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## G = C32×C24⋊C2order 432 = 24·33

### Direct product of C32 and C24⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C32×C24⋊C2
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C32×C12 — C32×D12 — C32×C24⋊C2
 Lower central C3 — C6 — C12 — C32×C24⋊C2
 Upper central C1 — C3×C6 — C3×C12 — C3×C24

Generators and relations for C32×C24⋊C2
G = < a,b,c,d | a3=b3=c24=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c11 >

Subgroups: 376 in 152 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, SD16, C3×S3, C3×C6, C3×C6, C3×C6, C24, C24, C24, Dic6, D12, C3×D4, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C62, C24⋊C2, C3×SD16, S3×C32, C32×C6, C3×C24, C3×C24, C3×C24, C3×Dic6, C3×D12, D4×C32, Q8×C32, C32×Dic3, C32×C12, S3×C3×C6, C3×C24⋊C2, C32×SD16, C32×C24, C32×Dic6, C32×D12, C32×C24⋊C2
Quotients: C1, C2, C3, C22, S3, C6, D4, C32, D6, C2×C6, SD16, C3×S3, C3×C6, D12, C3×D4, S3×C6, C62, C24⋊C2, C3×SD16, S3×C32, C3×D12, D4×C32, S3×C3×C6, C3×C24⋊C2, C32×SD16, C32×D12, C32×C24⋊C2

Smallest permutation representation of C32×C24⋊C2
On 144 points
Generators in S144
(1 51 74)(2 52 75)(3 53 76)(4 54 77)(5 55 78)(6 56 79)(7 57 80)(8 58 81)(9 59 82)(10 60 83)(11 61 84)(12 62 85)(13 63 86)(14 64 87)(15 65 88)(16 66 89)(17 67 90)(18 68 91)(19 69 92)(20 70 93)(21 71 94)(22 72 95)(23 49 96)(24 50 73)(25 135 113)(26 136 114)(27 137 115)(28 138 116)(29 139 117)(30 140 118)(31 141 119)(32 142 120)(33 143 97)(34 144 98)(35 121 99)(36 122 100)(37 123 101)(38 124 102)(39 125 103)(40 126 104)(41 127 105)(42 128 106)(43 129 107)(44 130 108)(45 131 109)(46 132 110)(47 133 111)(48 134 112)
(1 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(49 57 65)(50 58 66)(51 59 67)(52 60 68)(53 61 69)(54 62 70)(55 63 71)(56 64 72)(73 81 89)(74 82 90)(75 83 91)(76 84 92)(77 85 93)(78 86 94)(79 87 95)(80 88 96)(97 113 105)(98 114 106)(99 115 107)(100 116 108)(101 117 109)(102 118 110)(103 119 111)(104 120 112)(121 137 129)(122 138 130)(123 139 131)(124 140 132)(125 141 133)(126 142 134)(127 143 135)(128 144 136)
(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)
(1 123)(2 134)(3 121)(4 132)(5 143)(6 130)(7 141)(8 128)(9 139)(10 126)(11 137)(12 124)(13 135)(14 122)(15 133)(16 144)(17 131)(18 142)(19 129)(20 140)(21 127)(22 138)(23 125)(24 136)(25 86)(26 73)(27 84)(28 95)(29 82)(30 93)(31 80)(32 91)(33 78)(34 89)(35 76)(36 87)(37 74)(38 85)(39 96)(40 83)(41 94)(42 81)(43 92)(44 79)(45 90)(46 77)(47 88)(48 75)(49 103)(50 114)(51 101)(52 112)(53 99)(54 110)(55 97)(56 108)(57 119)(58 106)(59 117)(60 104)(61 115)(62 102)(63 113)(64 100)(65 111)(66 98)(67 109)(68 120)(69 107)(70 118)(71 105)(72 116)

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

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

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

135 conjugacy classes

 class 1 2A 2B 3A ··· 3H 3I ··· 3Q 4A 4B 6A ··· 6H 6I ··· 6Q 6R ··· 6Y 8A 8B 12A ··· 12Z 12AA ··· 12AH 24A ··· 24AZ order 1 2 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 8 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 12 1 ··· 1 2 ··· 2 2 12 1 ··· 1 2 ··· 2 12 ··· 12 2 2 2 ··· 2 12 ··· 12 2 ··· 2

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 SD16 C3×S3 D12 C3×D4 S3×C6 C24⋊C2 C3×SD16 C3×D12 C3×C24⋊C2 kernel C32×C24⋊C2 C32×C24 C32×Dic6 C32×D12 C3×C24⋊C2 C3×C24 C3×Dic6 C3×D12 C3×C24 C32×C6 C3×C12 C33 C24 C3×C6 C3×C6 C12 C32 C32 C6 C3 # reps 1 1 1 1 8 8 8 8 1 1 1 2 8 2 8 8 4 16 16 32

Matrix representation of C32×C24⋊C2 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 64 0 0 0 0 64
,
 64 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 3 0 0 0 71 49 0 0 0 0 1 18 0 0 48 60
,
 60 7 0 0 49 13 0 0 0 0 22 42 0 0 25 51
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[3,71,0,0,0,49,0,0,0,0,1,48,0,0,18,60],[60,49,0,0,7,13,0,0,0,0,22,25,0,0,42,51] >;

C32×C24⋊C2 in GAP, Magma, Sage, TeX

C_3^2\times C_{24}\rtimes C_2
% in TeX

G:=Group("C3^2xC24:C2");
// GroupNames label

G:=SmallGroup(432,466);
// by ID

G=gap.SmallGroup(432,466);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,533,260,3784,102,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^24=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^11>;
// generators/relations

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