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G = Q8xC18order 144 = 24·32

Direct product of C18 and Q8

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

Aliases: Q8xC18, C36.20C22, C18.12C23, C3.(C6xQ8), C6.4(C3xQ8), (C2xC4).3C18, C4.4(C2xC18), (C2xC36).9C2, (C3xQ8).9C6, (C6xQ8).2C3, (C2xC12).11C6, C12.21(C2xC6), C2.2(C22xC18), C22.4(C2xC18), C6.12(C22xC6), (C2xC18).15C22, (C2xC18)o(C6xQ8), (C2xC6).18(C2xC6), SmallGroup(144,49)

Series: Derived Chief Lower central Upper central

C1C2 — Q8xC18
C1C3C6C18C36Q8xC9 — Q8xC18
C1C2 — Q8xC18
C1C2xC18 — Q8xC18

Generators and relations for Q8xC18
 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, C3, C4, C22, C6, C6, C2xC4, Q8, C9, C12, C2xC6, C2xQ8, C18, C18, C2xC12, C3xQ8, C36, C2xC18, C6xQ8, C2xC36, Q8xC9, Q8xC18
Quotients: C1, C2, C3, C22, C6, Q8, C23, C9, C2xC6, C2xQ8, C18, C3xQ8, C22xC6, C2xC18, C6xQ8, Q8xC9, C22xC18, Q8xC18

Smallest permutation representation of Q8xC18
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 73 38 19)(2 74 39 20)(3 75 40 21)(4 76 41 22)(5 77 42 23)(6 78 43 24)(7 79 44 25)(8 80 45 26)(9 81 46 27)(10 82 47 28)(11 83 48 29)(12 84 49 30)(13 85 50 31)(14 86 51 32)(15 87 52 33)(16 88 53 34)(17 89 54 35)(18 90 37 36)(55 142 118 106)(56 143 119 107)(57 144 120 108)(58 127 121 91)(59 128 122 92)(60 129 123 93)(61 130 124 94)(62 131 125 95)(63 132 126 96)(64 133 109 97)(65 134 110 98)(66 135 111 99)(67 136 112 100)(68 137 113 101)(69 138 114 102)(70 139 115 103)(71 140 116 104)(72 141 117 105)
(1 72 38 117)(2 55 39 118)(3 56 40 119)(4 57 41 120)(5 58 42 121)(6 59 43 122)(7 60 44 123)(8 61 45 124)(9 62 46 125)(10 63 47 126)(11 64 48 109)(12 65 49 110)(13 66 50 111)(14 67 51 112)(15 68 52 113)(16 69 53 114)(17 70 54 115)(18 71 37 116)(19 141 73 105)(20 142 74 106)(21 143 75 107)(22 144 76 108)(23 127 77 91)(24 128 78 92)(25 129 79 93)(26 130 80 94)(27 131 81 95)(28 132 82 96)(29 133 83 97)(30 134 84 98)(31 135 85 99)(32 136 86 100)(33 137 87 101)(34 138 88 102)(35 139 89 103)(36 140 90 104)

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

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

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

Q8xC18 is a maximal subgroup of   C36.9D4  Q8:2Dic9  C36.C23  Dic9:Q8  D18:3Q8  C36.23D4  Q8.15D18

90 conjugacy classes

class 1 2A2B2C3A3B4A···4F6A···6F9A···9F12A···12L18A···18R36A···36AJ
order1222334···46···69···912···1218···1836···36
size1111112···21···11···12···21···12···2

90 irreducible representations

dim111111111222
type+++-
imageC1C2C2C3C6C6C9C18C18Q8C3xQ8Q8xC9
kernelQ8xC18C2xC36Q8xC9C6xQ8C2xC12C3xQ8C2xQ8C2xC4Q8C18C6C2
# reps134268618242412

Matrix representation of Q8xC18 in GL3(F37) generated by

3600
0160
0016
,
3600
012
03636
,
100
01827
01419
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] >;

Q8xC18 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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