direct product, metabelian, nilpotent (class 2), monomial
Aliases: C2×Q8×3- 1+2, C12.31C62, C9⋊3(C6×Q8), (Q8×C9)⋊9C6, C18⋊2(C3×Q8), (C2×C36).7C6, C32.(C6×Q8), (Q8×C18)⋊3C3, C36.20(C2×C6), (C6×C12).14C6, C6.8(Q8×C32), C6.24(C2×C62), (C2×C6).36C62, C62.41(C2×C6), (C6×Q8).8C32, C18.13(C22×C6), (Q8×C32).15C6, C4.4(C22×3- 1+2), C2.3(C23×3- 1+2), (C2×3- 1+2).13C23, (C4×3- 1+2).20C22, C22.6(C22×3- 1+2), (C22×3- 1+2).17C22, C3.3(Q8×C3×C6), (Q8×C3×C6).3C3, (C3×C6).8(C3×Q8), (C2×C12).21(C3×C6), (C3×C12).25(C2×C6), (C2×C18).19(C2×C6), (C3×Q8).23(C3×C6), (C3×C6).34(C22×C6), (C2×C4×3- 1+2).7C2, (C2×C4).3(C2×3- 1+2), SmallGroup(432,408)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8×3- 1+2
G = < a,b,c,d,e | a2=b4=d9=e3=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >
Subgroups: 190 in 152 conjugacy classes, 133 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, Q8, C9, C32, C12, C12, C2×C6, C2×C6, C2×Q8, C18, C3×C6, C3×C6, C2×C12, C2×C12, C3×Q8, C3×Q8, 3- 1+2, C36, C2×C18, C3×C12, C62, C6×Q8, C6×Q8, C2×3- 1+2, C2×3- 1+2, C2×C36, Q8×C9, C6×C12, Q8×C32, C4×3- 1+2, C22×3- 1+2, Q8×C18, Q8×C3×C6, C2×C4×3- 1+2, Q8×3- 1+2, C2×Q8×3- 1+2
Quotients: C1, C2, C3, C22, C6, Q8, C23, C32, C2×C6, C2×Q8, C3×C6, C3×Q8, C22×C6, 3- 1+2, C62, C6×Q8, C2×3- 1+2, Q8×C32, C2×C62, C22×3- 1+2, Q8×C3×C6, Q8×3- 1+2, C23×3- 1+2, C2×Q8×3- 1+2
►On 144 points
Generators in S144
(1 45)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 125)(11 126)(12 118)(13 119)(14 120)(15 121)(16 122)(17 123)(18 124)(19 35)(20 36)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 81)(53 73)(54 74)(55 71)(56 72)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(82 111)(83 112)(84 113)(85 114)(86 115)(87 116)(88 117)(89 109)(90 110)(91 107)(92 108)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(127 143)(128 144)(129 136)(130 137)(131 138)(132 139)(133 140)(134 141)(135 142)
(1 81 27 70)(2 73 19 71)(3 74 20 72)(4 75 21 64)(5 76 22 65)(6 77 23 66)(7 78 24 67)(8 79 25 68)(9 80 26 69)(10 89 143 91)(11 90 144 92)(12 82 136 93)(13 83 137 94)(14 84 138 95)(15 85 139 96)(16 86 140 97)(17 87 141 98)(18 88 142 99)(28 57 39 46)(29 58 40 47)(30 59 41 48)(31 60 42 49)(32 61 43 50)(33 62 44 51)(34 63 45 52)(35 55 37 53)(36 56 38 54)(100 118 111 129)(101 119 112 130)(102 120 113 131)(103 121 114 132)(104 122 115 133)(105 123 116 134)(106 124 117 135)(107 125 109 127)(108 126 110 128)
(1 117 27 106)(2 109 19 107)(3 110 20 108)(4 111 21 100)(5 112 22 101)(6 113 23 102)(7 114 24 103)(8 115 25 104)(9 116 26 105)(10 55 143 53)(11 56 144 54)(12 57 136 46)(13 58 137 47)(14 59 138 48)(15 60 139 49)(16 61 140 50)(17 62 141 51)(18 63 142 52)(28 93 39 82)(29 94 40 83)(30 95 41 84)(31 96 42 85)(32 97 43 86)(33 98 44 87)(34 99 45 88)(35 91 37 89)(36 92 38 90)(64 129 75 118)(65 130 76 119)(66 131 77 120)(67 132 78 121)(68 133 79 122)(69 134 80 123)(70 135 81 124)(71 127 73 125)(72 128 74 126)
(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)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(37 43 40)(38 41 44)(47 53 50)(48 51 54)(55 61 58)(56 59 62)(65 71 68)(66 69 72)(73 79 76)(74 77 80)(83 89 86)(84 87 90)(91 97 94)(92 95 98)(101 107 104)(102 105 108)(109 115 112)(110 113 116)(119 125 122)(120 123 126)(127 133 130)(128 131 134)(137 143 140)(138 141 144)
G:=sub<Sym(144)| (1,45)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,125)(11,126)(12,118)(13,119)(14,120)(15,121)(16,122)(17,123)(18,124)(19,35)(20,36)(21,28)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)(53,73)(54,74)(55,71)(56,72)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(82,111)(83,112)(84,113)(85,114)(86,115)(87,116)(88,117)(89,109)(90,110)(91,107)(92,108)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105)(99,106)(127,143)(128,144)(129,136)(130,137)(131,138)(132,139)(133,140)(134,141)(135,142), (1,81,27,70)(2,73,19,71)(3,74,20,72)(4,75,21,64)(5,76,22,65)(6,77,23,66)(7,78,24,67)(8,79,25,68)(9,80,26,69)(10,89,143,91)(11,90,144,92)(12,82,136,93)(13,83,137,94)(14,84,138,95)(15,85,139,96)(16,86,140,97)(17,87,141,98)(18,88,142,99)(28,57,39,46)(29,58,40,47)(30,59,41,48)(31,60,42,49)(32,61,43,50)(33,62,44,51)(34,63,45,52)(35,55,37,53)(36,56,38,54)(100,118,111,129)(101,119,112,130)(102,120,113,131)(103,121,114,132)(104,122,115,133)(105,123,116,134)(106,124,117,135)(107,125,109,127)(108,126,110,128), (1,117,27,106)(2,109,19,107)(3,110,20,108)(4,111,21,100)(5,112,22,101)(6,113,23,102)(7,114,24,103)(8,115,25,104)(9,116,26,105)(10,55,143,53)(11,56,144,54)(12,57,136,46)(13,58,137,47)(14,59,138,48)(15,60,139,49)(16,61,140,50)(17,62,141,51)(18,63,142,52)(28,93,39,82)(29,94,40,83)(30,95,41,84)(31,96,42,85)(32,97,43,86)(33,98,44,87)(34,99,45,88)(35,91,37,89)(36,92,38,90)(64,129,75,118)(65,130,76,119)(66,131,77,120)(67,132,78,121)(68,133,79,122)(69,134,80,123)(70,135,81,124)(71,127,73,125)(72,128,74,126), (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), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(37,43,40)(38,41,44)(47,53,50)(48,51,54)(55,61,58)(56,59,62)(65,71,68)(66,69,72)(73,79,76)(74,77,80)(83,89,86)(84,87,90)(91,97,94)(92,95,98)(101,107,104)(102,105,108)(109,115,112)(110,113,116)(119,125,122)(120,123,126)(127,133,130)(128,131,134)(137,143,140)(138,141,144)>;
G:=Group( (1,45)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,125)(11,126)(12,118)(13,119)(14,120)(15,121)(16,122)(17,123)(18,124)(19,35)(20,36)(21,28)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)(53,73)(54,74)(55,71)(56,72)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(82,111)(83,112)(84,113)(85,114)(86,115)(87,116)(88,117)(89,109)(90,110)(91,107)(92,108)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105)(99,106)(127,143)(128,144)(129,136)(130,137)(131,138)(132,139)(133,140)(134,141)(135,142), (1,81,27,70)(2,73,19,71)(3,74,20,72)(4,75,21,64)(5,76,22,65)(6,77,23,66)(7,78,24,67)(8,79,25,68)(9,80,26,69)(10,89,143,91)(11,90,144,92)(12,82,136,93)(13,83,137,94)(14,84,138,95)(15,85,139,96)(16,86,140,97)(17,87,141,98)(18,88,142,99)(28,57,39,46)(29,58,40,47)(30,59,41,48)(31,60,42,49)(32,61,43,50)(33,62,44,51)(34,63,45,52)(35,55,37,53)(36,56,38,54)(100,118,111,129)(101,119,112,130)(102,120,113,131)(103,121,114,132)(104,122,115,133)(105,123,116,134)(106,124,117,135)(107,125,109,127)(108,126,110,128), (1,117,27,106)(2,109,19,107)(3,110,20,108)(4,111,21,100)(5,112,22,101)(6,113,23,102)(7,114,24,103)(8,115,25,104)(9,116,26,105)(10,55,143,53)(11,56,144,54)(12,57,136,46)(13,58,137,47)(14,59,138,48)(15,60,139,49)(16,61,140,50)(17,62,141,51)(18,63,142,52)(28,93,39,82)(29,94,40,83)(30,95,41,84)(31,96,42,85)(32,97,43,86)(33,98,44,87)(34,99,45,88)(35,91,37,89)(36,92,38,90)(64,129,75,118)(65,130,76,119)(66,131,77,120)(67,132,78,121)(68,133,79,122)(69,134,80,123)(70,135,81,124)(71,127,73,125)(72,128,74,126), (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), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(37,43,40)(38,41,44)(47,53,50)(48,51,54)(55,61,58)(56,59,62)(65,71,68)(66,69,72)(73,79,76)(74,77,80)(83,89,86)(84,87,90)(91,97,94)(92,95,98)(101,107,104)(102,105,108)(109,115,112)(110,113,116)(119,125,122)(120,123,126)(127,133,130)(128,131,134)(137,143,140)(138,141,144) );
G=PermutationGroup([[(1,45),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,125),(11,126),(12,118),(13,119),(14,120),(15,121),(16,122),(17,123),(18,124),(19,35),(20,36),(21,28),(22,29),(23,30),(24,31),(25,32),(26,33),(27,34),(46,75),(47,76),(48,77),(49,78),(50,79),(51,80),(52,81),(53,73),(54,74),(55,71),(56,72),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(82,111),(83,112),(84,113),(85,114),(86,115),(87,116),(88,117),(89,109),(90,110),(91,107),(92,108),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105),(99,106),(127,143),(128,144),(129,136),(130,137),(131,138),(132,139),(133,140),(134,141),(135,142)], [(1,81,27,70),(2,73,19,71),(3,74,20,72),(4,75,21,64),(5,76,22,65),(6,77,23,66),(7,78,24,67),(8,79,25,68),(9,80,26,69),(10,89,143,91),(11,90,144,92),(12,82,136,93),(13,83,137,94),(14,84,138,95),(15,85,139,96),(16,86,140,97),(17,87,141,98),(18,88,142,99),(28,57,39,46),(29,58,40,47),(30,59,41,48),(31,60,42,49),(32,61,43,50),(33,62,44,51),(34,63,45,52),(35,55,37,53),(36,56,38,54),(100,118,111,129),(101,119,112,130),(102,120,113,131),(103,121,114,132),(104,122,115,133),(105,123,116,134),(106,124,117,135),(107,125,109,127),(108,126,110,128)], [(1,117,27,106),(2,109,19,107),(3,110,20,108),(4,111,21,100),(5,112,22,101),(6,113,23,102),(7,114,24,103),(8,115,25,104),(9,116,26,105),(10,55,143,53),(11,56,144,54),(12,57,136,46),(13,58,137,47),(14,59,138,48),(15,60,139,49),(16,61,140,50),(17,62,141,51),(18,63,142,52),(28,93,39,82),(29,94,40,83),(30,95,41,84),(31,96,42,85),(32,97,43,86),(33,98,44,87),(34,99,45,88),(35,91,37,89),(36,92,38,90),(64,129,75,118),(65,130,76,119),(66,131,77,120),(67,132,78,121),(68,133,79,122),(69,134,80,123),(70,135,81,124),(71,127,73,125),(72,128,74,126)], [(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)], [(2,8,5),(3,6,9),(10,16,13),(11,14,17),(19,25,22),(20,23,26),(29,35,32),(30,33,36),(37,43,40),(38,41,44),(47,53,50),(48,51,54),(55,61,58),(56,59,62),(65,71,68),(66,69,72),(73,79,76),(74,77,80),(83,89,86),(84,87,90),(91,97,94),(92,95,98),(101,107,104),(102,105,108),(109,115,112),(110,113,116),(119,125,122),(120,123,126),(127,133,130),(128,131,134),(137,143,140),(138,141,144)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | ··· | 4F | 6A | ··· | 6F | 6G | ··· | 6L | 9A | ··· | 9F | 12A | ··· | 12L | 12M | ··· | 12X | 18A | ··· | 18R | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 2 | ··· | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 2 | ··· | 2 | 6 | ··· | 6 | 3 | ··· | 3 | 6 | ··· | 6 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 6 |
| type | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C3 | C3 | C6 | C6 | C6 | C6 | Q8 | C3×Q8 | C3×Q8 | 3- 1+2 | C2×3- 1+2 | C2×3- 1+2 | Q8×3- 1+2 |
| kernel | C2×Q8×3- 1+2 | C2×C4×3- 1+2 | Q8×3- 1+2 | Q8×C18 | Q8×C3×C6 | C2×C36 | Q8×C9 | C6×C12 | Q8×C32 | C2×3- 1+2 | C18 | C3×C6 | C2×Q8 | C2×C4 | Q8 | C2 |
| # reps | 1 | 3 | 4 | 6 | 2 | 18 | 24 | 6 | 8 | 2 | 12 | 4 | 2 | 6 | 8 | 4 |
Matrix representation of C2×Q8×3- 1+2 ►in GL7(𝔽37)
| 36 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 36 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 |
| 5 | 14 | 0 | 0 | 0 | 0 | 0 |
| 14 | 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 14 | 0 | 0 | 0 |
| 0 | 0 | 14 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 |
| 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 26 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 26 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 26 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 26 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 |
G:=sub<GL(7,GF(37))| [36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[5,14,0,0,0,0,0,14,32,0,0,0,0,0,0,0,5,14,0,0,0,0,0,14,32,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[10,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,26,0],[10,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,10] >;
C2×Q8×3- 1+2 in GAP, Magma, Sage, TeX
C_2\times Q_8\times 3_-^{1+2} % in TeX
G:=Group("C2xQ8xES-(3,1)"); // GroupNames label
G:=SmallGroup(432,408);
// by ID
G=gap.SmallGroup(432,408);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,504,1037,512,528,760]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^9=e^3=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,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^4>;
// generators/relations