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## G = Q8×C18order 144 = 24·32

### Direct product of C18 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C18
 Chief series C1 — C3 — C6 — C18 — C36 — Q8×C9 — Q8×C18
 Lower central C1 — C2 — Q8×C18
 Upper central C1 — C2×C18 — Q8×C18

Generators and relations for Q8×C18
G = < a,b,c | a18=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 57, all normal (12 characteristic)
C1, C2, C2 [×2], C3, C4 [×6], C22, C6, C6 [×2], C2×C4 [×3], Q8 [×4], C9, C12 [×6], C2×C6, C2×Q8, C18, C18 [×2], C2×C12 [×3], C3×Q8 [×4], C36 [×6], C2×C18, C6×Q8, C2×C36 [×3], Q8×C9 [×4], Q8×C18
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], Q8 [×2], C23, C9, C2×C6 [×7], C2×Q8, C18 [×7], C3×Q8 [×2], C22×C6, C2×C18 [×7], C6×Q8, Q8×C9 [×2], C22×C18, Q8×C18

Smallest permutation representation of Q8×C18
Regular action 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 20 85 45)(2 21 86 46)(3 22 87 47)(4 23 88 48)(5 24 89 49)(6 25 90 50)(7 26 73 51)(8 27 74 52)(9 28 75 53)(10 29 76 54)(11 30 77 37)(12 31 78 38)(13 32 79 39)(14 33 80 40)(15 34 81 41)(16 35 82 42)(17 36 83 43)(18 19 84 44)(55 106 117 138)(56 107 118 139)(57 108 119 140)(58 91 120 141)(59 92 121 142)(60 93 122 143)(61 94 123 144)(62 95 124 127)(63 96 125 128)(64 97 126 129)(65 98 109 130)(66 99 110 131)(67 100 111 132)(68 101 112 133)(69 102 113 134)(70 103 114 135)(71 104 115 136)(72 105 116 137)
(1 131 85 99)(2 132 86 100)(3 133 87 101)(4 134 88 102)(5 135 89 103)(6 136 90 104)(7 137 73 105)(8 138 74 106)(9 139 75 107)(10 140 76 108)(11 141 77 91)(12 142 78 92)(13 143 79 93)(14 144 80 94)(15 127 81 95)(16 128 82 96)(17 129 83 97)(18 130 84 98)(19 109 44 65)(20 110 45 66)(21 111 46 67)(22 112 47 68)(23 113 48 69)(24 114 49 70)(25 115 50 71)(26 116 51 72)(27 117 52 55)(28 118 53 56)(29 119 54 57)(30 120 37 58)(31 121 38 59)(32 122 39 60)(33 123 40 61)(34 124 41 62)(35 125 42 63)(36 126 43 64)

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

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

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

Q8×C18 is a maximal subgroup of   C36.9D4  Q82Dic9  C36.C23  Dic9⋊Q8  D183Q8  C36.23D4  Q8.15D18

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A ··· 4F 6A ··· 6F 9A ··· 9F 12A ··· 12L 18A ··· 18R 36A ··· 36AJ order 1 2 2 2 3 3 4 ··· 4 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 type + + + - image C1 C2 C2 C3 C6 C6 C9 C18 C18 Q8 C3×Q8 Q8×C9 kernel Q8×C18 C2×C36 Q8×C9 C6×Q8 C2×C12 C3×Q8 C2×Q8 C2×C4 Q8 C18 C6 C2 # reps 1 3 4 2 6 8 6 18 24 2 4 12

Matrix representation of Q8×C18 in GL3(𝔽37) generated by

 36 0 0 0 16 0 0 0 16
,
 36 0 0 0 1 2 0 36 36
,
 1 0 0 0 18 27 0 14 19
G:=sub<GL(3,GF(37))| [36,0,0,0,16,0,0,0,16],[36,0,0,0,1,36,0,2,36],[1,0,0,0,18,14,0,27,19] >;

Q8×C18 in GAP, Magma, Sage, TeX

Q_8\times C_{18}
% in TeX

G:=Group("Q8xC18");
// GroupNames label

G:=SmallGroup(144,49);
// by ID

G=gap.SmallGroup(144,49);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,144,313,151,165]);
// Polycyclic

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

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