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G = C18×C3⋊C8order 432 = 24·33

Direct product of C18 and C3⋊C8

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C18×C3⋊C8, C6⋊C72, C12.3C36, C36.78D6, C62.17C12, C36.11Dic3, C32(C2×C72), (C3×C18)⋊1C8, (C2×C6).5C36, (C3×C36).2C4, (C6×C18).1C4, C6.5(C2×C36), (C6×C36).1C2, (C3×C6).8C24, C4.14(S3×C18), (C2×C12).6C18, (C6×C12).39C6, (C2×C36).21S3, C4.3(C9×Dic3), C12.118(S3×C6), C12.14(C2×C18), (C3×C12).20C12, C32.3(C2×C24), C2.1(Dic3×C18), (C2×C18).9Dic3, C6.29(C6×Dic3), (C3×C36).52C22, C12.23(C3×Dic3), C18.17(C2×Dic3), C22.2(C9×Dic3), (C6×C3⋊C8).C3, (C3×C9)⋊7(C2×C8), C6.9(C3×C3⋊C8), C3.4(C6×C3⋊C8), (C3×C3⋊C8).10C6, (C2×C4).5(S3×C9), (C2×C12).48(C3×S3), (C3×C12).89(C2×C6), (C3×C18).27(C2×C4), (C3×C6).51(C2×C12), (C2×C6).26(C3×Dic3), SmallGroup(432,126)

Series: Derived Chief Lower central Upper central

C1C3 — C18×C3⋊C8
C1C3C6C3×C6C3×C12C3×C36C9×C3⋊C8 — C18×C3⋊C8
C3 — C18×C3⋊C8
C1C2×C36

Generators and relations for C18×C3⋊C8
 G = < a,b,c | a18=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 116 in 82 conjugacy classes, 57 normal (39 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C22, C6 [×2], C6 [×4], C6 [×3], C8 [×2], C2×C4, C9, C9, C32, C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C2×C8, C18, C18 [×2], C18 [×3], C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×2], C2×C12 [×2], C2×C12, C3×C9, C36 [×2], C36 [×2], C2×C18, C2×C18, C3×C12 [×2], C62, C2×C3⋊C8, C2×C24, C3×C18, C3×C18 [×2], C72 [×2], C2×C36, C2×C36, C3×C3⋊C8 [×2], C6×C12, C3×C36 [×2], C6×C18, C2×C72, C6×C3⋊C8, C9×C3⋊C8 [×2], C6×C36, C18×C3⋊C8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, C9, Dic3 [×2], C12 [×2], D6, C2×C6, C2×C8, C18 [×3], C3×S3, C3⋊C8 [×2], C24 [×2], C2×Dic3, C2×C12, C36 [×2], C2×C18, C3×Dic3 [×2], S3×C6, C2×C3⋊C8, C2×C24, S3×C9, C72 [×2], C2×C36, C3×C3⋊C8 [×2], C6×Dic3, C9×Dic3 [×2], S3×C18, C2×C72, C6×C3⋊C8, C9×C3⋊C8 [×2], Dic3×C18, C18×C3⋊C8

Smallest permutation representation of C18×C3⋊C8
On 144 points
Generators in S144
(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 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 67 61)(56 68 62)(57 69 63)(58 70 64)(59 71 65)(60 72 66)(73 85 79)(74 86 80)(75 87 81)(76 88 82)(77 89 83)(78 90 84)(91 97 103)(92 98 104)(93 99 105)(94 100 106)(95 101 107)(96 102 108)(109 115 121)(110 116 122)(111 117 123)(112 118 124)(113 119 125)(114 120 126)(127 133 139)(128 134 140)(129 135 141)(130 136 142)(131 137 143)(132 138 144)
(1 72 144 46 96 23 109 87)(2 55 127 47 97 24 110 88)(3 56 128 48 98 25 111 89)(4 57 129 49 99 26 112 90)(5 58 130 50 100 27 113 73)(6 59 131 51 101 28 114 74)(7 60 132 52 102 29 115 75)(8 61 133 53 103 30 116 76)(9 62 134 54 104 31 117 77)(10 63 135 37 105 32 118 78)(11 64 136 38 106 33 119 79)(12 65 137 39 107 34 120 80)(13 66 138 40 108 35 121 81)(14 67 139 41 91 36 122 82)(15 68 140 42 92 19 123 83)(16 69 141 43 93 20 124 84)(17 70 142 44 94 21 125 85)(18 71 143 45 95 22 126 86)

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

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

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

216 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D6A···6F6G···6O8A···8H9A···9F9G···9L12A···12H12I···12T18A···18R18S···18AJ24A···24P36A···36X36Y···36AV72A···72AV
order12223333344446···66···68···89···99···912···1212···1218···1818···1824···2436···3636···3672···72
size11111122211111···12···23···31···12···21···12···21···12···23···31···12···23···3

216 irreducible representations

dim111111111111111111222222222222222
type++++-+-
imageC1C2C2C3C4C4C6C6C8C9C12C12C18C18C24C36C36C72S3Dic3D6Dic3C3×S3C3⋊C8C3×Dic3S3×C6C3×Dic3S3×C9C3×C3⋊C8C9×Dic3S3×C18C9×Dic3C9×C3⋊C8
kernelC18×C3⋊C8C9×C3⋊C8C6×C36C6×C3⋊C8C3×C36C6×C18C3×C3⋊C8C6×C12C3×C18C2×C3⋊C8C3×C12C62C3⋊C8C2×C12C3×C6C12C2×C6C6C2×C36C36C36C2×C18C2×C12C18C12C12C2×C6C2×C4C6C4C4C22C2
# reps121222428644126161212481111242226866624

Matrix representation of C18×C3⋊C8 in GL4(𝔽73) generated by

16000
01800
00160
00016
,
1000
0100
00640
0008
,
10000
07200
0001
00720
G:=sub<GL(4,GF(73))| [16,0,0,0,0,18,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,8],[10,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0] >;

C18×C3⋊C8 in GAP, Magma, Sage, TeX

C_{18}\times C_3\rtimes C_8
% in TeX

G:=Group("C18xC3:C8");
// GroupNames label

G:=SmallGroup(432,126);
// by ID

G=gap.SmallGroup(432,126);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,84,142,192,14118]);
// Polycyclic

G:=Group<a,b,c|a^18=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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