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G = D4.9D18order 288 = 25·32

4th non-split extension by D4 of D18 acting via D18/C18=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.9D18, C36.57D4, Q8.14D18, C36.19C23, D36.12C22, Dic18.12C22, D4⋊D97C2, C4○D44D9, C95(C4○D8), D4.D97C2, C9⋊Q167C2, Q82D97C2, (C3×D4).34D6, C18.61(C2×D4), (C2×C12).70D6, (C2×C18).10D4, (C2×C4).59D18, C9⋊C8.10C22, (C3×Q8).58D6, D365C24C2, C4.32(C9⋊D4), C3.(Q8.13D6), (D4×C9).9C22, C4.19(C22×D9), (Q8×C9).9C22, C12.58(C22×S3), (C2×C36).47C22, C22.1(C9⋊D4), C12.128(C3⋊D4), (C2×C9⋊C8)⋊8C2, (C9×C4○D4)⋊2C2, C2.25(C2×C9⋊D4), (C3×C4○D4).13S3, C6.109(C2×C3⋊D4), (C2×C6).9(C3⋊D4), SmallGroup(288,161)

Series: Derived Chief Lower central Upper central

C1C36 — D4.9D18
C1C3C9C18C36D36D365C2 — D4.9D18
C9C18C36 — D4.9D18
C1C4C2×C4C4○D4

Generators and relations for D4.9D18
 G = < a,b,c,d | a4=b2=1, c18=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c17 >

Subgroups: 368 in 93 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8, C9, Dic3, C12 [×2], C12, D6, C2×C6, C2×C6, C2×C8, D8, SD16 [×2], Q16, C4○D4, C4○D4, D9, C18, C18 [×2], C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4○D8, Dic9, C36 [×2], C36, D18, C2×C18, C2×C18, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C4○D12, C3×C4○D4, C9⋊C8 [×2], Dic18, C4×D9, D36, C9⋊D4, C2×C36, C2×C36, D4×C9, D4×C9, Q8×C9, Q8.13D6, C2×C9⋊C8, D4.D9, D4⋊D9, C9⋊Q16, Q82D9, D365C2, C9×C4○D4, D4.9D18
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C3⋊D4 [×2], C22×S3, C4○D8, D18 [×3], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, Q8.13D6, C2×C9⋊D4, D4.9D18

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

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

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

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

54 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E6A6B6C6D8A8B8C8D9A9B9C12A12B12C12D12E18A18B18C18D···18L36A···36F36G···36O
order1222234444466668888999121212121218181818···1836···3636···36
size1124362112436244418181818222224442224···42···24···4

54 irreducible representations

dim1111111122222222222222244
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6D9C3⋊D4C3⋊D4C4○D8D18D18D18C9⋊D4C9⋊D4Q8.13D6D4.9D18
kernelD4.9D18C2×C9⋊C8D4.D9D4⋊D9C9⋊Q16Q82D9D365C2C9×C4○D4C3×C4○D4C36C2×C18C2×C12C3×D4C3×Q8C4○D4C12C2×C6C9C2×C4D4Q8C4C22C3C1
# reps1111111111111132243336626

Matrix representation of D4.9D18 in GL4(𝔽73) generated by

1000
0100
0012
007272
,
72000
07200
003232
005741
,
704200
312800
00460
00046
,
454200
702800
002754
00046
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,72,0,0,2,72],[72,0,0,0,0,72,0,0,0,0,32,57,0,0,32,41],[70,31,0,0,42,28,0,0,0,0,46,0,0,0,0,46],[45,70,0,0,42,28,0,0,0,0,27,0,0,0,54,46] >;

D4.9D18 in GAP, Magma, Sage, TeX

D_4._9D_{18}
% in TeX

G:=Group("D4.9D18");
// GroupNames label

G:=SmallGroup(288,161);
// by ID

G=gap.SmallGroup(288,161);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,675,185,80,6725,292,9414]);
// Polycyclic

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

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