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## G = C32×C4⋊Dic3order 432 = 24·33

### Direct product of C32 and C4⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C32×C4⋊Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C3×C62 — Dic3×C3×C6 — C32×C4⋊Dic3
 Lower central C3 — C6 — C32×C4⋊Dic3
 Upper central C1 — C62 — C6×C12

Generators and relations for C32×C4⋊Dic3
G = < a,b,c,d,e | a3=b3=c4=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 392 in 220 conjugacy classes, 114 normal (26 characteristic)
C1, C2, C3, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C32, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C2×C12, C33, C3×Dic3, C3×C12, C3×C12, C62, C62, C62, C4⋊Dic3, C3×C4⋊C4, C32×C6, C6×Dic3, C6×C12, C6×C12, C6×C12, C32×Dic3, C32×C12, C3×C62, C3×C4⋊Dic3, C32×C4⋊C4, Dic3×C3×C6, C3×C6×C12, C32×C4⋊Dic3
Quotients:

Smallest permutation representation of C32×C4⋊Dic3
On 144 points
Generators in S144
(1 36 24)(2 31 19)(3 32 20)(4 33 21)(5 34 22)(6 35 23)(7 136 124)(8 137 125)(9 138 126)(10 133 121)(11 134 122)(12 135 123)(13 37 25)(14 38 26)(15 39 27)(16 40 28)(17 41 29)(18 42 30)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)(49 74 61)(50 75 62)(51 76 63)(52 77 64)(53 78 65)(54 73 66)(79 103 91)(80 104 92)(81 105 93)(82 106 94)(83 107 95)(84 108 96)(85 109 97)(86 110 98)(87 111 99)(88 112 100)(89 113 101)(90 114 102)(115 139 127)(116 140 128)(117 141 129)(118 142 130)(119 143 131)(120 144 132)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)(49 53 51)(50 54 52)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 77 75)(74 78 76)(79 81 83)(80 82 84)(85 87 89)(86 88 90)(91 93 95)(92 94 96)(97 99 101)(98 100 102)(103 105 107)(104 106 108)(109 111 113)(110 112 114)(115 117 119)(116 118 120)(121 123 125)(122 124 126)(127 129 131)(128 130 132)(133 135 137)(134 136 138)(139 141 143)(140 142 144)
(1 53 17 48)(2 54 18 43)(3 49 13 44)(4 50 14 45)(5 51 15 46)(6 52 16 47)(7 108 144 112)(8 103 139 113)(9 104 140 114)(10 105 141 109)(11 106 142 110)(12 107 143 111)(19 66 30 55)(20 61 25 56)(21 62 26 57)(22 63 27 58)(23 64 28 59)(24 65 29 60)(31 73 42 67)(32 74 37 68)(33 75 38 69)(34 76 39 70)(35 77 40 71)(36 78 41 72)(79 115 89 125)(80 116 90 126)(81 117 85 121)(82 118 86 122)(83 119 87 123)(84 120 88 124)(91 127 101 137)(92 128 102 138)(93 129 97 133)(94 130 98 134)(95 131 99 135)(96 132 100 136)
(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 81 4 84)(2 80 5 83)(3 79 6 82)(7 78 10 75)(8 77 11 74)(9 76 12 73)(13 89 16 86)(14 88 17 85)(15 87 18 90)(19 92 22 95)(20 91 23 94)(21 96 24 93)(25 101 28 98)(26 100 29 97)(27 99 30 102)(31 104 34 107)(32 103 35 106)(33 108 36 105)(37 113 40 110)(38 112 41 109)(39 111 42 114)(43 116 46 119)(44 115 47 118)(45 120 48 117)(49 125 52 122)(50 124 53 121)(51 123 54 126)(55 128 58 131)(56 127 59 130)(57 132 60 129)(61 137 64 134)(62 136 65 133)(63 135 66 138)(67 140 70 143)(68 139 71 142)(69 144 72 141)

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

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

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

162 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Q 4A 4B 4C 4D 4E 4F 6A ··· 6X 6Y ··· 6AY 12A ··· 12AZ 12BA ··· 12CF order 1 2 2 2 3 ··· 3 3 ··· 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 1 ··· 1 2 ··· 2 2 2 6 6 6 6 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - - + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Q8 Dic3 D6 C3×S3 Dic6 D12 C3×D4 C3×Q8 C3×Dic3 S3×C6 C3×Dic6 C3×D12 kernel C32×C4⋊Dic3 Dic3×C3×C6 C3×C6×C12 C3×C4⋊Dic3 C32×C12 C6×Dic3 C6×C12 C3×C12 C6×C12 C32×C6 C32×C6 C3×C12 C62 C2×C12 C3×C6 C3×C6 C3×C6 C3×C6 C12 C2×C6 C6 C6 # reps 1 2 1 8 4 16 8 32 1 1 1 2 1 8 2 2 8 8 16 8 16 16

Matrix representation of C32×C4⋊Dic3 in GL4(𝔽13) generated by

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

C32×C4⋊Dic3 in GAP, Magma, Sage, TeX

C_3^2\times C_4\rtimes {\rm Dic}_3
% in TeX

G:=Group("C3^2xC4:Dic3");
// GroupNames label

G:=SmallGroup(432,473);
// by ID

G=gap.SmallGroup(432,473);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,252,1037,512,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^4=d^6=1,e^2=d^3,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^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

׿
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