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G = D183Q8order 288 = 25·32

3rd semidirect product of D18 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D183Q8, C36.22D4, (C2×Q8)⋊5D9, C2.9(Q8×D9), (Q8×C18)⋊3C2, C95(C22⋊Q8), C6.40(S3×Q8), D18⋊C4.6C2, C4⋊Dic915C2, (C2×C4).19D18, C18.57(C2×D4), C3.(D63Q8), (C6×Q8).16S3, C18.17(C2×Q8), Dic9⋊C416C2, (C2×C12).219D6, C4.18(C9⋊D4), C18.36(C4○D4), C12.19(C3⋊D4), (C2×C36).63C22, (C2×C18).58C23, C2.8(Q83D9), C6.44(Q83S3), C22.64(C22×D9), (C2×Dic9).18C22, (C22×D9).27C22, (C2×C4×D9).4C2, C2.21(C2×C9⋊D4), C6.105(C2×C3⋊D4), (C2×C6).215(C22×S3), SmallGroup(288,156)

Series: Derived Chief Lower central Upper central

C1C2×C18 — D183Q8
C1C3C9C18C2×C18C22×D9C2×C4×D9 — D183Q8
C9C2×C18 — D183Q8
C1C22C2×Q8

Generators and relations for D183Q8
 G = < a,b,c,d | a18=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a9b, bd=db, dcd-1=c-1 >

Subgroups: 460 in 111 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], S3 [×2], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, C9, Dic3 [×3], C12 [×2], C12 [×2], D6 [×4], C2×C6, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, D9 [×2], C18 [×3], C4×S3 [×2], C2×Dic3 [×3], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3, C22⋊Q8, Dic9 [×3], C36 [×2], C36 [×2], D18 [×2], D18 [×2], C2×C18, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], S3×C2×C4, C6×Q8, C4×D9 [×2], C2×Dic9, C2×Dic9 [×2], C2×C36, C2×C36 [×2], Q8×C9 [×2], C22×D9, D63Q8, Dic9⋊C4 [×2], C4⋊Dic9, D18⋊C4 [×2], C2×C4×D9, Q8×C18, D183Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, D9, C3⋊D4 [×2], C22×S3, C22⋊Q8, D18 [×3], S3×Q8, Q83S3, C2×C3⋊D4, C9⋊D4 [×2], C22×D9, D63Q8, Q8×D9, Q83D9, C2×C9⋊D4, D183Q8

Smallest permutation representation of D183Q8
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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(35 36)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(73 75)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(91 92)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)(109 117)(110 116)(111 115)(112 114)(118 126)(119 125)(120 124)(121 123)(127 131)(128 130)(132 144)(133 143)(134 142)(135 141)(136 140)(137 139)
(1 134 101 88)(2 135 102 89)(3 136 103 90)(4 137 104 73)(5 138 105 74)(6 139 106 75)(7 140 107 76)(8 141 108 77)(9 142 91 78)(10 143 92 79)(11 144 93 80)(12 127 94 81)(13 128 95 82)(14 129 96 83)(15 130 97 84)(16 131 98 85)(17 132 99 86)(18 133 100 87)(19 61 47 110)(20 62 48 111)(21 63 49 112)(22 64 50 113)(23 65 51 114)(24 66 52 115)(25 67 53 116)(26 68 54 117)(27 69 37 118)(28 70 38 119)(29 71 39 120)(30 72 40 121)(31 55 41 122)(32 56 42 123)(33 57 43 124)(34 58 44 125)(35 59 45 126)(36 60 46 109)
(1 27 101 37)(2 28 102 38)(3 29 103 39)(4 30 104 40)(5 31 105 41)(6 32 106 42)(7 33 107 43)(8 34 108 44)(9 35 91 45)(10 36 92 46)(11 19 93 47)(12 20 94 48)(13 21 95 49)(14 22 96 50)(15 23 97 51)(16 24 98 52)(17 25 99 53)(18 26 100 54)(55 138 122 74)(56 139 123 75)(57 140 124 76)(58 141 125 77)(59 142 126 78)(60 143 109 79)(61 144 110 80)(62 127 111 81)(63 128 112 82)(64 129 113 83)(65 130 114 84)(66 131 115 85)(67 132 116 86)(68 133 117 87)(69 134 118 88)(70 135 119 89)(71 136 120 90)(72 137 121 73)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H6A6B6C9A9B9C12A···12F18A···18I36A···36R
order12222234444444466699912···1218···1836···36
size1111181822244181836362222224···42···24···4

54 irreducible representations

dim1111112222222224444
type++++++++-+++-+-+
imageC1C2C2C2C2C2S3D4Q8D6C4○D4D9C3⋊D4D18C9⋊D4S3×Q8Q83S3Q8×D9Q83D9
kernelD183Q8Dic9⋊C4C4⋊Dic9D18⋊C4C2×C4×D9Q8×C18C6×Q8C36D18C2×C12C18C2×Q8C12C2×C4C4C6C6C2C2
# reps12121112232349121133

Matrix representation of D183Q8 in GL4(𝔽37) generated by

61700
202600
00360
00036
,
61700
113100
00360
00361
,
71400
233000
00135
00136
,
1000
0100
0060
00631
G:=sub<GL(4,GF(37))| [6,20,0,0,17,26,0,0,0,0,36,0,0,0,0,36],[6,11,0,0,17,31,0,0,0,0,36,36,0,0,0,1],[7,23,0,0,14,30,0,0,0,0,1,1,0,0,35,36],[1,0,0,0,0,1,0,0,0,0,6,6,0,0,0,31] >;

D183Q8 in GAP, Magma, Sage, TeX

D_{18}\rtimes_3Q_8
% in TeX

G:=Group("D18:3Q8");
// GroupNames label

G:=SmallGroup(288,156);
// by ID

G=gap.SmallGroup(288,156);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,100,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^18=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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