direct product, non-abelian, soluble
Aliases: C3×Q8⋊D9, C32.2GL2(𝔽3), Q8⋊(C3×D9), Q8⋊C9⋊5C6, C6.4(C3×S4), (C3×Q8)⋊2D9, (C3×C6).12S4, C6.9(C3.S4), (Q8×C32).8S3, C3.2(C3×GL2(𝔽3)), (C3×Q8⋊C9)⋊3C2, C2.3(C3×C3.S4), (C3×Q8).4(C3×S3), SmallGroup(432,246)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8⋊C9 — C3×Q8⋊D9 |
Generators and relations for C3×Q8⋊D9
G = < a,b,c,d,e | a3=b4=d9=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=c, ebe=b-1c, dcd-1=bc, ece=b2c, ede=d-1 >
Subgroups: 314 in 57 conjugacy classes, 16 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C32, C12, D6, C2×C6, SD16, D9, C18, C3×S3, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×Q8, C3×Q8, C3×C9, D18, C3×C12, S3×C6, Q8⋊2S3, C3×SD16, C3×D9, C3×C18, Q8⋊C9, Q8⋊C9, C3×C3⋊C8, C3×D12, Q8×C32, C6×D9, Q8⋊D9, C3×Q8⋊2S3, C3×Q8⋊C9, C3×Q8⋊D9
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, S4, GL2(𝔽3), C3×D9, C3.S4, C3×S4, Q8⋊D9, C3×GL2(𝔽3), C3×C3.S4, C3×Q8⋊D9
►On 144 points
Generators in S144
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 106 103)(101 107 104)(102 108 105)(109 115 112)(110 116 113)(111 117 114)(118 124 121)(119 125 122)(120 126 123)(127 133 130)(128 134 131)(129 135 132)(136 139 142)(137 140 143)(138 141 144)
(1 81 93 122)(2 101 94 88)(3 134 95 112)(4 75 96 125)(5 104 97 82)(6 128 98 115)(7 78 99 119)(8 107 91 85)(9 131 92 109)(10 70 32 20)(11 56 33 43)(12 140 34 51)(13 64 35 23)(14 59 36 37)(15 143 28 54)(16 67 29 26)(17 62 30 40)(18 137 31 48)(19 63 69 41)(21 50 71 139)(22 57 72 44)(24 53 65 142)(25 60 66 38)(27 47 68 136)(39 144 61 46)(42 138 55 49)(45 141 58 52)(73 133 123 111)(74 89 124 102)(76 127 126 114)(77 83 118 105)(79 130 120 117)(80 86 121 108)(84 129 106 116)(87 132 100 110)(90 135 103 113)
(1 100 93 87)(2 133 94 111)(3 74 95 124)(4 103 96 90)(5 127 97 114)(6 77 98 118)(7 106 99 84)(8 130 91 117)(9 80 92 121)(10 55 32 42)(11 139 33 50)(12 72 34 22)(13 58 35 45)(14 142 36 53)(15 66 28 25)(16 61 29 39)(17 136 30 47)(18 69 31 19)(20 49 70 138)(21 56 71 43)(23 52 64 141)(24 59 65 37)(26 46 67 144)(27 62 68 40)(38 143 60 54)(41 137 63 48)(44 140 57 51)(73 88 123 101)(75 135 125 113)(76 82 126 104)(78 129 119 116)(79 85 120 107)(81 132 122 110)(83 128 105 115)(86 131 108 109)(89 134 102 112)
(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)
(1 140)(2 139)(3 138)(4 137)(5 136)(6 144)(7 143)(8 142)(9 141)(10 134)(11 133)(12 132)(13 131)(14 130)(15 129)(16 128)(17 127)(18 135)(19 125)(20 124)(21 123)(22 122)(23 121)(24 120)(25 119)(26 118)(27 126)(28 116)(29 115)(30 114)(31 113)(32 112)(33 111)(34 110)(35 109)(36 117)(37 107)(38 106)(39 105)(40 104)(41 103)(42 102)(43 101)(44 100)(45 108)(46 98)(47 97)(48 96)(49 95)(50 94)(51 93)(52 92)(53 91)(54 99)(55 89)(56 88)(57 87)(58 86)(59 85)(60 84)(61 83)(62 82)(63 90)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(72 81)
G:=sub<Sym(144)| (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,106,103)(101,107,104)(102,108,105)(109,115,112)(110,116,113)(111,117,114)(118,124,121)(119,125,122)(120,126,123)(127,133,130)(128,134,131)(129,135,132)(136,139,142)(137,140,143)(138,141,144), (1,81,93,122)(2,101,94,88)(3,134,95,112)(4,75,96,125)(5,104,97,82)(6,128,98,115)(7,78,99,119)(8,107,91,85)(9,131,92,109)(10,70,32,20)(11,56,33,43)(12,140,34,51)(13,64,35,23)(14,59,36,37)(15,143,28,54)(16,67,29,26)(17,62,30,40)(18,137,31,48)(19,63,69,41)(21,50,71,139)(22,57,72,44)(24,53,65,142)(25,60,66,38)(27,47,68,136)(39,144,61,46)(42,138,55,49)(45,141,58,52)(73,133,123,111)(74,89,124,102)(76,127,126,114)(77,83,118,105)(79,130,120,117)(80,86,121,108)(84,129,106,116)(87,132,100,110)(90,135,103,113), (1,100,93,87)(2,133,94,111)(3,74,95,124)(4,103,96,90)(5,127,97,114)(6,77,98,118)(7,106,99,84)(8,130,91,117)(9,80,92,121)(10,55,32,42)(11,139,33,50)(12,72,34,22)(13,58,35,45)(14,142,36,53)(15,66,28,25)(16,61,29,39)(17,136,30,47)(18,69,31,19)(20,49,70,138)(21,56,71,43)(23,52,64,141)(24,59,65,37)(26,46,67,144)(27,62,68,40)(38,143,60,54)(41,137,63,48)(44,140,57,51)(73,88,123,101)(75,135,125,113)(76,82,126,104)(78,129,119,116)(79,85,120,107)(81,132,122,110)(83,128,105,115)(86,131,108,109)(89,134,102,112), (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), (1,140)(2,139)(3,138)(4,137)(5,136)(6,144)(7,143)(8,142)(9,141)(10,134)(11,133)(12,132)(13,131)(14,130)(15,129)(16,128)(17,127)(18,135)(19,125)(20,124)(21,123)(22,122)(23,121)(24,120)(25,119)(26,118)(27,126)(28,116)(29,115)(30,114)(31,113)(32,112)(33,111)(34,110)(35,109)(36,117)(37,107)(38,106)(39,105)(40,104)(41,103)(42,102)(43,101)(44,100)(45,108)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,99)(55,89)(56,88)(57,87)(58,86)(59,85)(60,84)(61,83)(62,82)(63,90)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(72,81)>;
G:=Group( (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,106,103)(101,107,104)(102,108,105)(109,115,112)(110,116,113)(111,117,114)(118,124,121)(119,125,122)(120,126,123)(127,133,130)(128,134,131)(129,135,132)(136,139,142)(137,140,143)(138,141,144), (1,81,93,122)(2,101,94,88)(3,134,95,112)(4,75,96,125)(5,104,97,82)(6,128,98,115)(7,78,99,119)(8,107,91,85)(9,131,92,109)(10,70,32,20)(11,56,33,43)(12,140,34,51)(13,64,35,23)(14,59,36,37)(15,143,28,54)(16,67,29,26)(17,62,30,40)(18,137,31,48)(19,63,69,41)(21,50,71,139)(22,57,72,44)(24,53,65,142)(25,60,66,38)(27,47,68,136)(39,144,61,46)(42,138,55,49)(45,141,58,52)(73,133,123,111)(74,89,124,102)(76,127,126,114)(77,83,118,105)(79,130,120,117)(80,86,121,108)(84,129,106,116)(87,132,100,110)(90,135,103,113), (1,100,93,87)(2,133,94,111)(3,74,95,124)(4,103,96,90)(5,127,97,114)(6,77,98,118)(7,106,99,84)(8,130,91,117)(9,80,92,121)(10,55,32,42)(11,139,33,50)(12,72,34,22)(13,58,35,45)(14,142,36,53)(15,66,28,25)(16,61,29,39)(17,136,30,47)(18,69,31,19)(20,49,70,138)(21,56,71,43)(23,52,64,141)(24,59,65,37)(26,46,67,144)(27,62,68,40)(38,143,60,54)(41,137,63,48)(44,140,57,51)(73,88,123,101)(75,135,125,113)(76,82,126,104)(78,129,119,116)(79,85,120,107)(81,132,122,110)(83,128,105,115)(86,131,108,109)(89,134,102,112), (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), (1,140)(2,139)(3,138)(4,137)(5,136)(6,144)(7,143)(8,142)(9,141)(10,134)(11,133)(12,132)(13,131)(14,130)(15,129)(16,128)(17,127)(18,135)(19,125)(20,124)(21,123)(22,122)(23,121)(24,120)(25,119)(26,118)(27,126)(28,116)(29,115)(30,114)(31,113)(32,112)(33,111)(34,110)(35,109)(36,117)(37,107)(38,106)(39,105)(40,104)(41,103)(42,102)(43,101)(44,100)(45,108)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,99)(55,89)(56,88)(57,87)(58,86)(59,85)(60,84)(61,83)(62,82)(63,90)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(72,81) );
G=PermutationGroup([[(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,79,76),(74,80,77),(75,81,78),(82,88,85),(83,89,86),(84,90,87),(91,97,94),(92,98,95),(93,99,96),(100,106,103),(101,107,104),(102,108,105),(109,115,112),(110,116,113),(111,117,114),(118,124,121),(119,125,122),(120,126,123),(127,133,130),(128,134,131),(129,135,132),(136,139,142),(137,140,143),(138,141,144)], [(1,81,93,122),(2,101,94,88),(3,134,95,112),(4,75,96,125),(5,104,97,82),(6,128,98,115),(7,78,99,119),(8,107,91,85),(9,131,92,109),(10,70,32,20),(11,56,33,43),(12,140,34,51),(13,64,35,23),(14,59,36,37),(15,143,28,54),(16,67,29,26),(17,62,30,40),(18,137,31,48),(19,63,69,41),(21,50,71,139),(22,57,72,44),(24,53,65,142),(25,60,66,38),(27,47,68,136),(39,144,61,46),(42,138,55,49),(45,141,58,52),(73,133,123,111),(74,89,124,102),(76,127,126,114),(77,83,118,105),(79,130,120,117),(80,86,121,108),(84,129,106,116),(87,132,100,110),(90,135,103,113)], [(1,100,93,87),(2,133,94,111),(3,74,95,124),(4,103,96,90),(5,127,97,114),(6,77,98,118),(7,106,99,84),(8,130,91,117),(9,80,92,121),(10,55,32,42),(11,139,33,50),(12,72,34,22),(13,58,35,45),(14,142,36,53),(15,66,28,25),(16,61,29,39),(17,136,30,47),(18,69,31,19),(20,49,70,138),(21,56,71,43),(23,52,64,141),(24,59,65,37),(26,46,67,144),(27,62,68,40),(38,143,60,54),(41,137,63,48),(44,140,57,51),(73,88,123,101),(75,135,125,113),(76,82,126,104),(78,129,119,116),(79,85,120,107),(81,132,122,110),(83,128,105,115),(86,131,108,109),(89,134,102,112)], [(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)], [(1,140),(2,139),(3,138),(4,137),(5,136),(6,144),(7,143),(8,142),(9,141),(10,134),(11,133),(12,132),(13,131),(14,130),(15,129),(16,128),(17,127),(18,135),(19,125),(20,124),(21,123),(22,122),(23,121),(24,120),(25,119),(26,118),(27,126),(28,116),(29,115),(30,114),(31,113),(32,112),(33,111),(34,110),(35,109),(36,117),(37,107),(38,106),(39,105),(40,104),(41,103),(42,102),(43,101),(44,100),(45,108),(46,98),(47,97),(48,96),(49,95),(50,94),(51,93),(52,92),(53,91),(54,99),(55,89),(56,88),(57,87),(58,86),(59,85),(60,84),(61,83),(62,82),(63,90),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(72,81)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 9A | ··· | 9I | 12A | 12B | 12C | 12D | 12E | 18A | ··· | 18I | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 36 | 1 | 1 | 2 | 2 | 2 | 6 | 1 | 1 | 2 | 2 | 2 | 36 | 36 | 18 | 18 | 8 | ··· | 8 | 6 | 6 | 12 | 12 | 12 | 8 | ··· | 8 | 18 | 18 | 18 | 18 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C3 | C6 | S3 | D9 | C3×S3 | GL2(𝔽3) | C3×D9 | C3×GL2(𝔽3) | S4 | C3×S4 | GL2(𝔽3) | Q8⋊D9 | C3×GL2(𝔽3) | C3×Q8⋊D9 | C3.S4 | C3×C3.S4 |
| kernel | C3×Q8⋊D9 | C3×Q8⋊C9 | Q8⋊D9 | Q8⋊C9 | Q8×C32 | C3×Q8 | C3×Q8 | C32 | Q8 | C3 | C3×C6 | C6 | C32 | C3 | C3 | C1 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 3 | 2 | 2 | 6 | 4 | 2 | 4 | 1 | 3 | 2 | 6 | 1 | 2 |
Matrix representation of C3×Q8⋊D9 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 60 |
| 0 | 0 | 60 | 66 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 55 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 39 | 46 |
| 0 | 0 | 47 | 33 |
| 0 | 69 | 0 | 0 |
| 18 | 0 | 0 | 0 |
| 0 | 0 | 26 | 36 |
| 0 | 0 | 36 | 47 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,7,60,0,0,60,66],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[55,0,0,0,0,4,0,0,0,0,39,47,0,0,46,33],[0,18,0,0,69,0,0,0,0,0,26,36,0,0,36,47] >;
C3×Q8⋊D9 in GAP, Magma, Sage, TeX
C_3\times Q_8\rtimes D_9
% in TeX
G:=Group("C3xQ8:D9"); // GroupNames label
G:=SmallGroup(432,246);
// by ID
G=gap.SmallGroup(432,246);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,632,142,1011,3784,1908,172,2273,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=d^9=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,d*b*d^-1=c,e*b*e=b^-1*c,d*c*d^-1=b*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations