metabelian, supersoluble, monomial
Aliases: C62.242C23, (C2×C12).36D6, (C6×C12).19C22, C6.104(C4○D12), C12⋊Dic3⋊10C2, C6.52(Q8⋊3S3), C6.Dic6⋊24C2, C6.103(D4⋊2S3), C32⋊13(C42⋊2C2), C6.11D12.10C2, C2.7(C12.26D6), C2.14(C12.D6), C2.16(C12.59D6), (C3×C4⋊C4)⋊9S3, C4⋊C4⋊6(C3⋊S3), C3⋊6(C4⋊C4⋊S3), (C32×C4⋊C4)⋊18C2, (C4×C3⋊Dic3)⋊25C2, (C3×C6).119(C4○D4), (C2×C6).259(C22×S3), C22.53(C22×C3⋊S3), (C22×C3⋊S3).43C22, (C2×C3⋊Dic3).88C22, (C2×C4).33(C2×C3⋊S3), SmallGroup(288,755)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.242C23
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, dcd-1=b3c, ece-1=a3c, ede-1=b3d >
Subgroups: 684 in 180 conjugacy classes, 65 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C3⋊S3, C3×C6, C2×Dic3, C2×C12, C22×S3, C42⋊2C2, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C4⋊C4⋊S3, C4×C3⋊Dic3, C6.Dic6, C12⋊Dic3, C6.11D12, C32×C4⋊C4, C62.242C23
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, C42⋊2C2, C2×C3⋊S3, C4○D12, D4⋊2S3, Q8⋊3S3, C22×C3⋊S3, C4⋊C4⋊S3, C12.59D6, C12.D6, C12.26D6, C62.242C23
►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 15 58 33 39 61)(2 16 59 34 40 62)(3 17 60 35 41 63)(4 18 55 36 42 64)(5 13 56 31 37 65)(6 14 57 32 38 66)(7 22 123 116 29 140)(8 23 124 117 30 141)(9 24 125 118 25 142)(10 19 126 119 26 143)(11 20 121 120 27 144)(12 21 122 115 28 139)(43 92 67 73 101 50)(44 93 68 74 102 51)(45 94 69 75 97 52)(46 95 70 76 98 53)(47 96 71 77 99 54)(48 91 72 78 100 49)(79 128 103 109 137 86)(80 129 104 110 138 87)(81 130 105 111 133 88)(82 131 106 112 134 89)(83 132 107 113 135 90)(84 127 108 114 136 85)
(2 6)(3 5)(7 21)(8 20)(9 19)(10 24)(11 23)(12 22)(13 63)(14 62)(15 61)(16 66)(17 65)(18 64)(25 119)(26 118)(27 117)(28 116)(29 115)(30 120)(31 35)(32 34)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 52)(50 51)(53 54)(67 68)(69 72)(70 71)(73 93)(74 92)(75 91)(76 96)(77 95)(78 94)(79 132)(80 131)(81 130)(82 129)(83 128)(84 127)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(109 135)(110 134)(111 133)(112 138)(113 137)(114 136)(121 141)(122 140)(123 139)(124 144)(125 143)(126 142)
(1 85 4 88)(2 86 5 89)(3 87 6 90)(7 74 10 77)(8 75 11 78)(9 76 12 73)(13 82 16 79)(14 83 17 80)(15 84 18 81)(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 69 33 52)(2 70 34 53)(3 71 35 54)(4 72 36 49)(5 67 31 50)(6 68 32 51)(7 80 116 110)(8 81 117 111)(9 82 118 112)(10 83 119 113)(11 84 120 114)(12 79 115 109)(13 73 37 43)(14 74 38 44)(15 75 39 45)(16 76 40 46)(17 77 41 47)(18 78 42 48)(19 132 26 135)(20 127 27 136)(21 128 28 137)(22 129 29 138)(23 130 30 133)(24 131 25 134)(55 100 64 91)(56 101 65 92)(57 102 66 93)(58 97 61 94)(59 98 62 95)(60 99 63 96)(85 121 108 144)(86 122 103 139)(87 123 104 140)(88 124 105 141)(89 125 106 142)(90 126 107 143)
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), (1,15,58,33,39,61)(2,16,59,34,40,62)(3,17,60,35,41,63)(4,18,55,36,42,64)(5,13,56,31,37,65)(6,14,57,32,38,66)(7,22,123,116,29,140)(8,23,124,117,30,141)(9,24,125,118,25,142)(10,19,126,119,26,143)(11,20,121,120,27,144)(12,21,122,115,28,139)(43,92,67,73,101,50)(44,93,68,74,102,51)(45,94,69,75,97,52)(46,95,70,76,98,53)(47,96,71,77,99,54)(48,91,72,78,100,49)(79,128,103,109,137,86)(80,129,104,110,138,87)(81,130,105,111,133,88)(82,131,106,112,134,89)(83,132,107,113,135,90)(84,127,108,114,136,85), (2,6)(3,5)(7,21)(8,20)(9,19)(10,24)(11,23)(12,22)(13,63)(14,62)(15,61)(16,66)(17,65)(18,64)(25,119)(26,118)(27,117)(28,116)(29,115)(30,120)(31,35)(32,34)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,52)(50,51)(53,54)(67,68)(69,72)(70,71)(73,93)(74,92)(75,91)(76,96)(77,95)(78,94)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(109,135)(110,134)(111,133)(112,138)(113,137)(114,136)(121,141)(122,140)(123,139)(124,144)(125,143)(126,142), (1,85,4,88)(2,86,5,89)(3,87,6,90)(7,74,10,77)(8,75,11,78)(9,76,12,73)(13,82,16,79)(14,83,17,80)(15,84,18,81)(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), (1,69,33,52)(2,70,34,53)(3,71,35,54)(4,72,36,49)(5,67,31,50)(6,68,32,51)(7,80,116,110)(8,81,117,111)(9,82,118,112)(10,83,119,113)(11,84,120,114)(12,79,115,109)(13,73,37,43)(14,74,38,44)(15,75,39,45)(16,76,40,46)(17,77,41,47)(18,78,42,48)(19,132,26,135)(20,127,27,136)(21,128,28,137)(22,129,29,138)(23,130,30,133)(24,131,25,134)(55,100,64,91)(56,101,65,92)(57,102,66,93)(58,97,61,94)(59,98,62,95)(60,99,63,96)(85,121,108,144)(86,122,103,139)(87,123,104,140)(88,124,105,141)(89,125,106,142)(90,126,107,143)>;
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), (1,15,58,33,39,61)(2,16,59,34,40,62)(3,17,60,35,41,63)(4,18,55,36,42,64)(5,13,56,31,37,65)(6,14,57,32,38,66)(7,22,123,116,29,140)(8,23,124,117,30,141)(9,24,125,118,25,142)(10,19,126,119,26,143)(11,20,121,120,27,144)(12,21,122,115,28,139)(43,92,67,73,101,50)(44,93,68,74,102,51)(45,94,69,75,97,52)(46,95,70,76,98,53)(47,96,71,77,99,54)(48,91,72,78,100,49)(79,128,103,109,137,86)(80,129,104,110,138,87)(81,130,105,111,133,88)(82,131,106,112,134,89)(83,132,107,113,135,90)(84,127,108,114,136,85), (2,6)(3,5)(7,21)(8,20)(9,19)(10,24)(11,23)(12,22)(13,63)(14,62)(15,61)(16,66)(17,65)(18,64)(25,119)(26,118)(27,117)(28,116)(29,115)(30,120)(31,35)(32,34)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,52)(50,51)(53,54)(67,68)(69,72)(70,71)(73,93)(74,92)(75,91)(76,96)(77,95)(78,94)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(109,135)(110,134)(111,133)(112,138)(113,137)(114,136)(121,141)(122,140)(123,139)(124,144)(125,143)(126,142), (1,85,4,88)(2,86,5,89)(3,87,6,90)(7,74,10,77)(8,75,11,78)(9,76,12,73)(13,82,16,79)(14,83,17,80)(15,84,18,81)(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), (1,69,33,52)(2,70,34,53)(3,71,35,54)(4,72,36,49)(5,67,31,50)(6,68,32,51)(7,80,116,110)(8,81,117,111)(9,82,118,112)(10,83,119,113)(11,84,120,114)(12,79,115,109)(13,73,37,43)(14,74,38,44)(15,75,39,45)(16,76,40,46)(17,77,41,47)(18,78,42,48)(19,132,26,135)(20,127,27,136)(21,128,28,137)(22,129,29,138)(23,130,30,133)(24,131,25,134)(55,100,64,91)(56,101,65,92)(57,102,66,93)(58,97,61,94)(59,98,62,95)(60,99,63,96)(85,121,108,144)(86,122,103,139)(87,123,104,140)(88,124,105,141)(89,125,106,142)(90,126,107,143) );
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)], [(1,15,58,33,39,61),(2,16,59,34,40,62),(3,17,60,35,41,63),(4,18,55,36,42,64),(5,13,56,31,37,65),(6,14,57,32,38,66),(7,22,123,116,29,140),(8,23,124,117,30,141),(9,24,125,118,25,142),(10,19,126,119,26,143),(11,20,121,120,27,144),(12,21,122,115,28,139),(43,92,67,73,101,50),(44,93,68,74,102,51),(45,94,69,75,97,52),(46,95,70,76,98,53),(47,96,71,77,99,54),(48,91,72,78,100,49),(79,128,103,109,137,86),(80,129,104,110,138,87),(81,130,105,111,133,88),(82,131,106,112,134,89),(83,132,107,113,135,90),(84,127,108,114,136,85)], [(2,6),(3,5),(7,21),(8,20),(9,19),(10,24),(11,23),(12,22),(13,63),(14,62),(15,61),(16,66),(17,65),(18,64),(25,119),(26,118),(27,117),(28,116),(29,115),(30,120),(31,35),(32,34),(37,60),(38,59),(39,58),(40,57),(41,56),(42,55),(43,102),(44,101),(45,100),(46,99),(47,98),(48,97),(49,52),(50,51),(53,54),(67,68),(69,72),(70,71),(73,93),(74,92),(75,91),(76,96),(77,95),(78,94),(79,132),(80,131),(81,130),(82,129),(83,128),(84,127),(85,108),(86,107),(87,106),(88,105),(89,104),(90,103),(109,135),(110,134),(111,133),(112,138),(113,137),(114,136),(121,141),(122,140),(123,139),(124,144),(125,143),(126,142)], [(1,85,4,88),(2,86,5,89),(3,87,6,90),(7,74,10,77),(8,75,11,78),(9,76,12,73),(13,82,16,79),(14,83,17,80),(15,84,18,81),(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)], [(1,69,33,52),(2,70,34,53),(3,71,35,54),(4,72,36,49),(5,67,31,50),(6,68,32,51),(7,80,116,110),(8,81,117,111),(9,82,118,112),(10,83,119,113),(11,84,120,114),(12,79,115,109),(13,73,37,43),(14,74,38,44),(15,75,39,45),(16,76,40,46),(17,77,41,47),(18,78,42,48),(19,132,26,135),(20,127,27,136),(21,128,28,137),(22,129,29,138),(23,130,30,133),(24,131,25,134),(55,100,64,91),(56,101,65,92),(57,102,66,93),(58,97,61,94),(59,98,62,95),(60,99,63,96),(85,121,108,144),(86,122,103,139),(87,123,104,140),(88,124,105,141),(89,125,106,142),(90,126,107,143)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | ··· | 6L | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 18 | 18 | 36 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | C4○D4 | C4○D12 | D4⋊2S3 | Q8⋊3S3 |
| kernel | C62.242C23 | C4×C3⋊Dic3 | C6.Dic6 | C12⋊Dic3 | C6.11D12 | C32×C4⋊C4 | C3×C4⋊C4 | C2×C12 | C3×C6 | C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 3 | 1 | 4 | 12 | 6 | 16 | 4 | 4 |
Matrix representation of C62.242C23 ►in GL6(𝔽13)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 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 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 11 |
| 0 | 0 | 0 | 0 | 12 | 8 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 9 | 11 | 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 | 1 | 5 |
G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,5,0,0,0,0,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,12,0,0,0,0,11,8],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,1,0,0,0,0,0,5] >;
C62.242C23 in GAP, Magma, Sage, TeX
C_6^2._{242}C_2^3 % in TeX
G:=Group("C6^2.242C2^3"); // GroupNames label
G:=SmallGroup(288,755);
// by ID
G=gap.SmallGroup(288,755);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,254,219,100,2693,9414]);
// 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,d*c*d^-1=b^3*c,e*c*e^-1=a^3*c,e*d*e^-1=b^3*d>;
// generators/relations