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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, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C9, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, D9, C18, C18, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4○D8, Dic9, C36, C36, D18, C2×C18, C2×C18, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C4○D12, C3×C4○D4, C9⋊C8, 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, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, C4○D8, D18, C2×C3⋊D4, C9⋊D4, C22×D9, Q8.13D6, C2×C9⋊D4, D4.9D18

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

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

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

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

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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