metabelian, supersoluble, monomial
Aliases: C62.116D4, (C3×C6).41D8, (C3×D4)⋊1Dic3, (C6×D4).10S3, (C3×C12).56D4, (C2×C12).93D6, (D4×C32)⋊6C4, D4⋊1(C3⋊Dic3), C6.27(D4⋊S3), (C3×C6).24SD16, C3⋊3(D4⋊Dic3), C12.38(C3⋊D4), (C6×C12).60C22, C6.13(D4.S3), C12.13(C2×Dic3), C2.3(C32⋊7D8), C12⋊Dic3⋊14C2, C32⋊13(D4⋊C4), C2.3(C62⋊5C4), C4.12(C32⋊7D4), C2.3(C32⋊9SD16), C6.23(C6.D4), C22.17(C32⋊7D4), (D4×C3×C6).3C2, C4.1(C2×C3⋊Dic3), (C3×C12).52(C2×C4), (C2×D4).1(C3⋊S3), (C2×C32⋊4C8)⋊3C2, (C2×C6).92(C3⋊D4), (C3×C6).71(C22⋊C4), (C2×C4).40(C2×C3⋊S3), SmallGroup(288,307)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.116D4
G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a3b3c3 >
Subgroups: 452 in 150 conjugacy classes, 69 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×D4, C3×C6, C3×C6, C3⋊C8, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, D4⋊C4, C3⋊Dic3, C3×C12, C62, C62, C2×C3⋊C8, C4⋊Dic3, C6×D4, C32⋊4C8, C2×C3⋊Dic3, C6×C12, D4×C32, D4×C32, C2×C62, D4⋊Dic3, C2×C32⋊4C8, C12⋊Dic3, D4×C3×C6, C62.116D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D8, SD16, C3⋊S3, C2×Dic3, C3⋊D4, D4⋊C4, C3⋊Dic3, C2×C3⋊S3, D4⋊S3, D4.S3, C6.D4, C2×C3⋊Dic3, C32⋊7D4, D4⋊Dic3, C32⋊7D8, C32⋊9SD16, C62⋊5C4, C62.116D4
►On 144 points
Generators in S144
(1 128 141 26 108 113)(2 114 109 27 142 121)(3 122 143 28 110 115)(4 116 111 29 144 123)(5 124 137 30 112 117)(6 118 105 31 138 125)(7 126 139 32 106 119)(8 120 107 25 140 127)(9 100 41 65 78 129)(10 130 79 66 42 101)(11 102 43 67 80 131)(12 132 73 68 44 103)(13 104 45 69 74 133)(14 134 75 70 46 97)(15 98 47 71 76 135)(16 136 77 72 48 99)(17 89 81 59 50 34)(18 35 51 60 82 90)(19 91 83 61 52 36)(20 37 53 62 84 92)(21 93 85 63 54 38)(22 39 55 64 86 94)(23 95 87 57 56 40)(24 33 49 58 88 96)
(1 81 71 5 85 67)(2 68 86 6 72 82)(3 83 65 7 87 69)(4 70 88 8 66 84)(9 32 40 13 28 36)(10 37 29 14 33 25)(11 26 34 15 30 38)(12 39 31 16 35 27)(17 98 112 21 102 108)(18 109 103 22 105 99)(19 100 106 23 104 110)(20 111 97 24 107 101)(41 119 95 45 115 91)(42 92 116 46 96 120)(43 113 89 47 117 93)(44 94 118 48 90 114)(49 140 130 53 144 134)(50 135 137 54 131 141)(51 142 132 55 138 136)(52 129 139 56 133 143)(57 74 122 61 78 126)(58 127 79 62 123 75)(59 76 124 63 80 128)(60 121 73 64 125 77)
(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 25 26 8)(2 7 27 32)(3 31 28 6)(4 5 29 30)(9 68 65 12)(10 11 66 67)(13 72 69 16)(14 15 70 71)(17 58 59 24)(18 23 60 57)(19 64 61 22)(20 21 62 63)(33 34 88 81)(35 40 82 87)(36 86 83 39)(37 38 84 85)(41 132 129 44)(42 43 130 131)(45 136 133 48)(46 47 134 135)(49 50 96 89)(51 56 90 95)(52 94 91 55)(53 54 92 93)(73 78 103 100)(74 99 104 77)(75 76 97 98)(79 80 101 102)(105 110 125 122)(106 121 126 109)(107 108 127 128)(111 112 123 124)(113 140 141 120)(114 119 142 139)(115 138 143 118)(116 117 144 137)
G:=sub<Sym(144)| (1,128,141,26,108,113)(2,114,109,27,142,121)(3,122,143,28,110,115)(4,116,111,29,144,123)(5,124,137,30,112,117)(6,118,105,31,138,125)(7,126,139,32,106,119)(8,120,107,25,140,127)(9,100,41,65,78,129)(10,130,79,66,42,101)(11,102,43,67,80,131)(12,132,73,68,44,103)(13,104,45,69,74,133)(14,134,75,70,46,97)(15,98,47,71,76,135)(16,136,77,72,48,99)(17,89,81,59,50,34)(18,35,51,60,82,90)(19,91,83,61,52,36)(20,37,53,62,84,92)(21,93,85,63,54,38)(22,39,55,64,86,94)(23,95,87,57,56,40)(24,33,49,58,88,96), (1,81,71,5,85,67)(2,68,86,6,72,82)(3,83,65,7,87,69)(4,70,88,8,66,84)(9,32,40,13,28,36)(10,37,29,14,33,25)(11,26,34,15,30,38)(12,39,31,16,35,27)(17,98,112,21,102,108)(18,109,103,22,105,99)(19,100,106,23,104,110)(20,111,97,24,107,101)(41,119,95,45,115,91)(42,92,116,46,96,120)(43,113,89,47,117,93)(44,94,118,48,90,114)(49,140,130,53,144,134)(50,135,137,54,131,141)(51,142,132,55,138,136)(52,129,139,56,133,143)(57,74,122,61,78,126)(58,127,79,62,123,75)(59,76,124,63,80,128)(60,121,73,64,125,77), (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), (1,25,26,8)(2,7,27,32)(3,31,28,6)(4,5,29,30)(9,68,65,12)(10,11,66,67)(13,72,69,16)(14,15,70,71)(17,58,59,24)(18,23,60,57)(19,64,61,22)(20,21,62,63)(33,34,88,81)(35,40,82,87)(36,86,83,39)(37,38,84,85)(41,132,129,44)(42,43,130,131)(45,136,133,48)(46,47,134,135)(49,50,96,89)(51,56,90,95)(52,94,91,55)(53,54,92,93)(73,78,103,100)(74,99,104,77)(75,76,97,98)(79,80,101,102)(105,110,125,122)(106,121,126,109)(107,108,127,128)(111,112,123,124)(113,140,141,120)(114,119,142,139)(115,138,143,118)(116,117,144,137)>;
G:=Group( (1,128,141,26,108,113)(2,114,109,27,142,121)(3,122,143,28,110,115)(4,116,111,29,144,123)(5,124,137,30,112,117)(6,118,105,31,138,125)(7,126,139,32,106,119)(8,120,107,25,140,127)(9,100,41,65,78,129)(10,130,79,66,42,101)(11,102,43,67,80,131)(12,132,73,68,44,103)(13,104,45,69,74,133)(14,134,75,70,46,97)(15,98,47,71,76,135)(16,136,77,72,48,99)(17,89,81,59,50,34)(18,35,51,60,82,90)(19,91,83,61,52,36)(20,37,53,62,84,92)(21,93,85,63,54,38)(22,39,55,64,86,94)(23,95,87,57,56,40)(24,33,49,58,88,96), (1,81,71,5,85,67)(2,68,86,6,72,82)(3,83,65,7,87,69)(4,70,88,8,66,84)(9,32,40,13,28,36)(10,37,29,14,33,25)(11,26,34,15,30,38)(12,39,31,16,35,27)(17,98,112,21,102,108)(18,109,103,22,105,99)(19,100,106,23,104,110)(20,111,97,24,107,101)(41,119,95,45,115,91)(42,92,116,46,96,120)(43,113,89,47,117,93)(44,94,118,48,90,114)(49,140,130,53,144,134)(50,135,137,54,131,141)(51,142,132,55,138,136)(52,129,139,56,133,143)(57,74,122,61,78,126)(58,127,79,62,123,75)(59,76,124,63,80,128)(60,121,73,64,125,77), (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), (1,25,26,8)(2,7,27,32)(3,31,28,6)(4,5,29,30)(9,68,65,12)(10,11,66,67)(13,72,69,16)(14,15,70,71)(17,58,59,24)(18,23,60,57)(19,64,61,22)(20,21,62,63)(33,34,88,81)(35,40,82,87)(36,86,83,39)(37,38,84,85)(41,132,129,44)(42,43,130,131)(45,136,133,48)(46,47,134,135)(49,50,96,89)(51,56,90,95)(52,94,91,55)(53,54,92,93)(73,78,103,100)(74,99,104,77)(75,76,97,98)(79,80,101,102)(105,110,125,122)(106,121,126,109)(107,108,127,128)(111,112,123,124)(113,140,141,120)(114,119,142,139)(115,138,143,118)(116,117,144,137) );
G=PermutationGroup([[(1,128,141,26,108,113),(2,114,109,27,142,121),(3,122,143,28,110,115),(4,116,111,29,144,123),(5,124,137,30,112,117),(6,118,105,31,138,125),(7,126,139,32,106,119),(8,120,107,25,140,127),(9,100,41,65,78,129),(10,130,79,66,42,101),(11,102,43,67,80,131),(12,132,73,68,44,103),(13,104,45,69,74,133),(14,134,75,70,46,97),(15,98,47,71,76,135),(16,136,77,72,48,99),(17,89,81,59,50,34),(18,35,51,60,82,90),(19,91,83,61,52,36),(20,37,53,62,84,92),(21,93,85,63,54,38),(22,39,55,64,86,94),(23,95,87,57,56,40),(24,33,49,58,88,96)], [(1,81,71,5,85,67),(2,68,86,6,72,82),(3,83,65,7,87,69),(4,70,88,8,66,84),(9,32,40,13,28,36),(10,37,29,14,33,25),(11,26,34,15,30,38),(12,39,31,16,35,27),(17,98,112,21,102,108),(18,109,103,22,105,99),(19,100,106,23,104,110),(20,111,97,24,107,101),(41,119,95,45,115,91),(42,92,116,46,96,120),(43,113,89,47,117,93),(44,94,118,48,90,114),(49,140,130,53,144,134),(50,135,137,54,131,141),(51,142,132,55,138,136),(52,129,139,56,133,143),(57,74,122,61,78,126),(58,127,79,62,123,75),(59,76,124,63,80,128),(60,121,73,64,125,77)], [(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)], [(1,25,26,8),(2,7,27,32),(3,31,28,6),(4,5,29,30),(9,68,65,12),(10,11,66,67),(13,72,69,16),(14,15,70,71),(17,58,59,24),(18,23,60,57),(19,64,61,22),(20,21,62,63),(33,34,88,81),(35,40,82,87),(36,86,83,39),(37,38,84,85),(41,132,129,44),(42,43,130,131),(45,136,133,48),(46,47,134,135),(49,50,96,89),(51,56,90,95),(52,94,91,55),(53,54,92,93),(73,78,103,100),(74,99,104,77),(75,76,97,98),(79,80,101,102),(105,110,125,122),(106,121,126,109),(107,108,127,128),(111,112,123,124),(113,140,141,120),(114,119,142,139),(115,138,143,118),(116,117,144,137)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6L | 6M | ··· | 6AB | 8A | 8B | 8C | 8D | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | Dic3 | D8 | SD16 | C3⋊D4 | C3⋊D4 | D4⋊S3 | D4.S3 |
| kernel | C62.116D4 | C2×C32⋊4C8 | C12⋊Dic3 | D4×C3×C6 | D4×C32 | C6×D4 | C3×C12 | C62 | C2×C12 | C3×D4 | C3×C6 | C3×C6 | C12 | C2×C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 4 | 8 | 2 | 2 | 8 | 8 | 4 | 4 |
Matrix representation of C62.116D4 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 2 | 65 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 65 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 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 | 0 | 64 |
| 12 | 4 | 0 | 0 | 0 | 0 |
| 55 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 4 | 0 | 0 |
| 0 | 0 | 72 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 45 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 12 | 4 | 0 | 0 | 0 | 0 |
| 55 | 61 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 4 | 0 | 0 |
| 0 | 0 | 72 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 45 |
| 0 | 0 | 0 | 0 | 60 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,9,2,0,0,0,0,0,65,0,0,0,0,0,0,9,0,0,0,0,0,0,65],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,64],[12,55,0,0,0,0,4,0,0,0,0,0,0,0,21,72,0,0,0,0,4,52,0,0,0,0,0,0,0,13,0,0,0,0,45,0],[12,55,0,0,0,0,4,61,0,0,0,0,0,0,21,72,0,0,0,0,4,52,0,0,0,0,0,0,0,60,0,0,0,0,45,0] >;
C62.116D4 in GAP, Magma, Sage, TeX
C_6^2._{116}D_4 % in TeX
G:=Group("C6^2.116D4"); // GroupNames label
G:=SmallGroup(288,307);
// by ID
G=gap.SmallGroup(288,307);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,675,346,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^4=b^3,d^2=a^3,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^3*b^3*c^3>;
// generators/relations