metabelian, supersoluble, monomial
Aliases: C24.95D6, (C6×C24)⋊18C2, (C2×C24)⋊12S3, C3⋊4(C8○D12), C12.64(C4×S3), C24⋊S3⋊11C2, C12⋊S3.4C4, (C2×C12).398D6, C32⋊10(C8○D4), C62.84(C2×C4), C32⋊7D4.4C4, C32⋊4Q8.4C4, (C3×C24).72C22, C12.58D6⋊17C2, C12.208(C22×S3), (C6×C12).308C22, (C3×C12).177C23, C12.59D6.9C2, C32⋊4C8.39C22, C6.74(S3×C2×C4), (C8×C3⋊S3)⋊12C2, (C2×C8)⋊7(C3⋊S3), C4.10(C4×C3⋊S3), C8.22(C2×C3⋊S3), (C2×C6).54(C4×S3), C22.2(C4×C3⋊S3), (C3×C12).95(C2×C4), C4.37(C22×C3⋊S3), (C4×C3⋊S3).92C22, C3⋊Dic3.38(C2×C4), (C3×C6).105(C22×C4), C2.15(C2×C4×C3⋊S3), (C2×C4).78(C2×C3⋊S3), (C2×C3⋊S3).32(C2×C4), SmallGroup(288,758)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.95D6
G = < a,b,c | a24=b6=1, c2=a12, ab=ba, cac-1=a5, cbc-1=a12b-1 >
Subgroups: 572 in 186 conjugacy classes, 73 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C8○D4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C4○D12, C32⋊4C8, C3×C24, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C8○D12, C8×C3⋊S3, C24⋊S3, C12.58D6, C6×C24, C12.59D6, C24.95D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C3⋊S3, C4×S3, C22×S3, C8○D4, C2×C3⋊S3, S3×C2×C4, C4×C3⋊S3, C22×C3⋊S3, C8○D12, C2×C4×C3⋊S3, C24.95D6
►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 56 106 134 89 37)(2 57 107 135 90 38)(3 58 108 136 91 39)(4 59 109 137 92 40)(5 60 110 138 93 41)(6 61 111 139 94 42)(7 62 112 140 95 43)(8 63 113 141 96 44)(9 64 114 142 73 45)(10 65 115 143 74 46)(11 66 116 144 75 47)(12 67 117 121 76 48)(13 68 118 122 77 25)(14 69 119 123 78 26)(15 70 120 124 79 27)(16 71 97 125 80 28)(17 72 98 126 81 29)(18 49 99 127 82 30)(19 50 100 128 83 31)(20 51 101 129 84 32)(21 52 102 130 85 33)(22 53 103 131 86 34)(23 54 104 132 87 35)(24 55 105 133 88 36)
(1 112 13 100)(2 117 14 105)(3 98 15 110)(4 103 16 115)(5 108 17 120)(6 113 18 101)(7 118 19 106)(8 99 20 111)(9 104 21 116)(10 109 22 97)(11 114 23 102)(12 119 24 107)(25 140 37 128)(26 121 38 133)(27 126 39 138)(28 131 40 143)(29 136 41 124)(30 141 42 129)(31 122 43 134)(32 127 44 139)(33 132 45 144)(34 137 46 125)(35 142 47 130)(36 123 48 135)(49 63 61 51)(50 68 62 56)(52 54 64 66)(53 59 65 71)(55 69 67 57)(58 60 70 72)(73 87 85 75)(74 92 86 80)(76 78 88 90)(77 83 89 95)(79 93 91 81)(82 84 94 96)
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), (1,56,106,134,89,37)(2,57,107,135,90,38)(3,58,108,136,91,39)(4,59,109,137,92,40)(5,60,110,138,93,41)(6,61,111,139,94,42)(7,62,112,140,95,43)(8,63,113,141,96,44)(9,64,114,142,73,45)(10,65,115,143,74,46)(11,66,116,144,75,47)(12,67,117,121,76,48)(13,68,118,122,77,25)(14,69,119,123,78,26)(15,70,120,124,79,27)(16,71,97,125,80,28)(17,72,98,126,81,29)(18,49,99,127,82,30)(19,50,100,128,83,31)(20,51,101,129,84,32)(21,52,102,130,85,33)(22,53,103,131,86,34)(23,54,104,132,87,35)(24,55,105,133,88,36), (1,112,13,100)(2,117,14,105)(3,98,15,110)(4,103,16,115)(5,108,17,120)(6,113,18,101)(7,118,19,106)(8,99,20,111)(9,104,21,116)(10,109,22,97)(11,114,23,102)(12,119,24,107)(25,140,37,128)(26,121,38,133)(27,126,39,138)(28,131,40,143)(29,136,41,124)(30,141,42,129)(31,122,43,134)(32,127,44,139)(33,132,45,144)(34,137,46,125)(35,142,47,130)(36,123,48,135)(49,63,61,51)(50,68,62,56)(52,54,64,66)(53,59,65,71)(55,69,67,57)(58,60,70,72)(73,87,85,75)(74,92,86,80)(76,78,88,90)(77,83,89,95)(79,93,91,81)(82,84,94,96)>;
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), (1,56,106,134,89,37)(2,57,107,135,90,38)(3,58,108,136,91,39)(4,59,109,137,92,40)(5,60,110,138,93,41)(6,61,111,139,94,42)(7,62,112,140,95,43)(8,63,113,141,96,44)(9,64,114,142,73,45)(10,65,115,143,74,46)(11,66,116,144,75,47)(12,67,117,121,76,48)(13,68,118,122,77,25)(14,69,119,123,78,26)(15,70,120,124,79,27)(16,71,97,125,80,28)(17,72,98,126,81,29)(18,49,99,127,82,30)(19,50,100,128,83,31)(20,51,101,129,84,32)(21,52,102,130,85,33)(22,53,103,131,86,34)(23,54,104,132,87,35)(24,55,105,133,88,36), (1,112,13,100)(2,117,14,105)(3,98,15,110)(4,103,16,115)(5,108,17,120)(6,113,18,101)(7,118,19,106)(8,99,20,111)(9,104,21,116)(10,109,22,97)(11,114,23,102)(12,119,24,107)(25,140,37,128)(26,121,38,133)(27,126,39,138)(28,131,40,143)(29,136,41,124)(30,141,42,129)(31,122,43,134)(32,127,44,139)(33,132,45,144)(34,137,46,125)(35,142,47,130)(36,123,48,135)(49,63,61,51)(50,68,62,56)(52,54,64,66)(53,59,65,71)(55,69,67,57)(58,60,70,72)(73,87,85,75)(74,92,86,80)(76,78,88,90)(77,83,89,95)(79,93,91,81)(82,84,94,96) );
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)], [(1,56,106,134,89,37),(2,57,107,135,90,38),(3,58,108,136,91,39),(4,59,109,137,92,40),(5,60,110,138,93,41),(6,61,111,139,94,42),(7,62,112,140,95,43),(8,63,113,141,96,44),(9,64,114,142,73,45),(10,65,115,143,74,46),(11,66,116,144,75,47),(12,67,117,121,76,48),(13,68,118,122,77,25),(14,69,119,123,78,26),(15,70,120,124,79,27),(16,71,97,125,80,28),(17,72,98,126,81,29),(18,49,99,127,82,30),(19,50,100,128,83,31),(20,51,101,129,84,32),(21,52,102,130,85,33),(22,53,103,131,86,34),(23,54,104,132,87,35),(24,55,105,133,88,36)], [(1,112,13,100),(2,117,14,105),(3,98,15,110),(4,103,16,115),(5,108,17,120),(6,113,18,101),(7,118,19,106),(8,99,20,111),(9,104,21,116),(10,109,22,97),(11,114,23,102),(12,119,24,107),(25,140,37,128),(26,121,38,133),(27,126,39,138),(28,131,40,143),(29,136,41,124),(30,141,42,129),(31,122,43,134),(32,127,44,139),(33,132,45,144),(34,137,46,125),(35,142,47,130),(36,123,48,135),(49,63,61,51),(50,68,62,56),(52,54,64,66),(53,59,65,71),(55,69,67,57),(58,60,70,72),(73,87,85,75),(74,92,86,80),(76,78,88,90),(77,83,89,95),(79,93,91,81),(82,84,94,96)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | ··· | 12P | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 18 | 18 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 18 | 18 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | C8○D12 |
| kernel | C24.95D6 | C8×C3⋊S3 | C24⋊S3 | C12.58D6 | C6×C24 | C12.59D6 | C32⋊4Q8 | C12⋊S3 | C32⋊7D4 | C2×C24 | C24 | C2×C12 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 4 | 8 | 8 | 4 | 32 |
Matrix representation of C24.95D6 ►in GL4(𝔽73) generated by
| 0 | 27 | 0 | 0 |
| 46 | 46 | 0 | 0 |
| 0 | 0 | 8 | 16 |
| 0 | 0 | 57 | 65 |
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 30 | 43 |
| 0 | 0 | 30 | 60 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 27 | 27 |
| 0 | 0 | 0 | 46 |
G:=sub<GL(4,GF(73))| [0,46,0,0,27,46,0,0,0,0,8,57,0,0,16,65],[0,72,0,0,1,72,0,0,0,0,30,30,0,0,43,60],[0,72,0,0,72,0,0,0,0,0,27,0,0,0,27,46] >;
C24.95D6 in GAP, Magma, Sage, TeX
C_{24}._{95}D_6 % in TeX
G:=Group("C24.95D6"); // GroupNames label
G:=SmallGroup(288,758);
// by ID
G=gap.SmallGroup(288,758);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,58,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=1,c^2=a^12,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=a^12*b^-1>;
// generators/relations