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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 [×2], C3, C3 [×4], C3 [×4], C4 [×2], C22, C6, C6 [×14], C6 [×12], C8 [×2], C2×C4, C32, C32 [×4], C32 [×4], C12 [×2], C12 [×8], C12 [×8], C2×C6, C2×C6 [×4], C2×C6 [×4], C2×C8, C3×C6, C3×C6 [×14], C3×C6 [×12], C3⋊C8 [×8], C24 [×2], C2×C12, C2×C12 [×4], C2×C12 [×4], C33, C3×C12 [×2], C3×C12 [×8], C3×C12 [×8], C62, C62 [×4], C62 [×4], C2×C3⋊C8 [×4], C2×C24, C32×C6, C32×C6 [×2], C3×C3⋊C8 [×8], C324C8 [×2], C6×C12, C6×C12 [×4], C6×C12 [×4], C32×C12 [×2], C3×C62, C6×C3⋊C8 [×4], C2×C324C8, C3×C324C8 [×2], C3×C6×C12, C6×C324C8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3 [×4], C6 [×3], C8 [×2], C2×C4, Dic3 [×8], C12 [×2], D6 [×4], C2×C6, C2×C8, C3×S3 [×4], C3⋊S3, C3⋊C8 [×8], C24 [×2], C2×Dic3 [×4], C2×C12, C3×Dic3 [×8], C3⋊Dic3 [×2], S3×C6 [×4], C2×C3⋊S3, C2×C3⋊C8 [×4], C2×C24, C3×C3⋊S3, C3×C3⋊C8 [×8], C324C8 [×2], C6×Dic3 [×4], C2×C3⋊Dic3, C3×C3⋊Dic3 [×2], C6×C3⋊S3, C6×C3⋊C8 [×4], C2×C324C8, C3×C324C8 [×2], C6×C3⋊Dic3, C6×C324C8

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