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G = C2×D18⋊C4order 288 = 25·32

Direct product of C2 and D18⋊C4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D18⋊C4, C23.37D18, C22.16D36, (C2×C4)⋊8D18, D186(C2×C4), C2.3(C2×D36), (C22×C4)⋊3D9, (C22×C36)⋊1C2, (C2×C18).21D4, C6.45(C2×D12), (C2×C6).30D12, C18.41(C2×D4), (C22×D9)⋊3C4, C181(C22⋊C4), (C2×C36)⋊10C22, C6.19(D6⋊C4), (C2×C12).343D6, (C23×D9).2C2, C22.17(C4×D9), (C2×C18).45C23, (C22×C12).11S3, C18.19(C22×C4), (C22×Dic9)⋊3C2, (C2×Dic9)⋊6C22, (C22×C6).139D6, C22.20(C9⋊D4), C22.23(C22×D9), (C22×C18).37C22, (C22×D9).23C22, C3.(C2×D6⋊C4), C92(C2×C22⋊C4), C6.58(S3×C2×C4), C2.19(C2×C4×D9), C2.2(C2×C9⋊D4), (C2×C6).44(C4×S3), C6.88(C2×C3⋊D4), (C2×C18).18(C2×C4), (C2×C6).75(C3⋊D4), (C2×C6).202(C22×S3), SmallGroup(288,137)

Series: Derived Chief Lower central Upper central

C1C18 — C2×D18⋊C4
C1C3C9C18C2×C18C22×D9C23×D9 — C2×D18⋊C4
C9C18 — C2×D18⋊C4
C1C23C22×C4

Generators and relations for C2×D18⋊C4
 G = < a,b,c,d | a2=b18=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b9c >

Subgroups: 952 in 198 conjugacy classes, 76 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C22, C22 [×6], C22 [×16], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], C9, Dic3 [×2], C12 [×2], D6 [×16], C2×C6, C2×C6 [×6], C22⋊C4 [×4], C22×C4, C22×C4, C24, D9 [×4], C18 [×3], C18 [×4], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×2], C22×S3 [×10], C22×C6, C2×C22⋊C4, Dic9 [×2], C36 [×2], D18 [×4], D18 [×12], C2×C18, C2×C18 [×6], D6⋊C4 [×4], C22×Dic3, C22×C12, S3×C23, C2×Dic9 [×2], C2×Dic9 [×2], C2×C36 [×2], C2×C36 [×2], C22×D9 [×6], C22×D9 [×4], C22×C18, C2×D6⋊C4, D18⋊C4 [×4], C22×Dic9, C22×C36, C23×D9, C2×D18⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], D9, C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, D18 [×3], D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C4×D9 [×2], D36 [×2], C9⋊D4 [×2], C22×D9, C2×D6⋊C4, D18⋊C4 [×4], C2×C4×D9, C2×D36, C2×C9⋊D4, C2×D18⋊C4

Smallest permutation representation of C2×D18⋊C4
On 144 points
Generators in S144
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 105)(8 106)(9 107)(10 108)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 97)(18 98)(19 86)(20 87)(21 88)(22 89)(23 90)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 85)(37 136)(38 137)(39 138)(40 139)(41 140)(42 141)(43 142)(44 143)(45 144)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 133)(53 134)(54 135)(55 125)(56 126)(57 109)(58 110)(59 111)(60 112)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 121)(70 122)(71 123)(72 124)
(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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 90)(10 89)(11 88)(12 87)(13 86)(14 85)(15 84)(16 83)(17 82)(18 81)(19 93)(20 92)(21 91)(22 108)(23 107)(24 106)(25 105)(26 104)(27 103)(28 102)(29 101)(30 100)(31 99)(32 98)(33 97)(34 96)(35 95)(36 94)(37 115)(38 114)(39 113)(40 112)(41 111)(42 110)(43 109)(44 126)(45 125)(46 124)(47 123)(48 122)(49 121)(50 120)(51 119)(52 118)(53 117)(54 116)(55 144)(56 143)(57 142)(58 141)(59 140)(60 139)(61 138)(62 137)(63 136)(64 135)(65 134)(66 133)(67 132)(68 131)(69 130)(70 129)(71 128)(72 127)
(1 135 32 126)(2 136 33 109)(3 137 34 110)(4 138 35 111)(5 139 36 112)(6 140 19 113)(7 141 20 114)(8 142 21 115)(9 143 22 116)(10 144 23 117)(11 127 24 118)(12 128 25 119)(13 129 26 120)(14 130 27 121)(15 131 28 122)(16 132 29 123)(17 133 30 124)(18 134 31 125)(37 82 57 100)(38 83 58 101)(39 84 59 102)(40 85 60 103)(41 86 61 104)(42 87 62 105)(43 88 63 106)(44 89 64 107)(45 90 65 108)(46 73 66 91)(47 74 67 92)(48 75 68 93)(49 76 69 94)(50 77 70 95)(51 78 71 96)(52 79 72 97)(53 80 55 98)(54 81 56 99)

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

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

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

84 conjugacy classes

class 1 2A···2G2H2I2J2K 3 4A4B4C4D4E4F4G4H6A···6G9A9B9C12A···12H18A···18U36A···36X
order12···222223444444446···699912···1218···1836···36
size11···11818181822222181818182···22222···22···22···2

84 irreducible representations

dim1111112222222222222
type++++++++++++++
imageC1C2C2C2C2C4S3D4D6D6D9C4×S3D12C3⋊D4D18D18C4×D9D36C9⋊D4
kernelC2×D18⋊C4D18⋊C4C22×Dic9C22×C36C23×D9C22×D9C22×C12C2×C18C2×C12C22×C6C22×C4C2×C6C2×C6C2×C6C2×C4C23C22C22C22
# reps1411181421344463121212

Matrix representation of C2×D18⋊C4 in GL6(𝔽37)

3600000
0360000
001000
000100
0000360
0000036
,
3600000
0360000
0036000
0003600
00002031
0000626
,
100000
0360000
001000
0003600
0000176
00002620
,
0360000
3600000
0003600
001000
000010
000001

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,20,6,0,0,0,0,31,26],[1,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,17,26,0,0,0,0,6,20],[0,36,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×D18⋊C4 in GAP, Magma, Sage, TeX

C_2\times D_{18}\rtimes C_4
% in TeX

G:=Group("C2xD18:C4");
// GroupNames label

G:=SmallGroup(288,137);
// by ID

G=gap.SmallGroup(288,137);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,58,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^18=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^9*c>;
// generators/relations

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