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G = D8×C18order 288 = 25·32

Direct product of C18 and D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: D8×C18, C36.42D4, C7212C22, C36.44C23, C3.(C6×D8), (C6×D8).C3, C82(C2×C18), (C2×C8)⋊3C18, C4.6(D4×C9), (C2×C72)⋊11C2, D41(C2×C18), (C2×D4)⋊4C18, (C3×D8).6C6, C6.74(C6×D4), C6.18(C3×D8), (D4×C18)⋊13C2, C24.24(C2×C6), (C2×C24).14C6, (C6×D4).14C6, (C2×C18).52D4, C2.11(D4×C18), C18.74(C2×D4), C12.42(C3×D4), (D4×C9)⋊10C22, C4.1(C22×C18), C22.14(D4×C9), C12.44(C22×C6), (C2×C36).127C22, (C2×C6).61(C3×D4), (C2×C4).26(C2×C18), (C3×D4).11(C2×C6), (C2×C12).144(C2×C6), SmallGroup(288,182)

Series: Derived Chief Lower central Upper central

C1C4 — D8×C18
C1C2C6C12C36D4×C9C9×D8 — D8×C18
C1C2C4 — D8×C18
C1C2×C18C2×C36 — D8×C18

Generators and relations for D8×C18
 G = < a,b,c | a18=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 210 in 114 conjugacy classes, 66 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C6, C6 [×2], C6 [×4], C8 [×2], C2×C4, D4 [×4], D4 [×2], C23 [×2], C9, C12 [×2], C2×C6, C2×C6 [×8], C2×C8, D8 [×4], C2×D4 [×2], C18, C18 [×2], C18 [×4], C24 [×2], C2×C12, C3×D4 [×4], C3×D4 [×2], C22×C6 [×2], C2×D8, C36 [×2], C2×C18, C2×C18 [×8], C2×C24, C3×D8 [×4], C6×D4 [×2], C72 [×2], C2×C36, D4×C9 [×4], D4×C9 [×2], C22×C18 [×2], C6×D8, C2×C72, C9×D8 [×4], D4×C18 [×2], D8×C18
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C9, C2×C6 [×7], D8 [×2], C2×D4, C18 [×7], C3×D4 [×2], C22×C6, C2×D8, C2×C18 [×7], C3×D8 [×2], C6×D4, D4×C9 [×2], C22×C18, C6×D8, C9×D8 [×2], D4×C18, D8×C18

Smallest permutation representation of D8×C18
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)
(1 121 108 50 69 23 78 132)(2 122 91 51 70 24 79 133)(3 123 92 52 71 25 80 134)(4 124 93 53 72 26 81 135)(5 125 94 54 55 27 82 136)(6 126 95 37 56 28 83 137)(7 109 96 38 57 29 84 138)(8 110 97 39 58 30 85 139)(9 111 98 40 59 31 86 140)(10 112 99 41 60 32 87 141)(11 113 100 42 61 33 88 142)(12 114 101 43 62 34 89 143)(13 115 102 44 63 35 90 144)(14 116 103 45 64 36 73 127)(15 117 104 46 65 19 74 128)(16 118 105 47 66 20 75 129)(17 119 106 48 67 21 76 130)(18 120 107 49 68 22 77 131)
(1 132)(2 133)(3 134)(4 135)(5 136)(6 137)(7 138)(8 139)(9 140)(10 141)(11 142)(12 143)(13 144)(14 127)(15 128)(16 129)(17 130)(18 131)(19 104)(20 105)(21 106)(22 107)(23 108)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 56)(38 57)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 55)(73 116)(74 117)(75 118)(76 119)(77 120)(78 121)(79 122)(80 123)(81 124)(82 125)(83 126)(84 109)(85 110)(86 111)(87 112)(88 113)(89 114)(90 115)

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

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

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

126 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B6A···6F6G···6N8A8B8C8D9A···9F12A12B12C12D18A···18R18S···18AP24A···24H36A···36L72A···72X
order1222222233446···66···688889···91212121218···1818···1824···2436···3672···72
size1111444411221···14···422221···122221···14···42···22···22···2

126 irreducible representations

dim111111111111222222222
type+++++++
imageC1C2C2C2C3C6C6C6C9C18C18C18D4D4D8C3×D4C3×D4C3×D8D4×C9D4×C9C9×D8
kernelD8×C18C2×C72C9×D8D4×C18C6×D8C2×C24C3×D8C6×D4C2×D8C2×C8D8C2×D4C36C2×C18C18C12C2×C6C6C4C22C2
# reps114222846624121142286624

Matrix representation of D8×C18 in GL4(𝔽73) generated by

36000
03600
00570
00057
,
42400
153100
005716
005757
,
42400
523100
005716
001616
G:=sub<GL(4,GF(73))| [36,0,0,0,0,36,0,0,0,0,57,0,0,0,0,57],[42,15,0,0,4,31,0,0,0,0,57,57,0,0,16,57],[42,52,0,0,4,31,0,0,0,0,57,16,0,0,16,16] >;

D8×C18 in GAP, Magma, Sage, TeX

D_8\times C_{18}
% in TeX

G:=Group("D8xC18");
// GroupNames label

G:=SmallGroup(288,182);
// by ID

G=gap.SmallGroup(288,182);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,365,192,5884,2951,242]);
// Polycyclic

G:=Group<a,b,c|a^18=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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