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## G = C22×C9⋊C12order 432 = 24·33

### Direct product of C22 and C9⋊C12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C22×C9⋊C12
 Chief series C1 — C3 — C9 — C18 — C2×3- 1+2 — C9⋊C12 — C2×C9⋊C12 — C22×C9⋊C12
 Lower central C9 — C22×C9⋊C12
 Upper central C1 — C23

Generators and relations for C22×C9⋊C12
G = < a,b,c,d | a2=b2=c9=d12=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 446 in 178 conjugacy classes, 102 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C22×C4, C18, C18, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, 3- 1+2, Dic9, C2×C18, C2×C18, C3×Dic3, C62, C22×Dic3, C22×C12, C2×3- 1+2, C2×3- 1+2, C2×Dic9, C22×C18, C22×C18, C6×Dic3, C2×C62, C9⋊C12, C22×3- 1+2, C22×Dic9, Dic3×C2×C6, C2×C9⋊C12, C23×3- 1+2, C22×C9⋊C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, Dic3, C12, D6, C2×C6, C22×C4, C3×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C3×Dic3, S3×C6, C22×Dic3, C22×C12, C9⋊C6, C6×Dic3, S3×C2×C6, C9⋊C12, C2×C9⋊C6, Dic3×C2×C6, C2×C9⋊C12, C22×C9⋊C6, C22×C9⋊C12

Smallest permutation representation of C22×C9⋊C12
On 144 points
Generators in S144
(1 30)(2 31)(3 32)(4 29)(5 15)(6 16)(7 13)(8 14)(9 39)(10 40)(11 37)(12 38)(17 27)(18 28)(19 25)(20 26)(21 47)(22 48)(23 45)(24 46)(33 42)(34 43)(35 44)(36 41)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 73)(60 74)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 97)(69 98)(70 99)(71 100)(72 101)(85 141)(86 142)(87 143)(88 144)(89 133)(90 134)(91 135)(92 136)(93 137)(94 138)(95 139)(96 140)(109 131)(110 132)(111 121)(112 122)(113 123)(114 124)(115 125)(116 126)(117 127)(118 128)(119 129)(120 130)
(1 48)(2 45)(3 46)(4 47)(5 28)(6 25)(7 26)(8 27)(9 35)(10 36)(11 33)(12 34)(13 20)(14 17)(15 18)(16 19)(21 29)(22 30)(23 31)(24 32)(37 42)(38 43)(39 44)(40 41)(49 72)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 107)(82 108)(83 97)(84 98)(85 127)(86 128)(87 129)(88 130)(89 131)(90 132)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 141)(118 142)(119 143)(120 144)
(1 69 85 25 65 89 43 61 93)(2 90 70 44 86 62 26 94 66)(3 63 91 27 71 95 41 67 87)(4 96 64 42 92 68 28 88 72)(5 130 49 47 126 53 37 122 57)(6 54 131 38 50 123 48 58 127)(7 124 55 45 132 59 39 128 51)(8 60 125 40 56 129 46 52 121)(9 118 77 13 114 81 23 110 73)(10 82 119 24 78 111 14 74 115)(11 112 83 15 120 75 21 116 79)(12 76 113 22 84 117 16 80 109)(17 100 139 36 108 143 32 104 135)(18 144 101 29 140 105 33 136 97)(19 106 133 34 102 137 30 98 141)(20 138 107 31 134 99 35 142 103)
(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)| (1,30)(2,31)(3,32)(4,29)(5,15)(6,16)(7,13)(8,14)(9,39)(10,40)(11,37)(12,38)(17,27)(18,28)(19,25)(20,26)(21,47)(22,48)(23,45)(24,46)(33,42)(34,43)(35,44)(36,41)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,73)(60,74)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,97)(69,98)(70,99)(71,100)(72,101)(85,141)(86,142)(87,143)(88,144)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(109,131)(110,132)(111,121)(112,122)(113,123)(114,124)(115,125)(116,126)(117,127)(118,128)(119,129)(120,130), (1,48)(2,45)(3,46)(4,47)(5,28)(6,25)(7,26)(8,27)(9,35)(10,36)(11,33)(12,34)(13,20)(14,17)(15,18)(16,19)(21,29)(22,30)(23,31)(24,32)(37,42)(38,43)(39,44)(40,41)(49,72)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,97)(84,98)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144), (1,69,85,25,65,89,43,61,93)(2,90,70,44,86,62,26,94,66)(3,63,91,27,71,95,41,67,87)(4,96,64,42,92,68,28,88,72)(5,130,49,47,126,53,37,122,57)(6,54,131,38,50,123,48,58,127)(7,124,55,45,132,59,39,128,51)(8,60,125,40,56,129,46,52,121)(9,118,77,13,114,81,23,110,73)(10,82,119,24,78,111,14,74,115)(11,112,83,15,120,75,21,116,79)(12,76,113,22,84,117,16,80,109)(17,100,139,36,108,143,32,104,135)(18,144,101,29,140,105,33,136,97)(19,106,133,34,102,137,30,98,141)(20,138,107,31,134,99,35,142,103), (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( (1,30)(2,31)(3,32)(4,29)(5,15)(6,16)(7,13)(8,14)(9,39)(10,40)(11,37)(12,38)(17,27)(18,28)(19,25)(20,26)(21,47)(22,48)(23,45)(24,46)(33,42)(34,43)(35,44)(36,41)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,81)(56,82)(57,83)(58,84)(59,73)(60,74)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,97)(69,98)(70,99)(71,100)(72,101)(85,141)(86,142)(87,143)(88,144)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(109,131)(110,132)(111,121)(112,122)(113,123)(114,124)(115,125)(116,126)(117,127)(118,128)(119,129)(120,130), (1,48)(2,45)(3,46)(4,47)(5,28)(6,25)(7,26)(8,27)(9,35)(10,36)(11,33)(12,34)(13,20)(14,17)(15,18)(16,19)(21,29)(22,30)(23,31)(24,32)(37,42)(38,43)(39,44)(40,41)(49,72)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,97)(84,98)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144), (1,69,85,25,65,89,43,61,93)(2,90,70,44,86,62,26,94,66)(3,63,91,27,71,95,41,67,87)(4,96,64,42,92,68,28,88,72)(5,130,49,47,126,53,37,122,57)(6,54,131,38,50,123,48,58,127)(7,124,55,45,132,59,39,128,51)(8,60,125,40,56,129,46,52,121)(9,118,77,13,114,81,23,110,73)(10,82,119,24,78,111,14,74,115)(11,112,83,15,120,75,21,116,79)(12,76,113,22,84,117,16,80,109)(17,100,139,36,108,143,32,104,135)(18,144,101,29,140,105,33,136,97)(19,106,133,34,102,137,30,98,141)(20,138,107,31,134,99,35,142,103), (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([[(1,30),(2,31),(3,32),(4,29),(5,15),(6,16),(7,13),(8,14),(9,39),(10,40),(11,37),(12,38),(17,27),(18,28),(19,25),(20,26),(21,47),(22,48),(23,45),(24,46),(33,42),(34,43),(35,44),(36,41),(49,75),(50,76),(51,77),(52,78),(53,79),(54,80),(55,81),(56,82),(57,83),(58,84),(59,73),(60,74),(61,102),(62,103),(63,104),(64,105),(65,106),(66,107),(67,108),(68,97),(69,98),(70,99),(71,100),(72,101),(85,141),(86,142),(87,143),(88,144),(89,133),(90,134),(91,135),(92,136),(93,137),(94,138),(95,139),(96,140),(109,131),(110,132),(111,121),(112,122),(113,123),(114,124),(115,125),(116,126),(117,127),(118,128),(119,129),(120,130)], [(1,48),(2,45),(3,46),(4,47),(5,28),(6,25),(7,26),(8,27),(9,35),(10,36),(11,33),(12,34),(13,20),(14,17),(15,18),(16,19),(21,29),(22,30),(23,31),(24,32),(37,42),(38,43),(39,44),(40,41),(49,72),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104),(79,105),(80,106),(81,107),(82,108),(83,97),(84,98),(85,127),(86,128),(87,129),(88,130),(89,131),(90,132),(91,121),(92,122),(93,123),(94,124),(95,125),(96,126),(109,133),(110,134),(111,135),(112,136),(113,137),(114,138),(115,139),(116,140),(117,141),(118,142),(119,143),(120,144)], [(1,69,85,25,65,89,43,61,93),(2,90,70,44,86,62,26,94,66),(3,63,91,27,71,95,41,67,87),(4,96,64,42,92,68,28,88,72),(5,130,49,47,126,53,37,122,57),(6,54,131,38,50,123,48,58,127),(7,124,55,45,132,59,39,128,51),(8,60,125,40,56,129,46,52,121),(9,118,77,13,114,81,23,110,73),(10,82,119,24,78,111,14,74,115),(11,112,83,15,120,75,21,116,79),(12,76,113,22,84,117,16,80,109),(17,100,139,36,108,143,32,104,135),(18,144,101,29,140,105,33,136,97),(19,106,133,34,102,137,30,98,141),(20,138,107,31,134,99,35,142,103)], [(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 ··· 2G 3A 3B 3C 4A ··· 4H 6A ··· 6G 6H ··· 6U 9A 9B 9C 12A ··· 12P 18A ··· 18U order 1 2 ··· 2 3 3 3 4 ··· 4 6 ··· 6 6 ··· 6 9 9 9 12 ··· 12 18 ··· 18 size 1 1 ··· 1 2 3 3 9 ··· 9 2 ··· 2 3 ··· 3 6 6 6 9 ··· 9 6 ··· 6

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 6 6 6 type + + + + - + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 Dic3 D6 C3×S3 C3×Dic3 S3×C6 C9⋊C6 C9⋊C12 C2×C9⋊C6 kernel C22×C9⋊C12 C2×C9⋊C12 C23×3- 1+2 C22×Dic9 C22×3- 1+2 C2×Dic9 C22×C18 C2×C18 C2×C62 C62 C62 C22×C6 C2×C6 C2×C6 C23 C22 C22 # reps 1 6 1 2 8 12 2 16 1 4 3 2 8 6 1 4 3

Matrix representation of C22×C9⋊C12 in GL10(𝔽37)

 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 36
,
 36 0 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 36 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 0 36 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 36 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 29 4 0 0 0 0 0 0 0 0 33 8 0 0 0 0 0 0 0 0 0 0 29 4 0 0 0 0 0 0 0 0 33 8 0 0 0 0 0 0 0 0 0 0 30 7 0 0 0 0 0 0 0 0 14 7 0 0 0 0 0 0 0 0 0 0 0 0 14 7 0 0 0 0 0 0 0 0 30 23 0 0 0 0 0 0 30 7 0 0 0 0 0 0 0 0 14 7 0 0

G:=sub<GL(10,GF(37))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36],[36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,36,0,0],[29,33,0,0,0,0,0,0,0,0,4,8,0,0,0,0,0,0,0,0,0,0,29,33,0,0,0,0,0,0,0,0,4,8,0,0,0,0,0,0,0,0,0,0,30,14,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,0,0,0,0,0,0,30,14,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,14,30,0,0,0,0,0,0,0,0,7,23,0,0] >;

C22×C9⋊C12 in GAP, Magma, Sage, TeX

C_2^2\times C_9\rtimes C_{12}
% in TeX

G:=Group("C2^2xC9:C12");
// GroupNames label

G:=SmallGroup(432,378);
// by ID

G=gap.SmallGroup(432,378);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,10085,1034,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^9=d^12=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations

׿
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