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## G = C4×C32⋊C12order 432 = 24·33

### Direct product of C4 and C32⋊C12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C4×C32⋊C12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×He3 — C2×C32⋊C12 — C4×C32⋊C12
 Lower central C32 — C4×C32⋊C12
 Upper central C1 — C2×C4

Generators and relations for C4×C32⋊C12
G = < a,b,c,d | a4=b3=c3=d12=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=c-1 >

Subgroups: 397 in 121 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C62, C62, C4×Dic3, C4×C12, C2×He3, C2×He3, C6×Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C32⋊C12, C4×He3, C22×He3, Dic3×C12, C4×C3⋊Dic3, C2×C32⋊C12, C2×C4×He3, C4×C32⋊C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, C42, C3×S3, C4×S3, C2×Dic3, C2×C12, C3×Dic3, S3×C6, C4×Dic3, C4×C12, C32⋊C6, S3×C12, C6×Dic3, C32⋊C12, C2×C32⋊C6, Dic3×C12, C4×C32⋊C6, C2×C32⋊C12, C4×C32⋊C12

Smallest permutation representation of C4×C32⋊C12
On 144 points
Generators in S144
(1 62 120 15)(2 63 109 16)(3 64 110 17)(4 65 111 18)(5 66 112 19)(6 67 113 20)(7 68 114 21)(8 69 115 22)(9 70 116 23)(10 71 117 24)(11 72 118 13)(12 61 119 14)(25 93 41 130)(26 94 42 131)(27 95 43 132)(28 96 44 121)(29 85 45 122)(30 86 46 123)(31 87 47 124)(32 88 48 125)(33 89 37 126)(34 90 38 127)(35 91 39 128)(36 92 40 129)(49 137 106 78)(50 138 107 79)(51 139 108 80)(52 140 97 81)(53 141 98 82)(54 142 99 83)(55 143 100 84)(56 144 101 73)(57 133 102 74)(58 134 103 75)(59 135 104 76)(60 136 105 77)
(1 73 85)(2 74 86)(4 88 76)(5 89 77)(7 79 91)(8 80 92)(10 94 82)(11 95 83)(13 27 99)(15 101 29)(16 102 30)(18 32 104)(19 33 105)(21 107 35)(22 108 36)(24 26 98)(37 60 66)(39 68 50)(40 69 51)(42 53 71)(43 54 72)(45 62 56)(46 63 57)(48 59 65)(109 133 123)(111 125 135)(112 126 136)(114 138 128)(115 139 129)(117 131 141)(118 132 142)(120 144 122)
(1 85 73)(2 74 86)(3 87 75)(4 76 88)(5 89 77)(6 78 90)(7 91 79)(8 80 92)(9 93 81)(10 82 94)(11 95 83)(12 84 96)(13 27 99)(14 100 28)(15 29 101)(16 102 30)(17 31 103)(18 104 32)(19 33 105)(20 106 34)(21 35 107)(22 108 36)(23 25 97)(24 98 26)(37 60 66)(38 67 49)(39 50 68)(40 69 51)(41 52 70)(42 71 53)(43 54 72)(44 61 55)(45 56 62)(46 63 57)(47 58 64)(48 65 59)(109 133 123)(110 124 134)(111 135 125)(112 126 136)(113 137 127)(114 128 138)(115 139 129)(116 130 140)(117 141 131)(118 132 142)(119 143 121)(120 122 144)
(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)| (1,62,120,15)(2,63,109,16)(3,64,110,17)(4,65,111,18)(5,66,112,19)(6,67,113,20)(7,68,114,21)(8,69,115,22)(9,70,116,23)(10,71,117,24)(11,72,118,13)(12,61,119,14)(25,93,41,130)(26,94,42,131)(27,95,43,132)(28,96,44,121)(29,85,45,122)(30,86,46,123)(31,87,47,124)(32,88,48,125)(33,89,37,126)(34,90,38,127)(35,91,39,128)(36,92,40,129)(49,137,106,78)(50,138,107,79)(51,139,108,80)(52,140,97,81)(53,141,98,82)(54,142,99,83)(55,143,100,84)(56,144,101,73)(57,133,102,74)(58,134,103,75)(59,135,104,76)(60,136,105,77), (1,73,85)(2,74,86)(4,88,76)(5,89,77)(7,79,91)(8,80,92)(10,94,82)(11,95,83)(13,27,99)(15,101,29)(16,102,30)(18,32,104)(19,33,105)(21,107,35)(22,108,36)(24,26,98)(37,60,66)(39,68,50)(40,69,51)(42,53,71)(43,54,72)(45,62,56)(46,63,57)(48,59,65)(109,133,123)(111,125,135)(112,126,136)(114,138,128)(115,139,129)(117,131,141)(118,132,142)(120,144,122), (1,85,73)(2,74,86)(3,87,75)(4,76,88)(5,89,77)(6,78,90)(7,91,79)(8,80,92)(9,93,81)(10,82,94)(11,95,83)(12,84,96)(13,27,99)(14,100,28)(15,29,101)(16,102,30)(17,31,103)(18,104,32)(19,33,105)(20,106,34)(21,35,107)(22,108,36)(23,25,97)(24,98,26)(37,60,66)(38,67,49)(39,50,68)(40,69,51)(41,52,70)(42,71,53)(43,54,72)(44,61,55)(45,56,62)(46,63,57)(47,58,64)(48,65,59)(109,133,123)(110,124,134)(111,135,125)(112,126,136)(113,137,127)(114,128,138)(115,139,129)(116,130,140)(117,141,131)(118,132,142)(119,143,121)(120,122,144), (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( (1,62,120,15)(2,63,109,16)(3,64,110,17)(4,65,111,18)(5,66,112,19)(6,67,113,20)(7,68,114,21)(8,69,115,22)(9,70,116,23)(10,71,117,24)(11,72,118,13)(12,61,119,14)(25,93,41,130)(26,94,42,131)(27,95,43,132)(28,96,44,121)(29,85,45,122)(30,86,46,123)(31,87,47,124)(32,88,48,125)(33,89,37,126)(34,90,38,127)(35,91,39,128)(36,92,40,129)(49,137,106,78)(50,138,107,79)(51,139,108,80)(52,140,97,81)(53,141,98,82)(54,142,99,83)(55,143,100,84)(56,144,101,73)(57,133,102,74)(58,134,103,75)(59,135,104,76)(60,136,105,77), (1,73,85)(2,74,86)(4,88,76)(5,89,77)(7,79,91)(8,80,92)(10,94,82)(11,95,83)(13,27,99)(15,101,29)(16,102,30)(18,32,104)(19,33,105)(21,107,35)(22,108,36)(24,26,98)(37,60,66)(39,68,50)(40,69,51)(42,53,71)(43,54,72)(45,62,56)(46,63,57)(48,59,65)(109,133,123)(111,125,135)(112,126,136)(114,138,128)(115,139,129)(117,131,141)(118,132,142)(120,144,122), (1,85,73)(2,74,86)(3,87,75)(4,76,88)(5,89,77)(6,78,90)(7,91,79)(8,80,92)(9,93,81)(10,82,94)(11,95,83)(12,84,96)(13,27,99)(14,100,28)(15,29,101)(16,102,30)(17,31,103)(18,104,32)(19,33,105)(20,106,34)(21,35,107)(22,108,36)(23,25,97)(24,98,26)(37,60,66)(38,67,49)(39,50,68)(40,69,51)(41,52,70)(42,71,53)(43,54,72)(44,61,55)(45,56,62)(46,63,57)(47,58,64)(48,65,59)(109,133,123)(110,124,134)(111,135,125)(112,126,136)(113,137,127)(114,128,138)(115,139,129)(116,130,140)(117,141,131)(118,132,142)(119,143,121)(120,122,144), (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([[(1,62,120,15),(2,63,109,16),(3,64,110,17),(4,65,111,18),(5,66,112,19),(6,67,113,20),(7,68,114,21),(8,69,115,22),(9,70,116,23),(10,71,117,24),(11,72,118,13),(12,61,119,14),(25,93,41,130),(26,94,42,131),(27,95,43,132),(28,96,44,121),(29,85,45,122),(30,86,46,123),(31,87,47,124),(32,88,48,125),(33,89,37,126),(34,90,38,127),(35,91,39,128),(36,92,40,129),(49,137,106,78),(50,138,107,79),(51,139,108,80),(52,140,97,81),(53,141,98,82),(54,142,99,83),(55,143,100,84),(56,144,101,73),(57,133,102,74),(58,134,103,75),(59,135,104,76),(60,136,105,77)], [(1,73,85),(2,74,86),(4,88,76),(5,89,77),(7,79,91),(8,80,92),(10,94,82),(11,95,83),(13,27,99),(15,101,29),(16,102,30),(18,32,104),(19,33,105),(21,107,35),(22,108,36),(24,26,98),(37,60,66),(39,68,50),(40,69,51),(42,53,71),(43,54,72),(45,62,56),(46,63,57),(48,59,65),(109,133,123),(111,125,135),(112,126,136),(114,138,128),(115,139,129),(117,131,141),(118,132,142),(120,144,122)], [(1,85,73),(2,74,86),(3,87,75),(4,76,88),(5,89,77),(6,78,90),(7,91,79),(8,80,92),(9,93,81),(10,82,94),(11,95,83),(12,84,96),(13,27,99),(14,100,28),(15,29,101),(16,102,30),(17,31,103),(18,104,32),(19,33,105),(20,106,34),(21,35,107),(22,108,36),(23,25,97),(24,98,26),(37,60,66),(38,67,49),(39,50,68),(40,69,51),(41,52,70),(42,71,53),(43,54,72),(44,61,55),(45,56,62),(46,63,57),(47,58,64),(48,65,59),(109,133,123),(110,124,134),(111,135,125),(112,126,136),(113,137,127),(114,128,138),(115,139,129),(116,130,140),(117,141,131),(118,132,142),(119,143,121),(120,122,144)], [(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 4E ··· 4L 6A 6B 6C 6D ··· 6I 6J ··· 6R 12A 12B 12C 12D 12E ··· 12L 12M ··· 12X 12Y ··· 12AN order 1 2 2 2 3 3 3 3 3 3 4 4 4 4 4 ··· 4 6 6 6 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 2 3 3 6 6 6 1 1 1 1 9 ··· 9 2 2 2 3 ··· 3 6 ··· 6 2 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 6 6 6 6 type + + + + - + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 Dic3 D6 C3×S3 C4×S3 C3×Dic3 S3×C6 S3×C12 C32⋊C6 C32⋊C12 C2×C32⋊C6 C4×C32⋊C6 kernel C4×C32⋊C12 C2×C32⋊C12 C2×C4×He3 C4×C3⋊Dic3 C32⋊C12 C4×He3 C2×C3⋊Dic3 C6×C12 C3⋊Dic3 C3×C12 C6×C12 C3×C12 C62 C2×C12 C3×C6 C12 C2×C6 C6 C2×C4 C4 C22 C2 # reps 1 2 1 2 8 4 4 2 16 8 1 2 1 2 4 4 2 8 1 2 1 4

Matrix representation of C4×C32⋊C12 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5
,
 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 3 11 0 4 0 12 0 0 0 9 0 1 0 4 0 0 0 0 9 10 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 2 0 9 0 1 0 0 0 3 0 0 0 0 0 0 0 0 9 10 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 9 10 0 0 0 0 0 0 0 3
,
 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 6 9 0 0 11 8 7 0 8 0 0 0 6 7 0 12 0 10 0 0 12 4 0 5 0 2 0 0 0 0 6 9 0 0 0 0 0 0 12 7 0 0

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5],[12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,4,1,10,3,0,0,0,0,0,0,0,0,1,0,0,0,12,4,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,9,0,0,0,0,0,9,0,10,3,0,0,0,0,0,0,0,0,9,0,0,0,1,0,0,0,10,3],[0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,11,6,12,0,0,0,0,0,8,7,4,0,0,0,0,0,7,0,0,6,12,0,0,0,0,12,5,9,7,0,0,6,8,0,0,0,0,0,0,9,0,10,2,0,0] >;

C4×C32⋊C12 in GAP, Magma, Sage, TeX

C_4\times C_3^2\rtimes C_{12}
% in TeX

G:=Group("C4xC3^2:C12");
// GroupNames label

G:=SmallGroup(432,138);
// by ID

G=gap.SmallGroup(432,138);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,176,4037,2035,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^3=c^3=d^12=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=c^-1>;
// generators/relations

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