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G = D4.10D18order 288 = 25·32

The non-split extension by D4 of D18 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.10D18, Q8.16D18, C922- 1+4, C36.27C23, C18.13C24, D18.8C23, D36.14C22, Dic9.8C23, Dic18.14C22, C4○D46D9, (Q8×D9)⋊5C2, C3.(Q8○D12), C9⋊D4.C22, D42D95C2, (C3×D4).40D6, (C2×C4).23D18, (C3×Q8).64D6, D365C29C2, (C2×C12).106D6, C6.50(S3×C23), (C2×C18).5C23, (C4×D9).6C22, C2.14(C23×D9), C4.34(C22×D9), (C2×Dic18)⋊14C2, (C2×C36).52C22, (D4×C9).10C22, (Q8×C9).11C22, C22.4(C22×D9), C12.188(C22×S3), (C2×Dic9).19C22, (C9×C4○D4)⋊5C2, (C3×C4○D4).17S3, (C2×C6).11(C22×S3), SmallGroup(288,364)

Series: Derived Chief Lower central Upper central

C1C18 — D4.10D18
C1C3C9C18D18C4×D9Q8×D9 — D4.10D18
C9C18 — D4.10D18
C1C2C4○D4

Generators and relations for D4.10D18
 G = < a,b,c,d | a4=b2=1, c18=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c17 >

Subgroups: 784 in 219 conjugacy classes, 102 normal (17 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], S3 [×2], C6, C6 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×7], Q8, Q8 [×9], C9, Dic3 [×6], C12, C12 [×3], D6 [×2], C2×C6 [×3], C2×Q8 [×5], C4○D4, C4○D4 [×9], D9 [×2], C18, C18 [×3], Dic6 [×9], C4×S3 [×6], D12, C2×Dic3 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, 2- 1+4, Dic9 [×6], C36, C36 [×3], D18 [×2], C2×C18 [×3], C2×Dic6 [×3], C4○D12 [×3], D42S3 [×6], S3×Q8 [×2], C3×C4○D4, Dic18 [×9], C4×D9 [×6], D36, C2×Dic9 [×6], C9⋊D4 [×6], C2×C36 [×3], D4×C9 [×3], Q8×C9, Q8○D12, C2×Dic18 [×3], D365C2 [×3], D42D9 [×6], Q8×D9 [×2], C9×C4○D4, D4.10D18
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, D9, C22×S3 [×7], 2- 1+4, D18 [×7], S3×C23, C22×D9 [×7], Q8○D12, C23×D9, D4.10D18

Smallest permutation representation of D4.10D18
On 144 points
Generators in S144
(1 72 19 54)(2 55 20 37)(3 38 21 56)(4 57 22 39)(5 40 23 58)(6 59 24 41)(7 42 25 60)(8 61 26 43)(9 44 27 62)(10 63 28 45)(11 46 29 64)(12 65 30 47)(13 48 31 66)(14 67 32 49)(15 50 33 68)(16 69 34 51)(17 52 35 70)(18 71 36 53)(73 112 91 130)(74 131 92 113)(75 114 93 132)(76 133 94 115)(77 116 95 134)(78 135 96 117)(79 118 97 136)(80 137 98 119)(81 120 99 138)(82 139 100 121)(83 122 101 140)(84 141 102 123)(85 124 103 142)(86 143 104 125)(87 126 105 144)(88 109 106 127)(89 128 107 110)(90 111 108 129)
(1 117)(2 136)(3 119)(4 138)(5 121)(6 140)(7 123)(8 142)(9 125)(10 144)(11 127)(12 110)(13 129)(14 112)(15 131)(16 114)(17 133)(18 116)(19 135)(20 118)(21 137)(22 120)(23 139)(24 122)(25 141)(26 124)(27 143)(28 126)(29 109)(30 128)(31 111)(32 130)(33 113)(34 132)(35 115)(36 134)(37 79)(38 98)(39 81)(40 100)(41 83)(42 102)(43 85)(44 104)(45 87)(46 106)(47 89)(48 108)(49 91)(50 74)(51 93)(52 76)(53 95)(54 78)(55 97)(56 80)(57 99)(58 82)(59 101)(60 84)(61 103)(62 86)(63 105)(64 88)(65 107)(66 90)(67 73)(68 92)(69 75)(70 94)(71 77)(72 96)
(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 18 19 36)(2 35 20 17)(3 16 21 34)(4 33 22 15)(5 14 23 32)(6 31 24 13)(7 12 25 30)(8 29 26 11)(9 10 27 28)(37 70 55 52)(38 51 56 69)(39 68 57 50)(40 49 58 67)(41 66 59 48)(42 47 60 65)(43 64 61 46)(44 45 62 63)(53 54 71 72)(73 100 91 82)(74 81 92 99)(75 98 93 80)(76 79 94 97)(77 96 95 78)(83 90 101 108)(84 107 102 89)(85 88 103 106)(86 105 104 87)(109 124 127 142)(110 141 128 123)(111 122 129 140)(112 139 130 121)(113 120 131 138)(114 137 132 119)(115 118 133 136)(116 135 134 117)(125 144 143 126)

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

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

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

57 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E···4J6A6B6C6D9A9B9C12A12B12C12D12E18A18B18C18D···18L36A···36F36G···36O
order1222222344444···46666999121212121218181818···1836···3636···36
size1122218182222218···182444222224442224···42···24···4

57 irreducible representations

dim11111122222222444
type++++++++++++++---
imageC1C2C2C2C2C2S3D6D6D6D9D18D18D182- 1+4Q8○D12D4.10D18
kernelD4.10D18C2×Dic18D365C2D42D9Q8×D9C9×C4○D4C3×C4○D4C2×C12C3×D4C3×Q8C4○D4C2×C4D4Q8C9C3C1
# reps13362113313993126

Matrix representation of D4.10D18 in GL4(𝔽37) generated by

0010
0001
36000
03600
,
362149
3512823
149135
2823236
,
192416
35212120
4161835
2120216
,
35182133
1621716
2133219
17162135
G:=sub<GL(4,GF(37))| [0,0,36,0,0,0,0,36,1,0,0,0,0,1,0,0],[36,35,14,28,2,1,9,23,14,28,1,2,9,23,35,36],[19,35,4,21,2,21,16,20,4,21,18,2,16,20,35,16],[35,16,21,17,18,2,33,16,21,17,2,21,33,16,19,35] >;

D4.10D18 in GAP, Magma, Sage, TeX

D_4._{10}D_{18}
% in TeX

G:=Group("D4.10D18");
// GroupNames label

G:=SmallGroup(288,364);
// by ID

G=gap.SmallGroup(288,364);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^18=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^17>;
// generators/relations

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