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G = C2×C36.C6order 432 = 24·33

Direct product of C2 and C36.C6

direct product, metabelian, supersoluble, monomial

Aliases: C2×C36.C6, Dic185C6, C62.37D6, C18⋊(C3×Q8), C91(C6×Q8), (C2×C36).3C6, (C2×Dic18)⋊C3, C12.78(S3×C6), C36.11(C2×C6), (C6×C12).13S3, (C3×C12).51D6, C3.3(C6×Dic6), C32.(C2×Dic6), C9⋊C12.1C22, C18.1(C22×C6), (C2×Dic9).3C6, Dic9.1(C2×C6), C6.11(C3×Dic6), (C3×C6).11Dic6, (C2×3- 1+2)⋊Q8, 3- 1+21(C2×Q8), (C2×3- 1+2).1C23, (C4×3- 1+2).11C22, (C22×3- 1+2).7C22, C6.27(S3×C2×C6), C4.11(C2×C9⋊C6), (C2×C9⋊C12).3C2, (C2×C4).4(C9⋊C6), (C2×C18).7(C2×C6), (C2×C6).57(S3×C6), C2.3(C22×C9⋊C6), C22.8(C2×C9⋊C6), (C2×C12).18(C3×S3), (C3×C6).23(C22×S3), (C2×C4×3- 1+2).3C2, SmallGroup(432,352)

Series: Derived Chief Lower central Upper central

C1C18 — C2×C36.C6
C1C3C9C18C2×3- 1+2C9⋊C12C2×C9⋊C12 — C2×C36.C6
C9C18 — C2×C36.C6
C1C22C2×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 2A2B2C3A3B3C4A4B4C4D4E4F6A6B6C6D···6I9A9B9C12A12B12C12D12E12F12G12H12I···12P18A···18I36A···36L
order12223334444446666···6999121212121212121212···1218···1836···36
size111123322181818182223···36662222666618···186···66···6

62 irreducible representations

dim1111111122222222226666
type+++++-++-+++-
imageC1C2C2C2C3C6C6C6S3Q8D6D6C3×S3C3×Q8Dic6S3×C6S3×C6C3×Dic6C9⋊C6C2×C9⋊C6C2×C9⋊C6C36.C6
kernelC2×C36.C6C36.C6C2×C9⋊C12C2×C4×3- 1+2C2×Dic18Dic18C2×Dic9C2×C36C6×C12C2×3- 1+2C3×C12C62C2×C12C18C3×C6C12C2×C6C6C2×C4C4C22C2
# reps1421284212212444281214

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

10000000
01000000
003600000
000360000
000036000
000003600
000000360
000000036
,
363000000
241000000
00001100
000036000
00000011
000000360
000360000
00110000
,
2517000000
412000000
002940000
001280000
000000294
000000128
000012800
0000332500

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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