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## G = D18⋊3Q8order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — D18⋊3Q8
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — C2×C4×D9 — D18⋊3Q8
 Lower central C9 — C2×C18 — D18⋊3Q8
 Upper central C1 — C22 — C2×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, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C9, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, D9, C18, C4×S3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22⋊Q8, Dic9, C36, C36, D18, D18, C2×C18, Dic3⋊C4, C4⋊Dic3, D6⋊C4, S3×C2×C4, C6×Q8, C4×D9, C2×Dic9, C2×Dic9, C2×C36, C2×C36, Q8×C9, C22×D9, D63Q8, Dic9⋊C4, C4⋊Dic9, D18⋊C4, C2×C4×D9, Q8×C18, D183Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D9, C3⋊D4, C22×S3, C22⋊Q8, D18, S3×Q8, Q83S3, C2×C3⋊D4, C9⋊D4, 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 25)(20 24)(21 23)(26 36)(27 35)(28 34)(29 33)(30 32)(37 40)(38 39)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 60)(56 59)(57 58)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 87)(74 86)(75 85)(76 84)(77 83)(78 82)(79 81)(88 90)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)(109 121)(110 120)(111 119)(112 118)(113 117)(114 116)(122 126)(123 125)(127 134)(128 133)(129 132)(130 131)(135 144)(136 143)(137 142)(138 141)(139 140)
(1 105 48 27)(2 106 49 28)(3 107 50 29)(4 108 51 30)(5 91 52 31)(6 92 53 32)(7 93 54 33)(8 94 37 34)(9 95 38 35)(10 96 39 36)(11 97 40 19)(12 98 41 20)(13 99 42 21)(14 100 43 22)(15 101 44 23)(16 102 45 24)(17 103 46 25)(18 104 47 26)(55 73 137 126)(56 74 138 109)(57 75 139 110)(58 76 140 111)(59 77 141 112)(60 78 142 113)(61 79 143 114)(62 80 144 115)(63 81 127 116)(64 82 128 117)(65 83 129 118)(66 84 130 119)(67 85 131 120)(68 86 132 121)(69 87 133 122)(70 88 134 123)(71 89 135 124)(72 90 136 125)
(1 58 48 140)(2 59 49 141)(3 60 50 142)(4 61 51 143)(5 62 52 144)(6 63 53 127)(7 64 54 128)(8 65 37 129)(9 66 38 130)(10 67 39 131)(11 68 40 132)(12 69 41 133)(13 70 42 134)(14 71 43 135)(15 72 44 136)(16 55 45 137)(17 56 46 138)(18 57 47 139)(19 86 97 121)(20 87 98 122)(21 88 99 123)(22 89 100 124)(23 90 101 125)(24 73 102 126)(25 74 103 109)(26 75 104 110)(27 76 105 111)(28 77 106 112)(29 78 107 113)(30 79 108 114)(31 80 91 115)(32 81 92 116)(33 82 93 117)(34 83 94 118)(35 84 95 119)(36 85 96 120)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 9A 9B 9C 12A ··· 12F 18A ··· 18I 36A ··· 36R order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 18 18 2 2 2 4 4 18 18 36 36 2 2 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + + - + - + image C1 C2 C2 C2 C2 C2 S3 D4 Q8 D6 C4○D4 D9 C3⋊D4 D18 C9⋊D4 S3×Q8 Q8⋊3S3 Q8×D9 Q8⋊3D9 kernel D18⋊3Q8 Dic9⋊C4 C4⋊Dic9 D18⋊C4 C2×C4×D9 Q8×C18 C6×Q8 C36 D18 C2×C12 C18 C2×Q8 C12 C2×C4 C4 C6 C6 C2 C2 # reps 1 2 1 2 1 1 1 2 2 3 2 3 4 9 12 1 1 3 3

Matrix representation of D183Q8 in GL4(𝔽37) generated by

 6 17 0 0 20 26 0 0 0 0 36 0 0 0 0 36
,
 6 17 0 0 11 31 0 0 0 0 36 0 0 0 36 1
,
 7 14 0 0 23 30 0 0 0 0 1 35 0 0 1 36
,
 1 0 0 0 0 1 0 0 0 0 6 0 0 0 6 31
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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