direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C18, C36.20C22, C18.12C23, C3.(C6×Q8), C6.4(C3×Q8), (C2×C4).3C18, C4.4(C2×C18), (C2×C36).9C2, (C3×Q8).9C6, (C6×Q8).2C3, (C2×C12).11C6, C12.21(C2×C6), C2.2(C22×C18), C22.4(C2×C18), C6.12(C22×C6), (C2×C18).15C22, (C2×C18)○(C6×Q8), (C2×C6).18(C2×C6), SmallGroup(144,49)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C4, C22, C6, C6, C2×C4, Q8, C9, C12, C2×C6, C2×Q8, C18, C18, C2×C12, C3×Q8, C36, C2×C18, C6×Q8, C2×C36, Q8×C9, Q8×C18
Quotients: C1, C2, C3, C22, C6, Q8, C23, C9, C2×C6, C2×Q8, C18, C3×Q8, C22×C6, C2×C18, C6×Q8, Q8×C9, C22×C18, 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 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)]])
Q8×C18 is a maximal subgroup of
C36.9D4 Q8⋊2Dic9 C36.C23 Dic9⋊Q8 D18⋊3Q8 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