metabelian, supersoluble, monomial
Aliases: C62.229C23, C6.111(S3×D4), (C2×C12).209D6, C3⋊Dic3.46D4, (C22×C6).91D6, C62⋊5C4⋊13C2, (C6×C12).14C22, C6.100(C4○D12), C6.11D12⋊22C2, C6.96(D4⋊2S3), C32⋊14(C4.4D4), (C2×C62).68C22, C3⋊5(C23.11D6), C2.9(C12.D6), C2.12(C12.59D6), C2.10(D4×C3⋊S3), (C3×C22⋊C4)⋊7S3, C22⋊C4⋊5(C3⋊S3), (C4×C3⋊Dic3)⋊23C2, (C3×C6).233(C2×D4), C23.11(C2×C3⋊S3), (C2×C32⋊4Q8)⋊5C2, (C32×C22⋊C4)⋊8C2, (C3×C6).116(C4○D4), (C2×C6).246(C22×S3), (C2×C32⋊7D4).11C2, C22.44(C22×C3⋊S3), (C22×C3⋊S3).41C22, (C2×C3⋊Dic3).81C22, (C2×C4).29(C2×C3⋊S3), SmallGroup(288,742)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.229C23
G = < a,b,c,d,e | a6=b6=c2=e2=1, d2=a3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ece=a3b3c, ede=b3d >
Subgroups: 908 in 228 conjugacy classes, 67 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C2×D4, C2×Q8, C3⋊S3, C3×C6, C3×C6, Dic6, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4.4D4, C3⋊Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C62, C4×Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C32⋊4Q8, C2×C3⋊Dic3, C32⋊7D4, C6×C12, C22×C3⋊S3, C2×C62, C23.11D6, C4×C3⋊Dic3, C6.11D12, C62⋊5C4, C32×C22⋊C4, C2×C32⋊4Q8, C2×C32⋊7D4, C62.229C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊S3, C22×S3, C4.4D4, C2×C3⋊S3, C4○D12, S3×D4, D4⋊2S3, C22×C3⋊S3, C23.11D6, C12.59D6, D4×C3⋊S3, C12.D6, C62.229C23
►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 28)(8 27)(9 26)(10 25)(11 30)(12 29)(13 17)(14 16)(19 141)(20 140)(21 139)(22 144)(23 143)(24 142)(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)(79 89)(80 88)(81 87)(82 86)(83 85)(84 90)(103 131)(104 130)(105 129)(106 128)(107 127)(108 132)(109 135)(110 134)(111 133)(112 138)(113 137)(114 136)(115 118)(116 117)(119 120)(121 126)(122 125)(123 124)
(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 46)(2 47)(3 48)(4 43)(5 44)(6 45)(7 103)(8 104)(9 105)(10 106)(11 107)(12 108)(13 51)(14 52)(15 53)(16 54)(17 49)(18 50)(19 132)(20 127)(21 128)(22 129)(23 130)(24 131)(25 134)(26 135)(27 136)(28 137)(29 138)(30 133)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(79 125)(80 126)(81 121)(82 122)(83 123)(84 124)(85 117)(86 118)(87 119)(88 120)(89 115)(90 116)(109 144)(110 139)(111 140)(112 141)(113 142)(114 143)
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), (1,32,58,18,39,61)(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)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,17)(14,16)(19,141)(20,140)(21,139)(22,144)(23,143)(24,142)(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)(79,89)(80,88)(81,87)(82,86)(83,85)(84,90)(103,131)(104,130)(105,129)(106,128)(107,127)(108,132)(109,135)(110,134)(111,133)(112,138)(113,137)(114,136)(115,118)(116,117)(119,120)(121,126)(122,125)(123,124), (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), (1,46)(2,47)(3,48)(4,43)(5,44)(6,45)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,51)(14,52)(15,53)(16,54)(17,49)(18,50)(19,132)(20,127)(21,128)(22,129)(23,130)(24,131)(25,134)(26,135)(27,136)(28,137)(29,138)(30,133)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(79,125)(80,126)(81,121)(82,122)(83,123)(84,124)(85,117)(86,118)(87,119)(88,120)(89,115)(90,116)(109,144)(110,139)(111,140)(112,141)(113,142)(114,143)>;
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), (1,32,58,18,39,61)(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)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,17)(14,16)(19,141)(20,140)(21,139)(22,144)(23,143)(24,142)(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)(79,89)(80,88)(81,87)(82,86)(83,85)(84,90)(103,131)(104,130)(105,129)(106,128)(107,127)(108,132)(109,135)(110,134)(111,133)(112,138)(113,137)(114,136)(115,118)(116,117)(119,120)(121,126)(122,125)(123,124), (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), (1,46)(2,47)(3,48)(4,43)(5,44)(6,45)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,51)(14,52)(15,53)(16,54)(17,49)(18,50)(19,132)(20,127)(21,128)(22,129)(23,130)(24,131)(25,134)(26,135)(27,136)(28,137)(29,138)(30,133)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(79,125)(80,126)(81,121)(82,122)(83,123)(84,124)(85,117)(86,118)(87,119)(88,120)(89,115)(90,116)(109,144)(110,139)(111,140)(112,141)(113,142)(114,143) );
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)], [(1,32,58,18,39,61),(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),(7,28),(8,27),(9,26),(10,25),(11,30),(12,29),(13,17),(14,16),(19,141),(20,140),(21,139),(22,144),(23,143),(24,142),(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),(79,89),(80,88),(81,87),(82,86),(83,85),(84,90),(103,131),(104,130),(105,129),(106,128),(107,127),(108,132),(109,135),(110,134),(111,133),(112,138),(113,137),(114,136),(115,118),(116,117),(119,120),(121,126),(122,125),(123,124)], [(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)], [(1,46),(2,47),(3,48),(4,43),(5,44),(6,45),(7,103),(8,104),(9,105),(10,106),(11,107),(12,108),(13,51),(14,52),(15,53),(16,54),(17,49),(18,50),(19,132),(20,127),(21,128),(22,129),(23,130),(24,131),(25,134),(26,135),(27,136),(28,137),(29,138),(30,133),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(79,125),(80,126),(81,121),(82,122),(83,123),(84,124),(85,117),(86,118),(87,119),(88,120),(89,115),(90,116),(109,144),(110,139),(111,140),(112,141),(113,142),(114,143)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 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 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 18 | 18 | 18 | 18 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | C4○D12 | S3×D4 | D4⋊2S3 |
| kernel | C62.229C23 | C4×C3⋊Dic3 | C6.11D12 | C62⋊5C4 | C32×C22⋊C4 | C2×C32⋊4Q8 | C2×C32⋊7D4 | C3×C22⋊C4 | C3⋊Dic3 | C2×C12 | C22×C6 | C3×C6 | C6 | C6 | C6 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 8 | 4 | 4 | 16 | 4 | 4 |
Matrix representation of C62.229C23 ►in GL6(𝔽13)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 1 | 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 |
| 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 10 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 7 |
| 0 | 0 | 0 | 0 | 5 | 9 |
| 11 | 9 | 0 | 0 | 0 | 0 |
| 4 | 2 | 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 | 3 | 1 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,10,0,0,0,0,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,5,0,0,0,0,7,9],[11,4,0,0,0,0,9,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,3,0,0,0,0,0,1] >;
C62.229C23 in GAP, Magma, Sage, TeX
C_6^2._{229}C_2^3 % in TeX
G:=Group("C6^2.229C2^3"); // GroupNames label
G:=SmallGroup(288,742);
// by ID
G=gap.SmallGroup(288,742);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,590,219,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=c^2=e^2=1,d^2=a^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=a^3*b^3*c,e*d*e=b^3*d>;
// generators/relations