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## G = C6×C32⋊4C8order 432 = 24·33

### Direct product of C6 and C32⋊4C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C6×C32⋊4C8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C3×C32⋊4C8 — C6×C32⋊4C8
 Lower central C32 — C6×C32⋊4C8
 Upper central C1 — C2×C12

Generators and relations for C6×C324C8
G = < a,b,c,d | a6=b3=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 356 in 196 conjugacy classes, 102 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C32, C32, C32, C12, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C2×C12, C33, C3×C12, C3×C12, C3×C12, C62, C62, C62, C2×C3⋊C8, C2×C24, C32×C6, C32×C6, C3×C3⋊C8, C324C8, C6×C12, C6×C12, C6×C12, C32×C12, C3×C62, C6×C3⋊C8, C2×C324C8, C3×C324C8, C3×C6×C12, C6×C324C8
Quotients:

Smallest permutation representation of C6×C324C8
On 144 points
Generators in S144
(1 37 52 66 18 122)(2 38 53 67 19 123)(3 39 54 68 20 124)(4 40 55 69 21 125)(5 33 56 70 22 126)(6 34 49 71 23 127)(7 35 50 72 24 128)(8 36 51 65 17 121)(9 77 133 47 29 100)(10 78 134 48 30 101)(11 79 135 41 31 102)(12 80 136 42 32 103)(13 73 129 43 25 104)(14 74 130 44 26 97)(15 75 131 45 27 98)(16 76 132 46 28 99)(57 119 108 143 95 84)(58 120 109 144 96 85)(59 113 110 137 89 86)(60 114 111 138 90 87)(61 115 112 139 91 88)(62 116 105 140 92 81)(63 117 106 141 93 82)(64 118 107 142 94 83)
(1 63 11)(2 12 64)(3 57 13)(4 14 58)(5 59 15)(6 16 60)(7 61 9)(8 10 62)(17 30 92)(18 93 31)(19 32 94)(20 95 25)(21 26 96)(22 89 27)(23 28 90)(24 91 29)(33 113 75)(34 76 114)(35 115 77)(36 78 116)(37 117 79)(38 80 118)(39 119 73)(40 74 120)(41 66 141)(42 142 67)(43 68 143)(44 144 69)(45 70 137)(46 138 71)(47 72 139)(48 140 65)(49 132 111)(50 112 133)(51 134 105)(52 106 135)(53 136 107)(54 108 129)(55 130 109)(56 110 131)(81 121 101)(82 102 122)(83 123 103)(84 104 124)(85 125 97)(86 98 126)(87 127 99)(88 100 128)
(1 106 31)(2 32 107)(3 108 25)(4 26 109)(5 110 27)(6 28 111)(7 112 29)(8 30 105)(9 50 91)(10 92 51)(11 52 93)(12 94 53)(13 54 95)(14 96 55)(15 56 89)(16 90 49)(17 134 62)(18 63 135)(19 136 64)(20 57 129)(21 130 58)(22 59 131)(23 132 60)(24 61 133)(33 137 98)(34 99 138)(35 139 100)(36 101 140)(37 141 102)(38 103 142)(39 143 104)(40 97 144)(41 122 117)(42 118 123)(43 124 119)(44 120 125)(45 126 113)(46 114 127)(47 128 115)(48 116 121)(65 78 81)(66 82 79)(67 80 83)(68 84 73)(69 74 85)(70 86 75)(71 76 87)(72 88 77)
(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)| (1,37,52,66,18,122)(2,38,53,67,19,123)(3,39,54,68,20,124)(4,40,55,69,21,125)(5,33,56,70,22,126)(6,34,49,71,23,127)(7,35,50,72,24,128)(8,36,51,65,17,121)(9,77,133,47,29,100)(10,78,134,48,30,101)(11,79,135,41,31,102)(12,80,136,42,32,103)(13,73,129,43,25,104)(14,74,130,44,26,97)(15,75,131,45,27,98)(16,76,132,46,28,99)(57,119,108,143,95,84)(58,120,109,144,96,85)(59,113,110,137,89,86)(60,114,111,138,90,87)(61,115,112,139,91,88)(62,116,105,140,92,81)(63,117,106,141,93,82)(64,118,107,142,94,83), (1,63,11)(2,12,64)(3,57,13)(4,14,58)(5,59,15)(6,16,60)(7,61,9)(8,10,62)(17,30,92)(18,93,31)(19,32,94)(20,95,25)(21,26,96)(22,89,27)(23,28,90)(24,91,29)(33,113,75)(34,76,114)(35,115,77)(36,78,116)(37,117,79)(38,80,118)(39,119,73)(40,74,120)(41,66,141)(42,142,67)(43,68,143)(44,144,69)(45,70,137)(46,138,71)(47,72,139)(48,140,65)(49,132,111)(50,112,133)(51,134,105)(52,106,135)(53,136,107)(54,108,129)(55,130,109)(56,110,131)(81,121,101)(82,102,122)(83,123,103)(84,104,124)(85,125,97)(86,98,126)(87,127,99)(88,100,128), (1,106,31)(2,32,107)(3,108,25)(4,26,109)(5,110,27)(6,28,111)(7,112,29)(8,30,105)(9,50,91)(10,92,51)(11,52,93)(12,94,53)(13,54,95)(14,96,55)(15,56,89)(16,90,49)(17,134,62)(18,63,135)(19,136,64)(20,57,129)(21,130,58)(22,59,131)(23,132,60)(24,61,133)(33,137,98)(34,99,138)(35,139,100)(36,101,140)(37,141,102)(38,103,142)(39,143,104)(40,97,144)(41,122,117)(42,118,123)(43,124,119)(44,120,125)(45,126,113)(46,114,127)(47,128,115)(48,116,121)(65,78,81)(66,82,79)(67,80,83)(68,84,73)(69,74,85)(70,86,75)(71,76,87)(72,88,77), (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( (1,37,52,66,18,122)(2,38,53,67,19,123)(3,39,54,68,20,124)(4,40,55,69,21,125)(5,33,56,70,22,126)(6,34,49,71,23,127)(7,35,50,72,24,128)(8,36,51,65,17,121)(9,77,133,47,29,100)(10,78,134,48,30,101)(11,79,135,41,31,102)(12,80,136,42,32,103)(13,73,129,43,25,104)(14,74,130,44,26,97)(15,75,131,45,27,98)(16,76,132,46,28,99)(57,119,108,143,95,84)(58,120,109,144,96,85)(59,113,110,137,89,86)(60,114,111,138,90,87)(61,115,112,139,91,88)(62,116,105,140,92,81)(63,117,106,141,93,82)(64,118,107,142,94,83), (1,63,11)(2,12,64)(3,57,13)(4,14,58)(5,59,15)(6,16,60)(7,61,9)(8,10,62)(17,30,92)(18,93,31)(19,32,94)(20,95,25)(21,26,96)(22,89,27)(23,28,90)(24,91,29)(33,113,75)(34,76,114)(35,115,77)(36,78,116)(37,117,79)(38,80,118)(39,119,73)(40,74,120)(41,66,141)(42,142,67)(43,68,143)(44,144,69)(45,70,137)(46,138,71)(47,72,139)(48,140,65)(49,132,111)(50,112,133)(51,134,105)(52,106,135)(53,136,107)(54,108,129)(55,130,109)(56,110,131)(81,121,101)(82,102,122)(83,123,103)(84,104,124)(85,125,97)(86,98,126)(87,127,99)(88,100,128), (1,106,31)(2,32,107)(3,108,25)(4,26,109)(5,110,27)(6,28,111)(7,112,29)(8,30,105)(9,50,91)(10,92,51)(11,52,93)(12,94,53)(13,54,95)(14,96,55)(15,56,89)(16,90,49)(17,134,62)(18,63,135)(19,136,64)(20,57,129)(21,130,58)(22,59,131)(23,132,60)(24,61,133)(33,137,98)(34,99,138)(35,139,100)(36,101,140)(37,141,102)(38,103,142)(39,143,104)(40,97,144)(41,122,117)(42,118,123)(43,124,119)(44,120,125)(45,126,113)(46,114,127)(47,128,115)(48,116,121)(65,78,81)(66,82,79)(67,80,83)(68,84,73)(69,74,85)(70,86,75)(71,76,87)(72,88,77), (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([[(1,37,52,66,18,122),(2,38,53,67,19,123),(3,39,54,68,20,124),(4,40,55,69,21,125),(5,33,56,70,22,126),(6,34,49,71,23,127),(7,35,50,72,24,128),(8,36,51,65,17,121),(9,77,133,47,29,100),(10,78,134,48,30,101),(11,79,135,41,31,102),(12,80,136,42,32,103),(13,73,129,43,25,104),(14,74,130,44,26,97),(15,75,131,45,27,98),(16,76,132,46,28,99),(57,119,108,143,95,84),(58,120,109,144,96,85),(59,113,110,137,89,86),(60,114,111,138,90,87),(61,115,112,139,91,88),(62,116,105,140,92,81),(63,117,106,141,93,82),(64,118,107,142,94,83)], [(1,63,11),(2,12,64),(3,57,13),(4,14,58),(5,59,15),(6,16,60),(7,61,9),(8,10,62),(17,30,92),(18,93,31),(19,32,94),(20,95,25),(21,26,96),(22,89,27),(23,28,90),(24,91,29),(33,113,75),(34,76,114),(35,115,77),(36,78,116),(37,117,79),(38,80,118),(39,119,73),(40,74,120),(41,66,141),(42,142,67),(43,68,143),(44,144,69),(45,70,137),(46,138,71),(47,72,139),(48,140,65),(49,132,111),(50,112,133),(51,134,105),(52,106,135),(53,136,107),(54,108,129),(55,130,109),(56,110,131),(81,121,101),(82,102,122),(83,123,103),(84,104,124),(85,125,97),(86,98,126),(87,127,99),(88,100,128)], [(1,106,31),(2,32,107),(3,108,25),(4,26,109),(5,110,27),(6,28,111),(7,112,29),(8,30,105),(9,50,91),(10,92,51),(11,52,93),(12,94,53),(13,54,95),(14,96,55),(15,56,89),(16,90,49),(17,134,62),(18,63,135),(19,136,64),(20,57,129),(21,130,58),(22,59,131),(23,132,60),(24,61,133),(33,137,98),(34,99,138),(35,139,100),(36,101,140),(37,141,102),(38,103,142),(39,143,104),(40,97,144),(41,122,117),(42,118,123),(43,124,119),(44,120,125),(45,126,113),(46,114,127),(47,128,115),(48,116,121),(65,78,81),(66,82,79),(67,80,83),(68,84,73),(69,74,85),(70,86,75),(71,76,87),(72,88,77)], [(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)]])

144 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3N 4A 4B 4C 4D 6A ··· 6F 6G ··· 6AP 8A ··· 8H 12A ··· 12H 12I ··· 12BD 24A ··· 24P order 1 2 2 2 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 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 9 ··· 9 1 ··· 1 2 ··· 2 9 ··· 9

144 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 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 kernel C6×C32⋊4C8 C3×C32⋊4C8 C3×C6×C12 C2×C32⋊4C8 C32×C12 C3×C62 C32⋊4C8 C6×C12 C32×C6 C3×C12 C62 C3×C6 C6×C12 C3×C12 C3×C12 C62 C2×C12 C3×C6 C12 C12 C2×C6 C6 # reps 1 2 1 2 2 2 4 2 8 4 4 16 4 4 4 4 8 16 8 8 8 32

Matrix representation of C6×C324C8 in GL5(𝔽73)

 64 0 0 0 0 0 64 0 0 0 0 0 64 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 1 0 0 0 72 0
,
 1 0 0 0 0 0 8 0 0 0 0 0 64 0 0 0 0 0 72 1 0 0 0 72 0
,
 51 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 34 50 0 0 0 11 39

G:=sub<GL(5,GF(73))| [64,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,1,0],[1,0,0,0,0,0,8,0,0,0,0,0,64,0,0,0,0,0,72,72,0,0,0,1,0],[51,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,34,11,0,0,0,50,39] >;

C6×C324C8 in GAP, Magma, Sage, TeX

C_6\times C_3^2\rtimes_4C_8
% in TeX

G:=Group("C6xC3^2:4C8");
// GroupNames label

G:=SmallGroup(432,485);
// by ID

G=gap.SmallGroup(432,485);
# by ID

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

G:=Group<a,b,c,d|a^6=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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