metabelian, supersoluble, monomial
Aliases: C62.134D4, (C6×Q8)⋊6S3, (C3×Q8).65D6, (C2×C12).157D6, (C3×C12).102D4, C32⋊7Q16⋊9C2, C32⋊11SD16⋊9C2, C12.61(C3⋊D4), C12.58D6⋊13C2, C3⋊5(Q8.11D6), (C6×C12).148C22, C12.101(C22×S3), (C3×C12).105C23, C12.59D6.8C2, C4.17(C32⋊7D4), C32⋊23(C8.C22), C12⋊S3.30C22, C32⋊4C8.17C22, (Q8×C32).27C22, C32⋊4Q8.30C22, C22.11(C32⋊7D4), (Q8×C3×C6)⋊6C2, (C2×Q8)⋊4(C3⋊S3), Q8.11(C2×C3⋊S3), (C3×C6).288(C2×D4), C6.129(C2×C3⋊D4), C4.15(C22×C3⋊S3), C2.18(C2×C32⋊7D4), (C2×C6).102(C3⋊D4), (C2×C4).20(C2×C3⋊S3), SmallGroup(288,799)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.134D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=dad=a-1b3, cbc-1=dbd=b-1, dcd=c3 >
Subgroups: 620 in 180 conjugacy classes, 65 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8, C3×Q8, C8.C22, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C62, C4.Dic3, Q8⋊2S3, C3⋊Q16, C4○D12, C6×Q8, C32⋊4C8, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C6×C12, Q8×C32, Q8×C32, Q8.11D6, C12.58D6, C32⋊11SD16, C32⋊7Q16, C12.59D6, Q8×C3×C6, C62.134D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C3⋊D4, C22×S3, C8.C22, C2×C3⋊S3, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, Q8.11D6, C2×C32⋊7D4, C62.134D4
►On 144 points
Generators in S144
(1 38 99 18 68 15)(2 12 69 23 100 35)(3 40 101 20 70 9)(4 14 71 17 102 37)(5 34 103 22 72 11)(6 16 65 19 104 39)(7 36 97 24 66 13)(8 10 67 21 98 33)(25 116 74 90 62 110)(26 107 63 95 75 113)(27 118 76 92 64 112)(28 109 57 89 77 115)(29 120 78 94 58 106)(30 111 59 91 79 117)(31 114 80 96 60 108)(32 105 61 93 73 119)(41 83 52 127 140 130)(42 135 141 124 53 88)(43 85 54 121 142 132)(44 129 143 126 55 82)(45 87 56 123 144 134)(46 131 137 128 49 84)(47 81 50 125 138 136)(48 133 139 122 51 86)
(1 130 96 5 134 92)(2 93 135 6 89 131)(3 132 90 7 136 94)(4 95 129 8 91 133)(9 142 74 13 138 78)(10 79 139 14 75 143)(11 144 76 15 140 80)(12 73 141 16 77 137)(17 26 55 21 30 51)(18 52 31 22 56 27)(19 28 49 23 32 53)(20 54 25 24 50 29)(33 59 48 37 63 44)(34 45 64 38 41 60)(35 61 42 39 57 46)(36 47 58 40 43 62)(65 115 128 69 119 124)(66 125 120 70 121 116)(67 117 122 71 113 126)(68 127 114 72 123 118)(81 106 101 85 110 97)(82 98 111 86 102 107)(83 108 103 87 112 99)(84 100 105 88 104 109)
(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 4)(3 7)(6 8)(9 40)(10 35)(11 38)(12 33)(13 36)(14 39)(15 34)(16 37)(17 19)(18 22)(21 23)(25 50)(26 53)(27 56)(28 51)(29 54)(30 49)(31 52)(32 55)(41 80)(42 75)(43 78)(44 73)(45 76)(46 79)(47 74)(48 77)(57 139)(58 142)(59 137)(60 140)(61 143)(62 138)(63 141)(64 144)(65 98)(66 101)(67 104)(68 99)(69 102)(70 97)(71 100)(72 103)(81 120)(82 115)(83 118)(84 113)(85 116)(86 119)(87 114)(88 117)(89 129)(90 132)(91 135)(92 130)(93 133)(94 136)(95 131)(96 134)(105 122)(106 125)(107 128)(108 123)(109 126)(110 121)(111 124)(112 127)
G:=sub<Sym(144)| (1,38,99,18,68,15)(2,12,69,23,100,35)(3,40,101,20,70,9)(4,14,71,17,102,37)(5,34,103,22,72,11)(6,16,65,19,104,39)(7,36,97,24,66,13)(8,10,67,21,98,33)(25,116,74,90,62,110)(26,107,63,95,75,113)(27,118,76,92,64,112)(28,109,57,89,77,115)(29,120,78,94,58,106)(30,111,59,91,79,117)(31,114,80,96,60,108)(32,105,61,93,73,119)(41,83,52,127,140,130)(42,135,141,124,53,88)(43,85,54,121,142,132)(44,129,143,126,55,82)(45,87,56,123,144,134)(46,131,137,128,49,84)(47,81,50,125,138,136)(48,133,139,122,51,86), (1,130,96,5,134,92)(2,93,135,6,89,131)(3,132,90,7,136,94)(4,95,129,8,91,133)(9,142,74,13,138,78)(10,79,139,14,75,143)(11,144,76,15,140,80)(12,73,141,16,77,137)(17,26,55,21,30,51)(18,52,31,22,56,27)(19,28,49,23,32,53)(20,54,25,24,50,29)(33,59,48,37,63,44)(34,45,64,38,41,60)(35,61,42,39,57,46)(36,47,58,40,43,62)(65,115,128,69,119,124)(66,125,120,70,121,116)(67,117,122,71,113,126)(68,127,114,72,123,118)(81,106,101,85,110,97)(82,98,111,86,102,107)(83,108,103,87,112,99)(84,100,105,88,104,109), (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), (2,4)(3,7)(6,8)(9,40)(10,35)(11,38)(12,33)(13,36)(14,39)(15,34)(16,37)(17,19)(18,22)(21,23)(25,50)(26,53)(27,56)(28,51)(29,54)(30,49)(31,52)(32,55)(41,80)(42,75)(43,78)(44,73)(45,76)(46,79)(47,74)(48,77)(57,139)(58,142)(59,137)(60,140)(61,143)(62,138)(63,141)(64,144)(65,98)(66,101)(67,104)(68,99)(69,102)(70,97)(71,100)(72,103)(81,120)(82,115)(83,118)(84,113)(85,116)(86,119)(87,114)(88,117)(89,129)(90,132)(91,135)(92,130)(93,133)(94,136)(95,131)(96,134)(105,122)(106,125)(107,128)(108,123)(109,126)(110,121)(111,124)(112,127)>;
G:=Group( (1,38,99,18,68,15)(2,12,69,23,100,35)(3,40,101,20,70,9)(4,14,71,17,102,37)(5,34,103,22,72,11)(6,16,65,19,104,39)(7,36,97,24,66,13)(8,10,67,21,98,33)(25,116,74,90,62,110)(26,107,63,95,75,113)(27,118,76,92,64,112)(28,109,57,89,77,115)(29,120,78,94,58,106)(30,111,59,91,79,117)(31,114,80,96,60,108)(32,105,61,93,73,119)(41,83,52,127,140,130)(42,135,141,124,53,88)(43,85,54,121,142,132)(44,129,143,126,55,82)(45,87,56,123,144,134)(46,131,137,128,49,84)(47,81,50,125,138,136)(48,133,139,122,51,86), (1,130,96,5,134,92)(2,93,135,6,89,131)(3,132,90,7,136,94)(4,95,129,8,91,133)(9,142,74,13,138,78)(10,79,139,14,75,143)(11,144,76,15,140,80)(12,73,141,16,77,137)(17,26,55,21,30,51)(18,52,31,22,56,27)(19,28,49,23,32,53)(20,54,25,24,50,29)(33,59,48,37,63,44)(34,45,64,38,41,60)(35,61,42,39,57,46)(36,47,58,40,43,62)(65,115,128,69,119,124)(66,125,120,70,121,116)(67,117,122,71,113,126)(68,127,114,72,123,118)(81,106,101,85,110,97)(82,98,111,86,102,107)(83,108,103,87,112,99)(84,100,105,88,104,109), (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), (2,4)(3,7)(6,8)(9,40)(10,35)(11,38)(12,33)(13,36)(14,39)(15,34)(16,37)(17,19)(18,22)(21,23)(25,50)(26,53)(27,56)(28,51)(29,54)(30,49)(31,52)(32,55)(41,80)(42,75)(43,78)(44,73)(45,76)(46,79)(47,74)(48,77)(57,139)(58,142)(59,137)(60,140)(61,143)(62,138)(63,141)(64,144)(65,98)(66,101)(67,104)(68,99)(69,102)(70,97)(71,100)(72,103)(81,120)(82,115)(83,118)(84,113)(85,116)(86,119)(87,114)(88,117)(89,129)(90,132)(91,135)(92,130)(93,133)(94,136)(95,131)(96,134)(105,122)(106,125)(107,128)(108,123)(109,126)(110,121)(111,124)(112,127) );
G=PermutationGroup([[(1,38,99,18,68,15),(2,12,69,23,100,35),(3,40,101,20,70,9),(4,14,71,17,102,37),(5,34,103,22,72,11),(6,16,65,19,104,39),(7,36,97,24,66,13),(8,10,67,21,98,33),(25,116,74,90,62,110),(26,107,63,95,75,113),(27,118,76,92,64,112),(28,109,57,89,77,115),(29,120,78,94,58,106),(30,111,59,91,79,117),(31,114,80,96,60,108),(32,105,61,93,73,119),(41,83,52,127,140,130),(42,135,141,124,53,88),(43,85,54,121,142,132),(44,129,143,126,55,82),(45,87,56,123,144,134),(46,131,137,128,49,84),(47,81,50,125,138,136),(48,133,139,122,51,86)], [(1,130,96,5,134,92),(2,93,135,6,89,131),(3,132,90,7,136,94),(4,95,129,8,91,133),(9,142,74,13,138,78),(10,79,139,14,75,143),(11,144,76,15,140,80),(12,73,141,16,77,137),(17,26,55,21,30,51),(18,52,31,22,56,27),(19,28,49,23,32,53),(20,54,25,24,50,29),(33,59,48,37,63,44),(34,45,64,38,41,60),(35,61,42,39,57,46),(36,47,58,40,43,62),(65,115,128,69,119,124),(66,125,120,70,121,116),(67,117,122,71,113,126),(68,127,114,72,123,118),(81,106,101,85,110,97),(82,98,111,86,102,107),(83,108,103,87,112,99),(84,100,105,88,104,109)], [(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)], [(2,4),(3,7),(6,8),(9,40),(10,35),(11,38),(12,33),(13,36),(14,39),(15,34),(16,37),(17,19),(18,22),(21,23),(25,50),(26,53),(27,56),(28,51),(29,54),(30,49),(31,52),(32,55),(41,80),(42,75),(43,78),(44,73),(45,76),(46,79),(47,74),(48,77),(57,139),(58,142),(59,137),(60,140),(61,143),(62,138),(63,141),(64,144),(65,98),(66,101),(67,104),(68,99),(69,102),(70,97),(71,100),(72,103),(81,120),(82,115),(83,118),(84,113),(85,116),(86,119),(87,114),(88,117),(89,129),(90,132),(91,135),(92,130),(93,133),(94,136),(95,131),(96,134),(105,122),(106,125),(107,128),(108,123),(109,126),(110,121),(111,124),(112,127)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6L | 8A | 8B | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 36 | 2 | ··· | 2 | 36 | 36 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C3⋊D4 | C3⋊D4 | C8.C22 | Q8.11D6 |
| kernel | C62.134D4 | C12.58D6 | C32⋊11SD16 | C32⋊7Q16 | C12.59D6 | Q8×C3×C6 | C6×Q8 | C3×C12 | C62 | C2×C12 | C3×Q8 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 1 | 1 | 4 | 8 | 8 | 8 | 1 | 8 |
Matrix representation of C62.134D4 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 30 | 0 | 0 |
| 0 | 0 | 43 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 30 |
| 0 | 0 | 0 | 0 | 43 | 30 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 62 | 45 | 12 |
| 0 | 0 | 11 | 22 | 40 | 28 |
| 0 | 0 | 63 | 46 | 19 | 0 |
| 0 | 0 | 56 | 10 | 54 | 54 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 25 | 25 | 72 | 0 |
| 0 | 0 | 0 | 48 | 1 | 1 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,60,43,0,0,0,0,30,30,0,0,0,0,0,0,60,43,0,0,0,0,30,30],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,51,11,63,56,0,0,62,22,46,10,0,0,45,40,19,54,0,0,12,28,0,54],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,25,0,0,0,1,0,25,48,0,0,0,0,72,1,0,0,0,0,0,1] >;
C62.134D4 in GAP, Magma, Sage, TeX
C_6^2._{134}D_4 % in TeX
G:=Group("C6^2.134D4"); // GroupNames label
G:=SmallGroup(288,799);
// by ID
G=gap.SmallGroup(288,799);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,100,675,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=d^2=1,c^4=b^3,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=c^3>;
// generators/relations