Copied to
clipboard

G = D18⋊C8order 288 = 25·32

The semidirect product of D18 and C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D18⋊C8, C36.35D4, C4.19D36, C12.54D12, C18.3M4(2), (C2×C8)⋊1D9, (C2×C72)⋊1C2, C3.(D6⋊C8), C2.5(C8×D9), C6.10(S3×C8), C91(C22⋊C8), C18.5(C2×C8), (C2×C24).3S3, (C2×C4).93D18, C6.6(C8⋊S3), C2.1(D18⋊C4), C6.13(D6⋊C4), (C2×C12).407D6, C2.3(C8⋊D9), C4.27(C9⋊D4), (C2×Dic9).4C4, (C22×D9).2C4, C22.11(C4×D9), C18.6(C22⋊C4), C12.122(C3⋊D4), (C2×C36).105C22, (C2×C9⋊C8)⋊9C2, (C2×C4×D9).6C2, (C2×C6).37(C4×S3), (C2×C18).12(C2×C4), SmallGroup(288,27)

Series: Derived Chief Lower central Upper central

C1C18 — D18⋊C8
C1C3C9C18C36C2×C36C2×C4×D9 — D18⋊C8
C9C18 — D18⋊C8
C1C2×C4C2×C8

Generators and relations for D18⋊C8
 G = < a,b,c | a18=b2=c8=1, bab=a-1, ac=ca, cbc-1=a9b >

Subgroups: 332 in 75 conjugacy classes, 34 normal (32 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], S3 [×2], C6 [×3], C8 [×2], C2×C4, C2×C4 [×3], C23, C9, Dic3, C12 [×2], D6 [×4], C2×C6, C2×C8, C2×C8, C22×C4, D9 [×2], C18 [×3], C3⋊C8, C24, C4×S3 [×2], C2×Dic3, C2×C12, C22×S3, C22⋊C8, Dic9, C36 [×2], D18 [×2], D18 [×2], C2×C18, C2×C3⋊C8, C2×C24, S3×C2×C4, C9⋊C8, C72, C4×D9 [×2], C2×Dic9, C2×C36, C22×D9, D6⋊C8, C2×C9⋊C8, C2×C72, C2×C4×D9, D18⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4 [×2], D6, C22⋊C4, C2×C8, M4(2), D9, C4×S3, D12, C3⋊D4, C22⋊C8, D18, S3×C8, C8⋊S3, D6⋊C4, C4×D9, D36, C9⋊D4, D6⋊C8, C8×D9, C8⋊D9, D18⋊C4, D18⋊C8

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

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F6A6B6C8A8B8C8D8E8F8G8H9A9B9C12A12B12C12D18A···18I24A···24H36A···36L72A···72X
order1222223444444666888888889991212121218···1824···2436···3672···72
size1111181821111181822222221818181822222222···22···22···22···2

84 irreducible representations

dim11111112222222222222222
type+++++++++++
imageC1C2C2C2C4C4C8S3D4D6M4(2)D9D12C3⋊D4C4×S3D18S3×C8C8⋊S3D36C9⋊D4C4×D9C8×D9C8⋊D9
kernelD18⋊C8C2×C9⋊C8C2×C72C2×C4×D9C2×Dic9C22×D9D18C2×C24C36C2×C12C18C2×C8C12C12C2×C6C2×C4C6C6C4C4C22C2C2
# reps1111228121232223446661212

Matrix representation of D18⋊C8 in GL3(𝔽73) generated by

100
0328
04531
,
100
02842
07045
,
1000
06516
0578
G:=sub<GL(3,GF(73))| [1,0,0,0,3,45,0,28,31],[1,0,0,0,28,70,0,42,45],[10,0,0,0,65,57,0,16,8] >;

D18⋊C8 in GAP, Magma, Sage, TeX

D_{18}\rtimes C_8
% in TeX

G:=Group("D18:C8");
// GroupNames label

G:=SmallGroup(288,27);
// by ID

G=gap.SmallGroup(288,27);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^18=b^2=c^8=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^9*b>;
// generators/relations

׿
×
𝔽