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## G = C2×C36.C6order 432 = 24·33

### Direct product of C2 and C36.C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C2×C36.C6
 Chief series C1 — C3 — C9 — C18 — C2×3- 1+2 — C9⋊C12 — C2×C9⋊C12 — C2×C36.C6
 Lower central C9 — C18 — C2×C36.C6
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C36.C6
G = < a,b,c | a2=b36=1, c6=b18, ab=ba, ac=ca, cbc-1=b11 >

Subgroups: 366 in 122 conjugacy classes, 62 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C9, C9, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×Q8, C18, C18, C18, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, 3- 1+2, Dic9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, C2×Dic6, C6×Q8, C2×3- 1+2, C2×3- 1+2, Dic18, C2×Dic9, C2×C36, C2×C36, C3×Dic6, C6×Dic3, C6×C12, C9⋊C12, C4×3- 1+2, C22×3- 1+2, C2×Dic18, C6×Dic6, C36.C6, C2×C9⋊C12, C2×C4×3- 1+2, C2×C36.C6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C3×S3, Dic6, C3×Q8, C22×S3, C22×C6, S3×C6, C2×Dic6, C6×Q8, C9⋊C6, C3×Dic6, S3×C2×C6, C2×C9⋊C6, C6×Dic6, C36.C6, C22×C9⋊C6, C2×C36.C6

Smallest permutation representation of C2×C36.C6
On 144 points
Generators in S144
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)(97 109)(98 110)(99 111)(100 112)(101 113)(102 114)(103 115)(104 116)(105 117)(106 118)(107 119)(108 120)
(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)
(1 117 19 135)(2 140 8 134 14 128 20 122 26 116 32 110)(3 127 33 133 27 139 21 109 15 115 9 121)(4 114 22 132)(5 137 11 131 17 125 23 119 29 113 35 143)(6 124 36 130 30 136 24 142 18 112 12 118)(7 111 25 129)(10 144 28 126)(13 141 31 123)(16 138 34 120)(37 101 43 95 49 89 55 83 61 77 67 107)(38 88 68 94 62 100 56 106 50 76 44 82)(39 75 57 93)(40 98 46 92 52 86 58 80 64 74 70 104)(41 85 71 91 65 97 59 103 53 73 47 79)(42 108 60 90)(45 105 63 87)(48 102 66 84)(51 99 69 81)(54 96 72 78)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D ··· 6I 9A 9B 9C 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12P 18A ··· 18I 36A ··· 36L order 1 2 2 2 3 3 3 4 4 4 4 4 4 6 6 6 6 ··· 6 9 9 9 12 12 12 12 12 12 12 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 2 3 3 2 2 18 18 18 18 2 2 2 3 ··· 3 6 6 6 2 2 2 2 6 6 6 6 18 ··· 18 6 ··· 6 6 ··· 6

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 6 6 6 6 type + + + + + - + + - + + + - image C1 C2 C2 C2 C3 C6 C6 C6 S3 Q8 D6 D6 C3×S3 C3×Q8 Dic6 S3×C6 S3×C6 C3×Dic6 C9⋊C6 C2×C9⋊C6 C2×C9⋊C6 C36.C6 kernel C2×C36.C6 C36.C6 C2×C9⋊C12 C2×C4×3- 1+2 C2×Dic18 Dic18 C2×Dic9 C2×C36 C6×C12 C2×3- 1+2 C3×C12 C62 C2×C12 C18 C3×C6 C12 C2×C6 C6 C2×C4 C4 C22 C2 # reps 1 4 2 1 2 8 4 2 1 2 2 1 2 4 4 4 2 8 1 2 1 4

Matrix representation of C2×C36.C6 in GL8(𝔽37)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36
,
 36 3 0 0 0 0 0 0 24 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 36 0 0 0 0 36 0 0 0 0 0 0 1 1 0 0 0 0
,
 25 17 0 0 0 0 0 0 4 12 0 0 0 0 0 0 0 0 29 4 0 0 0 0 0 0 12 8 0 0 0 0 0 0 0 0 0 0 29 4 0 0 0 0 0 0 12 8 0 0 0 0 12 8 0 0 0 0 0 0 33 25 0 0

G:=sub<GL(8,GF(37))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36],[36,24,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,1,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,1,0,0,0],[25,4,0,0,0,0,0,0,17,12,0,0,0,0,0,0,0,0,29,12,0,0,0,0,0,0,4,8,0,0,0,0,0,0,0,0,0,0,12,33,0,0,0,0,0,0,8,25,0,0,0,0,29,12,0,0,0,0,0,0,4,8,0,0] >;

C2×C36.C6 in GAP, Magma, Sage, TeX

C_2\times C_{36}.C_6
% in TeX

G:=Group("C2xC36.C6");
// GroupNames label

G:=SmallGroup(432,352);
// by ID

G=gap.SmallGroup(432,352);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,590,142,10085,1034,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^2=b^36=1,c^6=b^18,a*b=b*a,a*c=c*a,c*b*c^-1=b^11>;
// generators/relations

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