metabelian, supersoluble, monomial
Aliases: C62.221C23, (C2×C12).204D6, C62.62(C2×C4), (C22×C6).83D6, C62⋊5C4.4C2, C6.91(D4⋊2S3), (C6×C12).250C22, C6.Dic6⋊18C2, (C2×C62).60C22, C32⋊14(C42⋊C2), C2.1(C12.D6), C3⋊4(C23.16D6), C6.65(S3×C2×C4), (C2×C6).21(C4×S3), C22.6(C4×C3⋊S3), (C4×C3⋊Dic3)⋊20C2, (C2×C3⋊Dic3)⋊10C4, C23.16(C2×C3⋊S3), C22⋊C4.3(C3⋊S3), (C3×C22⋊C4).14S3, C3⋊Dic3.47(C2×C4), (C3×C6).96(C22×C4), (C3×C6).141(C4○D4), (C2×C6).238(C22×S3), (C32×C22⋊C4).6C2, C22.12(C22×C3⋊S3), (C22×C3⋊Dic3).9C2, (C2×C3⋊Dic3).153C22, C2.7(C2×C4×C3⋊S3), (C2×C4).25(C2×C3⋊S3), SmallGroup(288,734)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C62.221C23 |
Generators and relations for C62.221C23
G = < a,b,c,d,e | a6=b6=d2=1, c2=b3, e2=a3, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b3d >
Subgroups: 644 in 228 conjugacy classes, 85 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C42⋊C2, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C62, C62, C4×Dic3, Dic3⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C2×C62, C23.16D6, C4×C3⋊Dic3, C6.Dic6, C62⋊5C4, C32×C22⋊C4, C22×C3⋊Dic3, C62.221C23
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C3⋊S3, C4×S3, C22×S3, C42⋊C2, C2×C3⋊S3, S3×C2×C4, D4⋊2S3, C4×C3⋊S3, C22×C3⋊S3, C23.16D6, C2×C4×C3⋊S3, C12.D6, C62.221C23
►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 35 58 15 39 61)(2 36 59 16 40 62)(3 31 60 17 41 63)(4 32 55 18 42 64)(5 33 56 13 37 65)(6 34 57 14 38 66)(7 123 27 120 140 24)(8 124 28 115 141 19)(9 125 29 116 142 20)(10 126 30 117 143 21)(11 121 25 118 144 22)(12 122 26 119 139 23)(43 96 51 77 101 71)(44 91 52 78 102 72)(45 92 53 73 97 67)(46 93 54 74 98 68)(47 94 49 75 99 69)(48 95 50 76 100 70)(79 105 135 130 88 111)(80 106 136 131 89 112)(81 107 137 132 90 113)(82 108 138 127 85 114)(83 103 133 128 86 109)(84 104 134 129 87 110)
(1 90 15 107)(2 89 16 106)(3 88 17 105)(4 87 18 104)(5 86 13 103)(6 85 14 108)(7 76 120 48)(8 75 115 47)(9 74 116 46)(10 73 117 45)(11 78 118 44)(12 77 119 43)(19 99 28 94)(20 98 29 93)(21 97 30 92)(22 102 25 91)(23 101 26 96)(24 100 27 95)(31 130 41 79)(32 129 42 84)(33 128 37 83)(34 127 38 82)(35 132 39 81)(36 131 40 80)(49 141 69 124)(50 140 70 123)(51 139 71 122)(52 144 72 121)(53 143 67 126)(54 142 68 125)(55 134 64 110)(56 133 65 109)(57 138 66 114)(58 137 61 113)(59 136 62 112)(60 135 63 111)
(7 120)(8 115)(9 116)(10 117)(11 118)(12 119)(19 28)(20 29)(21 30)(22 25)(23 26)(24 27)(43 77)(44 78)(45 73)(46 74)(47 75)(48 76)(49 69)(50 70)(51 71)(52 72)(53 67)(54 68)(91 102)(92 97)(93 98)(94 99)(95 100)(96 101)(121 144)(122 139)(123 140)(124 141)(125 142)(126 143)
(1 70 4 67)(2 71 5 68)(3 72 6 69)(7 129 10 132)(8 130 11 127)(9 131 12 128)(13 54 16 51)(14 49 17 52)(15 50 18 53)(19 135 22 138)(20 136 23 133)(21 137 24 134)(25 114 28 111)(26 109 29 112)(27 110 30 113)(31 44 34 47)(32 45 35 48)(33 46 36 43)(37 74 40 77)(38 75 41 78)(39 76 42 73)(55 92 58 95)(56 93 59 96)(57 94 60 91)(61 100 64 97)(62 101 65 98)(63 102 66 99)(79 118 82 115)(80 119 83 116)(81 120 84 117)(85 124 88 121)(86 125 89 122)(87 126 90 123)(103 142 106 139)(104 143 107 140)(105 144 108 141)
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), (1,35,58,15,39,61)(2,36,59,16,40,62)(3,31,60,17,41,63)(4,32,55,18,42,64)(5,33,56,13,37,65)(6,34,57,14,38,66)(7,123,27,120,140,24)(8,124,28,115,141,19)(9,125,29,116,142,20)(10,126,30,117,143,21)(11,121,25,118,144,22)(12,122,26,119,139,23)(43,96,51,77,101,71)(44,91,52,78,102,72)(45,92,53,73,97,67)(46,93,54,74,98,68)(47,94,49,75,99,69)(48,95,50,76,100,70)(79,105,135,130,88,111)(80,106,136,131,89,112)(81,107,137,132,90,113)(82,108,138,127,85,114)(83,103,133,128,86,109)(84,104,134,129,87,110), (1,90,15,107)(2,89,16,106)(3,88,17,105)(4,87,18,104)(5,86,13,103)(6,85,14,108)(7,76,120,48)(8,75,115,47)(9,74,116,46)(10,73,117,45)(11,78,118,44)(12,77,119,43)(19,99,28,94)(20,98,29,93)(21,97,30,92)(22,102,25,91)(23,101,26,96)(24,100,27,95)(31,130,41,79)(32,129,42,84)(33,128,37,83)(34,127,38,82)(35,132,39,81)(36,131,40,80)(49,141,69,124)(50,140,70,123)(51,139,71,122)(52,144,72,121)(53,143,67,126)(54,142,68,125)(55,134,64,110)(56,133,65,109)(57,138,66,114)(58,137,61,113)(59,136,62,112)(60,135,63,111), (7,120)(8,115)(9,116)(10,117)(11,118)(12,119)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27)(43,77)(44,78)(45,73)(46,74)(47,75)(48,76)(49,69)(50,70)(51,71)(52,72)(53,67)(54,68)(91,102)(92,97)(93,98)(94,99)(95,100)(96,101)(121,144)(122,139)(123,140)(124,141)(125,142)(126,143), (1,70,4,67)(2,71,5,68)(3,72,6,69)(7,129,10,132)(8,130,11,127)(9,131,12,128)(13,54,16,51)(14,49,17,52)(15,50,18,53)(19,135,22,138)(20,136,23,133)(21,137,24,134)(25,114,28,111)(26,109,29,112)(27,110,30,113)(31,44,34,47)(32,45,35,48)(33,46,36,43)(37,74,40,77)(38,75,41,78)(39,76,42,73)(55,92,58,95)(56,93,59,96)(57,94,60,91)(61,100,64,97)(62,101,65,98)(63,102,66,99)(79,118,82,115)(80,119,83,116)(81,120,84,117)(85,124,88,121)(86,125,89,122)(87,126,90,123)(103,142,106,139)(104,143,107,140)(105,144,108,141)>;
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), (1,35,58,15,39,61)(2,36,59,16,40,62)(3,31,60,17,41,63)(4,32,55,18,42,64)(5,33,56,13,37,65)(6,34,57,14,38,66)(7,123,27,120,140,24)(8,124,28,115,141,19)(9,125,29,116,142,20)(10,126,30,117,143,21)(11,121,25,118,144,22)(12,122,26,119,139,23)(43,96,51,77,101,71)(44,91,52,78,102,72)(45,92,53,73,97,67)(46,93,54,74,98,68)(47,94,49,75,99,69)(48,95,50,76,100,70)(79,105,135,130,88,111)(80,106,136,131,89,112)(81,107,137,132,90,113)(82,108,138,127,85,114)(83,103,133,128,86,109)(84,104,134,129,87,110), (1,90,15,107)(2,89,16,106)(3,88,17,105)(4,87,18,104)(5,86,13,103)(6,85,14,108)(7,76,120,48)(8,75,115,47)(9,74,116,46)(10,73,117,45)(11,78,118,44)(12,77,119,43)(19,99,28,94)(20,98,29,93)(21,97,30,92)(22,102,25,91)(23,101,26,96)(24,100,27,95)(31,130,41,79)(32,129,42,84)(33,128,37,83)(34,127,38,82)(35,132,39,81)(36,131,40,80)(49,141,69,124)(50,140,70,123)(51,139,71,122)(52,144,72,121)(53,143,67,126)(54,142,68,125)(55,134,64,110)(56,133,65,109)(57,138,66,114)(58,137,61,113)(59,136,62,112)(60,135,63,111), (7,120)(8,115)(9,116)(10,117)(11,118)(12,119)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27)(43,77)(44,78)(45,73)(46,74)(47,75)(48,76)(49,69)(50,70)(51,71)(52,72)(53,67)(54,68)(91,102)(92,97)(93,98)(94,99)(95,100)(96,101)(121,144)(122,139)(123,140)(124,141)(125,142)(126,143), (1,70,4,67)(2,71,5,68)(3,72,6,69)(7,129,10,132)(8,130,11,127)(9,131,12,128)(13,54,16,51)(14,49,17,52)(15,50,18,53)(19,135,22,138)(20,136,23,133)(21,137,24,134)(25,114,28,111)(26,109,29,112)(27,110,30,113)(31,44,34,47)(32,45,35,48)(33,46,36,43)(37,74,40,77)(38,75,41,78)(39,76,42,73)(55,92,58,95)(56,93,59,96)(57,94,60,91)(61,100,64,97)(62,101,65,98)(63,102,66,99)(79,118,82,115)(80,119,83,116)(81,120,84,117)(85,124,88,121)(86,125,89,122)(87,126,90,123)(103,142,106,139)(104,143,107,140)(105,144,108,141) );
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)], [(1,35,58,15,39,61),(2,36,59,16,40,62),(3,31,60,17,41,63),(4,32,55,18,42,64),(5,33,56,13,37,65),(6,34,57,14,38,66),(7,123,27,120,140,24),(8,124,28,115,141,19),(9,125,29,116,142,20),(10,126,30,117,143,21),(11,121,25,118,144,22),(12,122,26,119,139,23),(43,96,51,77,101,71),(44,91,52,78,102,72),(45,92,53,73,97,67),(46,93,54,74,98,68),(47,94,49,75,99,69),(48,95,50,76,100,70),(79,105,135,130,88,111),(80,106,136,131,89,112),(81,107,137,132,90,113),(82,108,138,127,85,114),(83,103,133,128,86,109),(84,104,134,129,87,110)], [(1,90,15,107),(2,89,16,106),(3,88,17,105),(4,87,18,104),(5,86,13,103),(6,85,14,108),(7,76,120,48),(8,75,115,47),(9,74,116,46),(10,73,117,45),(11,78,118,44),(12,77,119,43),(19,99,28,94),(20,98,29,93),(21,97,30,92),(22,102,25,91),(23,101,26,96),(24,100,27,95),(31,130,41,79),(32,129,42,84),(33,128,37,83),(34,127,38,82),(35,132,39,81),(36,131,40,80),(49,141,69,124),(50,140,70,123),(51,139,71,122),(52,144,72,121),(53,143,67,126),(54,142,68,125),(55,134,64,110),(56,133,65,109),(57,138,66,114),(58,137,61,113),(59,136,62,112),(60,135,63,111)], [(7,120),(8,115),(9,116),(10,117),(11,118),(12,119),(19,28),(20,29),(21,30),(22,25),(23,26),(24,27),(43,77),(44,78),(45,73),(46,74),(47,75),(48,76),(49,69),(50,70),(51,71),(52,72),(53,67),(54,68),(91,102),(92,97),(93,98),(94,99),(95,100),(96,101),(121,144),(122,139),(123,140),(124,141),(125,142),(126,143)], [(1,70,4,67),(2,71,5,68),(3,72,6,69),(7,129,10,132),(8,130,11,127),(9,131,12,128),(13,54,16,51),(14,49,17,52),(15,50,18,53),(19,135,22,138),(20,136,23,133),(21,137,24,134),(25,114,28,111),(26,109,29,112),(27,110,30,113),(31,44,34,47),(32,45,35,48),(33,46,36,43),(37,74,40,77),(38,75,41,78),(39,76,42,73),(55,92,58,95),(56,93,59,96),(57,94,60,91),(61,100,64,97),(62,101,65,98),(63,102,66,99),(79,118,82,115),(80,119,83,116),(81,120,84,117),(85,124,88,121),(86,125,89,122),(87,126,90,123),(103,142,106,139),(104,143,107,140),(105,144,108,141)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | ··· | 6L | 6M | ··· | 6T | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | C4○D4 | C4×S3 | D4⋊2S3 |
| kernel | C62.221C23 | C4×C3⋊Dic3 | C6.Dic6 | C62⋊5C4 | C32×C22⋊C4 | C22×C3⋊Dic3 | C2×C3⋊Dic3 | C3×C22⋊C4 | C2×C12 | C22×C6 | C3×C6 | C2×C6 | C6 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 8 | 4 | 4 | 16 | 8 |
Matrix representation of C62.221C23 ►in GL8(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6],[1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,12,0,0,0,0,0,0,11,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12] >;
C62.221C23 in GAP, Magma, Sage, TeX
C_6^2._{221}C_2^3 % in TeX
G:=Group("C6^2.221C2^3"); // GroupNames label
G:=SmallGroup(288,734);
// by ID
G=gap.SmallGroup(288,734);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,219,58,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=d^2=1,c^2=b^3,e^2=a^3,a*b=b*a,c*a*c^-1=a^-1,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=b^3*d>;
// generators/relations