metabelian, supersoluble, monomial
Aliases: C62.240C23, C6.46(S3×Q8), C6.117(S3×D4), C3⋊5(D6⋊Q8), (C2×C12).249D6, C3⋊Dic3.64D4, (C6×C12).17C22, C6.103(C4○D12), C32⋊20(C22⋊Q8), C6.Dic6⋊23C2, C6.11D12.2C2, C2.15(C12.59D6), (C3×C4⋊C4)⋊7S3, C4⋊C4⋊4(C3⋊S3), (C2×C3⋊S3)⋊7Q8, C2.6(Q8×C3⋊S3), C2.14(D4×C3⋊S3), (C3×C6).73(C2×Q8), (C32×C4⋊C4)⋊16C2, (C3×C6).238(C2×D4), (C2×C32⋊4Q8)⋊6C2, (C3×C6).118(C4○D4), (C2×C6).257(C22×S3), C22.51(C22×C3⋊S3), (C22×C3⋊S3).88C22, (C2×C3⋊Dic3).86C22, (C2×C4×C3⋊S3).23C2, (C2×C4).13(C2×C3⋊S3), SmallGroup(288,753)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.240C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=a3, e2=b3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=a3b3c, ede-1=b3d >
Subgroups: 860 in 222 conjugacy classes, 69 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, Q8, C23, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C3⋊S3, C3×C6, Dic6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22⋊Q8, C3⋊Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, D6⋊Q8, C6.Dic6, C6.11D12, C32×C4⋊C4, C2×C32⋊4Q8, C2×C4×C3⋊S3, C62.240C23
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C3⋊S3, C22×S3, C22⋊Q8, C2×C3⋊S3, C4○D12, S3×D4, S3×Q8, C22×C3⋊S3, D6⋊Q8, C12.59D6, D4×C3⋊S3, Q8×C3⋊S3, C62.240C23
►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 32 58 18 39 61)(2 33 59 13 40 62)(3 34 60 14 41 63)(4 35 55 15 42 64)(5 36 56 16 37 65)(6 31 57 17 38 66)(7 22 117 139 29 121)(8 23 118 140 30 122)(9 24 119 141 25 123)(10 19 120 142 26 124)(11 20 115 143 27 125)(12 21 116 144 28 126)(43 71 91 53 78 100)(44 72 92 54 73 101)(45 67 93 49 74 102)(46 68 94 50 75 97)(47 69 95 51 76 98)(48 70 96 52 77 99)(79 107 127 89 114 136)(80 108 128 90 109 137)(81 103 129 85 110 138)(82 104 130 86 111 133)(83 105 131 87 112 134)(84 106 132 88 113 135)
(2 6)(3 5)(7 21)(8 20)(9 19)(10 24)(11 23)(12 22)(13 17)(14 16)(25 142)(26 141)(27 140)(28 139)(29 144)(30 143)(31 62)(32 61)(33 66)(34 65)(35 64)(36 63)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 50)(44 49)(45 54)(46 53)(47 52)(48 51)(67 92)(68 91)(69 96)(70 95)(71 94)(72 93)(73 102)(74 101)(75 100)(76 99)(77 98)(78 97)(80 84)(81 83)(85 87)(88 90)(103 134)(104 133)(105 138)(106 137)(107 136)(108 135)(109 132)(110 131)(111 130)(112 129)(113 128)(114 127)(115 122)(116 121)(117 126)(118 125)(119 124)(120 123)
(1 79 4 82)(2 80 5 83)(3 81 6 84)(7 74 10 77)(8 75 11 78)(9 76 12 73)(13 90 16 87)(14 85 17 88)(15 86 18 89)(19 99 22 102)(20 100 23 97)(21 101 24 98)(25 95 28 92)(26 96 29 93)(27 91 30 94)(31 106 34 103)(32 107 35 104)(33 108 36 105)(37 112 40 109)(38 113 41 110)(39 114 42 111)(43 118 46 115)(44 119 47 116)(45 120 48 117)(49 124 52 121)(50 125 53 122)(51 126 54 123)(55 130 58 127)(56 131 59 128)(57 132 60 129)(61 136 64 133)(62 137 65 134)(63 138 66 135)(67 142 70 139)(68 143 71 140)(69 144 72 141)
(1 50 18 46)(2 51 13 47)(3 52 14 48)(4 53 15 43)(5 54 16 44)(6 49 17 45)(7 103 139 110)(8 104 140 111)(9 105 141 112)(10 106 142 113)(11 107 143 114)(12 108 144 109)(19 132 26 135)(20 127 27 136)(21 128 28 137)(22 129 29 138)(23 130 30 133)(24 131 25 134)(31 74 38 67)(32 75 39 68)(33 76 40 69)(34 77 41 70)(35 78 42 71)(36 73 37 72)(55 100 64 91)(56 101 65 92)(57 102 66 93)(58 97 61 94)(59 98 62 95)(60 99 63 96)(79 115 89 125)(80 116 90 126)(81 117 85 121)(82 118 86 122)(83 119 87 123)(84 120 88 124)
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), (1,32,58,18,39,61)(2,33,59,13,40,62)(3,34,60,14,41,63)(4,35,55,15,42,64)(5,36,56,16,37,65)(6,31,57,17,38,66)(7,22,117,139,29,121)(8,23,118,140,30,122)(9,24,119,141,25,123)(10,19,120,142,26,124)(11,20,115,143,27,125)(12,21,116,144,28,126)(43,71,91,53,78,100)(44,72,92,54,73,101)(45,67,93,49,74,102)(46,68,94,50,75,97)(47,69,95,51,76,98)(48,70,96,52,77,99)(79,107,127,89,114,136)(80,108,128,90,109,137)(81,103,129,85,110,138)(82,104,130,86,111,133)(83,105,131,87,112,134)(84,106,132,88,113,135), (2,6)(3,5)(7,21)(8,20)(9,19)(10,24)(11,23)(12,22)(13,17)(14,16)(25,142)(26,141)(27,140)(28,139)(29,144)(30,143)(31,62)(32,61)(33,66)(34,65)(35,64)(36,63)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,50)(44,49)(45,54)(46,53)(47,52)(48,51)(67,92)(68,91)(69,96)(70,95)(71,94)(72,93)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(80,84)(81,83)(85,87)(88,90)(103,134)(104,133)(105,138)(106,137)(107,136)(108,135)(109,132)(110,131)(111,130)(112,129)(113,128)(114,127)(115,122)(116,121)(117,126)(118,125)(119,124)(120,123), (1,79,4,82)(2,80,5,83)(3,81,6,84)(7,74,10,77)(8,75,11,78)(9,76,12,73)(13,90,16,87)(14,85,17,88)(15,86,18,89)(19,99,22,102)(20,100,23,97)(21,101,24,98)(25,95,28,92)(26,96,29,93)(27,91,30,94)(31,106,34,103)(32,107,35,104)(33,108,36,105)(37,112,40,109)(38,113,41,110)(39,114,42,111)(43,118,46,115)(44,119,47,116)(45,120,48,117)(49,124,52,121)(50,125,53,122)(51,126,54,123)(55,130,58,127)(56,131,59,128)(57,132,60,129)(61,136,64,133)(62,137,65,134)(63,138,66,135)(67,142,70,139)(68,143,71,140)(69,144,72,141), (1,50,18,46)(2,51,13,47)(3,52,14,48)(4,53,15,43)(5,54,16,44)(6,49,17,45)(7,103,139,110)(8,104,140,111)(9,105,141,112)(10,106,142,113)(11,107,143,114)(12,108,144,109)(19,132,26,135)(20,127,27,136)(21,128,28,137)(22,129,29,138)(23,130,30,133)(24,131,25,134)(31,74,38,67)(32,75,39,68)(33,76,40,69)(34,77,41,70)(35,78,42,71)(36,73,37,72)(55,100,64,91)(56,101,65,92)(57,102,66,93)(58,97,61,94)(59,98,62,95)(60,99,63,96)(79,115,89,125)(80,116,90,126)(81,117,85,121)(82,118,86,122)(83,119,87,123)(84,120,88,124)>;
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), (1,32,58,18,39,61)(2,33,59,13,40,62)(3,34,60,14,41,63)(4,35,55,15,42,64)(5,36,56,16,37,65)(6,31,57,17,38,66)(7,22,117,139,29,121)(8,23,118,140,30,122)(9,24,119,141,25,123)(10,19,120,142,26,124)(11,20,115,143,27,125)(12,21,116,144,28,126)(43,71,91,53,78,100)(44,72,92,54,73,101)(45,67,93,49,74,102)(46,68,94,50,75,97)(47,69,95,51,76,98)(48,70,96,52,77,99)(79,107,127,89,114,136)(80,108,128,90,109,137)(81,103,129,85,110,138)(82,104,130,86,111,133)(83,105,131,87,112,134)(84,106,132,88,113,135), (2,6)(3,5)(7,21)(8,20)(9,19)(10,24)(11,23)(12,22)(13,17)(14,16)(25,142)(26,141)(27,140)(28,139)(29,144)(30,143)(31,62)(32,61)(33,66)(34,65)(35,64)(36,63)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,50)(44,49)(45,54)(46,53)(47,52)(48,51)(67,92)(68,91)(69,96)(70,95)(71,94)(72,93)(73,102)(74,101)(75,100)(76,99)(77,98)(78,97)(80,84)(81,83)(85,87)(88,90)(103,134)(104,133)(105,138)(106,137)(107,136)(108,135)(109,132)(110,131)(111,130)(112,129)(113,128)(114,127)(115,122)(116,121)(117,126)(118,125)(119,124)(120,123), (1,79,4,82)(2,80,5,83)(3,81,6,84)(7,74,10,77)(8,75,11,78)(9,76,12,73)(13,90,16,87)(14,85,17,88)(15,86,18,89)(19,99,22,102)(20,100,23,97)(21,101,24,98)(25,95,28,92)(26,96,29,93)(27,91,30,94)(31,106,34,103)(32,107,35,104)(33,108,36,105)(37,112,40,109)(38,113,41,110)(39,114,42,111)(43,118,46,115)(44,119,47,116)(45,120,48,117)(49,124,52,121)(50,125,53,122)(51,126,54,123)(55,130,58,127)(56,131,59,128)(57,132,60,129)(61,136,64,133)(62,137,65,134)(63,138,66,135)(67,142,70,139)(68,143,71,140)(69,144,72,141), (1,50,18,46)(2,51,13,47)(3,52,14,48)(4,53,15,43)(5,54,16,44)(6,49,17,45)(7,103,139,110)(8,104,140,111)(9,105,141,112)(10,106,142,113)(11,107,143,114)(12,108,144,109)(19,132,26,135)(20,127,27,136)(21,128,28,137)(22,129,29,138)(23,130,30,133)(24,131,25,134)(31,74,38,67)(32,75,39,68)(33,76,40,69)(34,77,41,70)(35,78,42,71)(36,73,37,72)(55,100,64,91)(56,101,65,92)(57,102,66,93)(58,97,61,94)(59,98,62,95)(60,99,63,96)(79,115,89,125)(80,116,90,126)(81,117,85,121)(82,118,86,122)(83,119,87,123)(84,120,88,124) );
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)], [(1,32,58,18,39,61),(2,33,59,13,40,62),(3,34,60,14,41,63),(4,35,55,15,42,64),(5,36,56,16,37,65),(6,31,57,17,38,66),(7,22,117,139,29,121),(8,23,118,140,30,122),(9,24,119,141,25,123),(10,19,120,142,26,124),(11,20,115,143,27,125),(12,21,116,144,28,126),(43,71,91,53,78,100),(44,72,92,54,73,101),(45,67,93,49,74,102),(46,68,94,50,75,97),(47,69,95,51,76,98),(48,70,96,52,77,99),(79,107,127,89,114,136),(80,108,128,90,109,137),(81,103,129,85,110,138),(82,104,130,86,111,133),(83,105,131,87,112,134),(84,106,132,88,113,135)], [(2,6),(3,5),(7,21),(8,20),(9,19),(10,24),(11,23),(12,22),(13,17),(14,16),(25,142),(26,141),(27,140),(28,139),(29,144),(30,143),(31,62),(32,61),(33,66),(34,65),(35,64),(36,63),(37,60),(38,59),(39,58),(40,57),(41,56),(42,55),(43,50),(44,49),(45,54),(46,53),(47,52),(48,51),(67,92),(68,91),(69,96),(70,95),(71,94),(72,93),(73,102),(74,101),(75,100),(76,99),(77,98),(78,97),(80,84),(81,83),(85,87),(88,90),(103,134),(104,133),(105,138),(106,137),(107,136),(108,135),(109,132),(110,131),(111,130),(112,129),(113,128),(114,127),(115,122),(116,121),(117,126),(118,125),(119,124),(120,123)], [(1,79,4,82),(2,80,5,83),(3,81,6,84),(7,74,10,77),(8,75,11,78),(9,76,12,73),(13,90,16,87),(14,85,17,88),(15,86,18,89),(19,99,22,102),(20,100,23,97),(21,101,24,98),(25,95,28,92),(26,96,29,93),(27,91,30,94),(31,106,34,103),(32,107,35,104),(33,108,36,105),(37,112,40,109),(38,113,41,110),(39,114,42,111),(43,118,46,115),(44,119,47,116),(45,120,48,117),(49,124,52,121),(50,125,53,122),(51,126,54,123),(55,130,58,127),(56,131,59,128),(57,132,60,129),(61,136,64,133),(62,137,65,134),(63,138,66,135),(67,142,70,139),(68,143,71,140),(69,144,72,141)], [(1,50,18,46),(2,51,13,47),(3,52,14,48),(4,53,15,43),(5,54,16,44),(6,49,17,45),(7,103,139,110),(8,104,140,111),(9,105,141,112),(10,106,142,113),(11,107,143,114),(12,108,144,109),(19,132,26,135),(20,127,27,136),(21,128,28,137),(22,129,29,138),(23,130,30,133),(24,131,25,134),(31,74,38,67),(32,75,39,68),(33,76,40,69),(34,77,41,70),(35,78,42,71),(36,73,37,72),(55,100,64,91),(56,101,65,92),(57,102,66,93),(58,97,61,94),(59,98,62,95),(60,99,63,96),(79,115,89,125),(80,116,90,126),(81,117,85,121),(82,118,86,122),(83,119,87,123),(84,120,88,124)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6L | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | C4○D4 | C4○D12 | S3×D4 | S3×Q8 |
| kernel | C62.240C23 | C6.Dic6 | C6.11D12 | C32×C4⋊C4 | C2×C32⋊4Q8 | C2×C4×C3⋊S3 | C3×C4⋊C4 | C3⋊Dic3 | C2×C3⋊S3 | C2×C12 | C3×C6 | C6 | C6 | C6 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 12 | 2 | 16 | 4 | 4 |
Matrix representation of C62.240C23 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 10 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 3 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,11,1,0,0,0,0,10,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,5,0,0,0,0,0,5],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,3,5] >;
C62.240C23 in GAP, Magma, Sage, TeX
C_6^2._{240}C_2^3 % in TeX
G:=Group("C6^2.240C2^3"); // GroupNames label
G:=SmallGroup(288,753);
// by ID
G=gap.SmallGroup(288,753);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,219,100,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=c^2=1,d^2=a^3,e^2=b^3,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^3*b^3*c,e*d*e^-1=b^3*d>;
// generators/relations