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## G = C32×C8⋊S3order 432 = 24·33

### Direct product of C32 and C8⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C32×C8⋊S3
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C32×C12 — S3×C3×C12 — C32×C8⋊S3
 Lower central C3 — C6 — C32×C8⋊S3
 Upper central C1 — C3×C12 — C3×C24

Generators and relations for C32×C8⋊S3
G = < a,b,c,d,e | a3=b3=c8=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c5, ede=d-1 >

Subgroups: 280 in 152 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, C8, C2×C4, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, M4(2), C3×S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C24, C24, C4×S3, C2×C12, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C62, C8⋊S3, C3×M4(2), S3×C32, C32×C6, C3×C3⋊C8, C3×C24, C3×C24, C3×C24, S3×C12, C6×C12, C32×Dic3, C32×C12, S3×C3×C6, C3×C8⋊S3, C32×M4(2), C32×C3⋊C8, C32×C24, S3×C3×C12, C32×C8⋊S3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C32, C12, D6, C2×C6, M4(2), C3×S3, C3×C6, C4×S3, C2×C12, C3×C12, S3×C6, C62, C8⋊S3, C3×M4(2), S3×C32, S3×C12, C6×C12, S3×C3×C6, C3×C8⋊S3, C32×M4(2), S3×C3×C12, C32×C8⋊S3

Smallest permutation representation of C32×C8⋊S3
On 144 points
Generators in S144
(1 85 73)(2 86 74)(3 87 75)(4 88 76)(5 81 77)(6 82 78)(7 83 79)(8 84 80)(9 98 95)(10 99 96)(11 100 89)(12 101 90)(13 102 91)(14 103 92)(15 104 93)(16 97 94)(17 113 60)(18 114 61)(19 115 62)(20 116 63)(21 117 64)(22 118 57)(23 119 58)(24 120 59)(25 55 67)(26 56 68)(27 49 69)(28 50 70)(29 51 71)(30 52 72)(31 53 65)(32 54 66)(33 111 46)(34 112 47)(35 105 48)(36 106 41)(37 107 42)(38 108 43)(39 109 44)(40 110 45)(121 144 134)(122 137 135)(123 138 136)(124 139 129)(125 140 130)(126 141 131)(127 142 132)(128 143 133)
(1 21 71)(2 22 72)(3 23 65)(4 24 66)(5 17 67)(6 18 68)(7 19 69)(8 20 70)(9 135 47)(10 136 48)(11 129 41)(12 130 42)(13 131 43)(14 132 44)(15 133 45)(16 134 46)(25 81 113)(26 82 114)(27 83 115)(28 84 116)(29 85 117)(30 86 118)(31 87 119)(32 88 120)(33 97 121)(34 98 122)(35 99 123)(36 100 124)(37 101 125)(38 102 126)(39 103 127)(40 104 128)(49 79 62)(50 80 63)(51 73 64)(52 74 57)(53 75 58)(54 76 59)(55 77 60)(56 78 61)(89 139 106)(90 140 107)(91 141 108)(92 142 109)(93 143 110)(94 144 111)(95 137 112)(96 138 105)
(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 71 21)(2 72 22)(3 65 23)(4 66 24)(5 67 17)(6 68 18)(7 69 19)(8 70 20)(9 135 47)(10 136 48)(11 129 41)(12 130 42)(13 131 43)(14 132 44)(15 133 45)(16 134 46)(25 113 81)(26 114 82)(27 115 83)(28 116 84)(29 117 85)(30 118 86)(31 119 87)(32 120 88)(33 97 121)(34 98 122)(35 99 123)(36 100 124)(37 101 125)(38 102 126)(39 103 127)(40 104 128)(49 62 79)(50 63 80)(51 64 73)(52 57 74)(53 58 75)(54 59 76)(55 60 77)(56 61 78)(89 139 106)(90 140 107)(91 141 108)(92 142 109)(93 143 110)(94 144 111)(95 137 112)(96 138 105)
(1 106)(2 111)(3 108)(4 105)(5 110)(6 107)(7 112)(8 109)(9 115)(10 120)(11 117)(12 114)(13 119)(14 116)(15 113)(16 118)(17 93)(18 90)(19 95)(20 92)(21 89)(22 94)(23 91)(24 96)(25 133)(26 130)(27 135)(28 132)(29 129)(30 134)(31 131)(32 136)(33 74)(34 79)(35 76)(36 73)(37 78)(38 75)(39 80)(40 77)(41 85)(42 82)(43 87)(44 84)(45 81)(46 86)(47 83)(48 88)(49 122)(50 127)(51 124)(52 121)(53 126)(54 123)(55 128)(56 125)(57 97)(58 102)(59 99)(60 104)(61 101)(62 98)(63 103)(64 100)(65 141)(66 138)(67 143)(68 140)(69 137)(70 142)(71 139)(72 144)

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

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

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

162 conjugacy classes

 class 1 2A 2B 3A ··· 3H 3I ··· 3Q 4A 4B 4C 6A ··· 6H 6I ··· 6Q 6R ··· 6Y 8A 8B 8C 8D 12A ··· 12P 12Q ··· 12AH 12AI ··· 12AP 24A ··· 24AZ 24BA ··· 24BP order 1 2 2 3 ··· 3 3 ··· 3 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 8 8 8 8 12 ··· 12 12 ··· 12 12 ··· 12 24 ··· 24 24 ··· 24 size 1 1 6 1 ··· 1 2 ··· 2 1 1 6 1 ··· 1 2 ··· 2 6 ··· 6 2 2 6 6 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 6 ··· 6

162 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 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 S3 D6 M4(2) C3×S3 C4×S3 S3×C6 C8⋊S3 C3×M4(2) S3×C12 C3×C8⋊S3 kernel C32×C8⋊S3 C32×C3⋊C8 C32×C24 S3×C3×C12 C3×C8⋊S3 C32×Dic3 S3×C3×C6 C3×C3⋊C8 C3×C24 S3×C12 C3×Dic3 S3×C6 C3×C24 C3×C12 C33 C24 C3×C6 C12 C32 C32 C6 C3 # reps 1 1 1 1 8 2 2 8 8 8 16 16 1 1 2 8 2 8 4 16 16 32

Matrix representation of C32×C8⋊S3 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 1 0 0 0 0 1 0 0 0 0 64 0 0 0 0 64
,
 0 1 0 0 27 0 0 0 0 0 63 0 0 0 5 10
,
 1 0 0 0 0 1 0 0 0 0 8 0 0 0 14 64
,
 70 21 0 0 17 3 0 0 0 0 55 1 0 0 42 18
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[0,27,0,0,1,0,0,0,0,0,63,5,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,8,14,0,0,0,64],[70,17,0,0,21,3,0,0,0,0,55,42,0,0,1,18] >;

C32×C8⋊S3 in GAP, Magma, Sage, TeX

C_3^2\times C_8\rtimes S_3
% in TeX

G:=Group("C3^2xC8:S3");
// GroupNames label

G:=SmallGroup(432,465);
// by ID

G=gap.SmallGroup(432,465);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,1037,260,102,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^3=e^2=1,a*b=b*a,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^5,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽