direct product, non-abelian, soluble
Aliases: C2×Q8⋊C9, Q8⋊C18, C6.SL2(𝔽3), (C2×Q8)⋊C9, (C6×Q8).C3, C6.5(C2×A4), (C2×C6).4A4, (C3×Q8).2C6, C3.(C2×SL2(𝔽3)), C22.2(C3.A4), C2.2(C2×C3.A4), SmallGroup(144,35)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C2×Q8⋊C9 |
Generators and relations for C2×Q8⋊C9
G = < a,b,c,d | a2=b4=d9=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >
Smallest permutation representation of C2×Q8⋊C9
►Regular action on 144 points
Generators in S144
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 28)(9 29)(10 93)(11 94)(12 95)(13 96)(14 97)(15 98)(16 99)(17 91)(18 92)(19 112)(20 113)(21 114)(22 115)(23 116)(24 117)(25 109)(26 110)(27 111)(37 119)(38 120)(39 121)(40 122)(41 123)(42 124)(43 125)(44 126)(45 118)(46 106)(47 107)(48 108)(49 100)(50 101)(51 102)(52 103)(53 104)(54 105)(55 81)(56 73)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 133)(65 134)(66 135)(67 127)(68 128)(69 129)(70 130)(71 131)(72 132)(82 144)(83 136)(84 137)(85 138)(86 139)(87 140)(88 141)(89 142)(90 143)
(1 133 85 119)(2 114 86 76)(3 104 87 91)(4 127 88 122)(5 117 89 79)(6 107 90 94)(7 130 82 125)(8 111 83 73)(9 101 84 97)(10 68 46 41)(11 35 47 143)(12 55 48 26)(13 71 49 44)(14 29 50 137)(15 58 51 20)(16 65 52 38)(17 32 53 140)(18 61 54 23)(19 72 57 45)(21 139 59 31)(22 66 60 39)(24 142 62 34)(25 69 63 42)(27 136 56 28)(30 64 138 37)(33 67 141 40)(36 70 144 43)(74 118 112 132)(75 102 113 98)(77 121 115 135)(78 105 116 92)(80 124 109 129)(81 108 110 95)(93 128 106 123)(96 131 100 126)(99 134 103 120)
(1 113 85 75)(2 103 86 99)(3 135 87 121)(4 116 88 78)(5 106 89 93)(6 129 90 124)(7 110 82 81)(8 100 83 96)(9 132 84 118)(10 34 46 142)(11 63 47 25)(12 70 48 43)(13 28 49 136)(14 57 50 19)(15 64 51 37)(16 31 52 139)(17 60 53 22)(18 67 54 40)(20 138 58 30)(21 65 59 38)(23 141 61 33)(24 68 62 41)(26 144 55 36)(27 71 56 44)(29 72 137 45)(32 66 140 39)(35 69 143 42)(73 126 111 131)(74 101 112 97)(76 120 114 134)(77 104 115 91)(79 123 117 128)(80 107 109 94)(92 127 105 122)(95 130 108 125)(98 133 102 119)
(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)
G:=sub<Sym(144)| (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,28)(9,29)(10,93)(11,94)(12,95)(13,96)(14,97)(15,98)(16,99)(17,91)(18,92)(19,112)(20,113)(21,114)(22,115)(23,116)(24,117)(25,109)(26,110)(27,111)(37,119)(38,120)(39,121)(40,122)(41,123)(42,124)(43,125)(44,126)(45,118)(46,106)(47,107)(48,108)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,81)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,133)(65,134)(66,135)(67,127)(68,128)(69,129)(70,130)(71,131)(72,132)(82,144)(83,136)(84,137)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143), (1,133,85,119)(2,114,86,76)(3,104,87,91)(4,127,88,122)(5,117,89,79)(6,107,90,94)(7,130,82,125)(8,111,83,73)(9,101,84,97)(10,68,46,41)(11,35,47,143)(12,55,48,26)(13,71,49,44)(14,29,50,137)(15,58,51,20)(16,65,52,38)(17,32,53,140)(18,61,54,23)(19,72,57,45)(21,139,59,31)(22,66,60,39)(24,142,62,34)(25,69,63,42)(27,136,56,28)(30,64,138,37)(33,67,141,40)(36,70,144,43)(74,118,112,132)(75,102,113,98)(77,121,115,135)(78,105,116,92)(80,124,109,129)(81,108,110,95)(93,128,106,123)(96,131,100,126)(99,134,103,120), (1,113,85,75)(2,103,86,99)(3,135,87,121)(4,116,88,78)(5,106,89,93)(6,129,90,124)(7,110,82,81)(8,100,83,96)(9,132,84,118)(10,34,46,142)(11,63,47,25)(12,70,48,43)(13,28,49,136)(14,57,50,19)(15,64,51,37)(16,31,52,139)(17,60,53,22)(18,67,54,40)(20,138,58,30)(21,65,59,38)(23,141,61,33)(24,68,62,41)(26,144,55,36)(27,71,56,44)(29,72,137,45)(32,66,140,39)(35,69,143,42)(73,126,111,131)(74,101,112,97)(76,120,114,134)(77,104,115,91)(79,123,117,128)(80,107,109,94)(92,127,105,122)(95,130,108,125)(98,133,102,119), (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)>;
G:=Group( (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,28)(9,29)(10,93)(11,94)(12,95)(13,96)(14,97)(15,98)(16,99)(17,91)(18,92)(19,112)(20,113)(21,114)(22,115)(23,116)(24,117)(25,109)(26,110)(27,111)(37,119)(38,120)(39,121)(40,122)(41,123)(42,124)(43,125)(44,126)(45,118)(46,106)(47,107)(48,108)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,81)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,133)(65,134)(66,135)(67,127)(68,128)(69,129)(70,130)(71,131)(72,132)(82,144)(83,136)(84,137)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143), (1,133,85,119)(2,114,86,76)(3,104,87,91)(4,127,88,122)(5,117,89,79)(6,107,90,94)(7,130,82,125)(8,111,83,73)(9,101,84,97)(10,68,46,41)(11,35,47,143)(12,55,48,26)(13,71,49,44)(14,29,50,137)(15,58,51,20)(16,65,52,38)(17,32,53,140)(18,61,54,23)(19,72,57,45)(21,139,59,31)(22,66,60,39)(24,142,62,34)(25,69,63,42)(27,136,56,28)(30,64,138,37)(33,67,141,40)(36,70,144,43)(74,118,112,132)(75,102,113,98)(77,121,115,135)(78,105,116,92)(80,124,109,129)(81,108,110,95)(93,128,106,123)(96,131,100,126)(99,134,103,120), (1,113,85,75)(2,103,86,99)(3,135,87,121)(4,116,88,78)(5,106,89,93)(6,129,90,124)(7,110,82,81)(8,100,83,96)(9,132,84,118)(10,34,46,142)(11,63,47,25)(12,70,48,43)(13,28,49,136)(14,57,50,19)(15,64,51,37)(16,31,52,139)(17,60,53,22)(18,67,54,40)(20,138,58,30)(21,65,59,38)(23,141,61,33)(24,68,62,41)(26,144,55,36)(27,71,56,44)(29,72,137,45)(32,66,140,39)(35,69,143,42)(73,126,111,131)(74,101,112,97)(76,120,114,134)(77,104,115,91)(79,123,117,128)(80,107,109,94)(92,127,105,122)(95,130,108,125)(98,133,102,119), (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) );
G=PermutationGroup([[(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,28),(9,29),(10,93),(11,94),(12,95),(13,96),(14,97),(15,98),(16,99),(17,91),(18,92),(19,112),(20,113),(21,114),(22,115),(23,116),(24,117),(25,109),(26,110),(27,111),(37,119),(38,120),(39,121),(40,122),(41,123),(42,124),(43,125),(44,126),(45,118),(46,106),(47,107),(48,108),(49,100),(50,101),(51,102),(52,103),(53,104),(54,105),(55,81),(56,73),(57,74),(58,75),(59,76),(60,77),(61,78),(62,79),(63,80),(64,133),(65,134),(66,135),(67,127),(68,128),(69,129),(70,130),(71,131),(72,132),(82,144),(83,136),(84,137),(85,138),(86,139),(87,140),(88,141),(89,142),(90,143)], [(1,133,85,119),(2,114,86,76),(3,104,87,91),(4,127,88,122),(5,117,89,79),(6,107,90,94),(7,130,82,125),(8,111,83,73),(9,101,84,97),(10,68,46,41),(11,35,47,143),(12,55,48,26),(13,71,49,44),(14,29,50,137),(15,58,51,20),(16,65,52,38),(17,32,53,140),(18,61,54,23),(19,72,57,45),(21,139,59,31),(22,66,60,39),(24,142,62,34),(25,69,63,42),(27,136,56,28),(30,64,138,37),(33,67,141,40),(36,70,144,43),(74,118,112,132),(75,102,113,98),(77,121,115,135),(78,105,116,92),(80,124,109,129),(81,108,110,95),(93,128,106,123),(96,131,100,126),(99,134,103,120)], [(1,113,85,75),(2,103,86,99),(3,135,87,121),(4,116,88,78),(5,106,89,93),(6,129,90,124),(7,110,82,81),(8,100,83,96),(9,132,84,118),(10,34,46,142),(11,63,47,25),(12,70,48,43),(13,28,49,136),(14,57,50,19),(15,64,51,37),(16,31,52,139),(17,60,53,22),(18,67,54,40),(20,138,58,30),(21,65,59,38),(23,141,61,33),(24,68,62,41),(26,144,55,36),(27,71,56,44),(29,72,137,45),(32,66,140,39),(35,69,143,42),(73,126,111,131),(74,101,112,97),(76,120,114,134),(77,104,115,91),(79,123,117,128),(80,107,109,94),(92,127,105,122),(95,130,108,125),(98,133,102,119)], [(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)]])
C2×Q8⋊C9 is a maximal subgroup of
Q8⋊Dic9 Q8.D18 2- 1+4⋊C9 C18×SL2(𝔽3)
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 6A | ··· | 6F | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 18A | ··· | 18R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | + | - | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | SL2(𝔽3) | SL2(𝔽3) | Q8⋊C9 | A4 | C2×A4 | C3.A4 | C2×C3.A4 |
| kernel | C2×Q8⋊C9 | Q8⋊C9 | C6×Q8 | C3×Q8 | C2×Q8 | Q8 | C6 | C6 | C2 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 4 | 12 | 1 | 1 | 2 | 2 |
Matrix representation of C2×Q8⋊C9 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 31 | 12 |
| 0 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 31 | 6 |
| 10 | 0 | 0 | 0 |
| 0 | 33 | 0 | 0 |
| 0 | 0 | 23 | 4 |
| 0 | 0 | 35 | 4 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,31,0,0,0,12,6],[1,0,0,0,0,1,0,0,0,0,31,31,0,0,0,6],[10,0,0,0,0,33,0,0,0,0,23,35,0,0,4,4] >;
C2×Q8⋊C9 in GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes C_9
% in TeX
G:=Group("C2xQ8:C9"); // GroupNames label
G:=SmallGroup(144,35);
// by ID
G=gap.SmallGroup(144,35);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,2,-2,43,441,117,820,202,88]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=d^9=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations
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