in TeX

G:=Group("C2xHe3:4C8");
// GroupNames label

G:=SmallGroup(432,184);
// by ID

G=gap.SmallGroup(432,184);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,58,1124,4037,537]);
// 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,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽