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G = C24.5D6order 288 = 25·32

5th non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C24.5D6, C12.33D12, C62.71D4, C242S34C2, C6.61(C2×D12), (C3×C12).96D4, (C2×C6).18D12, C325Q163C2, (C2×C12).148D6, C33(C8.D6), (C3×M4(2))⋊2S3, (C3×C24).5C22, M4(2)⋊2(C3⋊S3), C4.15(C12⋊S3), (C6×C12).139C22, C12.194(C22×S3), (C3×C12).156C23, C12.59D6.7C2, C3217(C8.C22), (C32×M4(2))⋊4C2, C12⋊S3.24C22, C22.6(C12⋊S3), C324Q8.25C22, C8.1(C2×C3⋊S3), (C3×C6).201(C2×D4), C4.31(C22×C3⋊S3), C2.16(C2×C12⋊S3), (C2×C324Q8)⋊14C2, (C2×C4).16(C2×C3⋊S3), SmallGroup(288,766)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C24.5D6
C1C3C32C3×C6C3×C12C12⋊S3C12.59D6 — C24.5D6
C32C3×C6C3×C12 — C24.5D6
C1C2C2×C4M4(2)

Generators and relations for C24.5D6
 G = < a,b,c | a24=1, b6=c2=a12, bab-1=a13, cac-1=a-1, cbc-1=b5 >

Subgroups: 748 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, C32, Dic3, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C8.C22, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C24⋊C2, Dic12, C3×M4(2), C2×Dic6, C4○D12, C3×C24, C324Q8, C324Q8, C324Q8, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C327D4, C6×C12, C8.D6, C242S3, C325Q16, C32×M4(2), C2×C324Q8, C12.59D6, C24.5D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, D12, C22×S3, C8.C22, C2×C3⋊S3, C2×D12, C12⋊S3, C22×C3⋊S3, C8.D6, C2×C12⋊S3, C24.5D6

Smallest permutation representation of C24.5D6
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 26 136 77 70 102 13 38 124 89 58 114)(2 39 137 90 71 115 14 27 125 78 59 103)(3 28 138 79 72 104 15 40 126 91 60 116)(4 41 139 92 49 117 16 29 127 80 61 105)(5 30 140 81 50 106 17 42 128 93 62 118)(6 43 141 94 51 119 18 31 129 82 63 107)(7 32 142 83 52 108 19 44 130 95 64 120)(8 45 143 96 53 97 20 33 131 84 65 109)(9 34 144 85 54 110 21 46 132 73 66 98)(10 47 121 74 55 99 22 35 133 86 67 111)(11 36 122 87 56 112 23 48 134 75 68 100)(12 25 123 76 57 101 24 37 135 88 69 113)
(1 102 13 114)(2 101 14 113)(3 100 15 112)(4 99 16 111)(5 98 17 110)(6 97 18 109)(7 120 19 108)(8 119 20 107)(9 118 21 106)(10 117 22 105)(11 116 23 104)(12 115 24 103)(25 59 37 71)(26 58 38 70)(27 57 39 69)(28 56 40 68)(29 55 41 67)(30 54 42 66)(31 53 43 65)(32 52 44 64)(33 51 45 63)(34 50 46 62)(35 49 47 61)(36 72 48 60)(73 128 85 140)(74 127 86 139)(75 126 87 138)(76 125 88 137)(77 124 89 136)(78 123 90 135)(79 122 91 134)(80 121 92 133)(81 144 93 132)(82 143 94 131)(83 142 95 130)(84 141 96 129)

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), (1,26,136,77,70,102,13,38,124,89,58,114)(2,39,137,90,71,115,14,27,125,78,59,103)(3,28,138,79,72,104,15,40,126,91,60,116)(4,41,139,92,49,117,16,29,127,80,61,105)(5,30,140,81,50,106,17,42,128,93,62,118)(6,43,141,94,51,119,18,31,129,82,63,107)(7,32,142,83,52,108,19,44,130,95,64,120)(8,45,143,96,53,97,20,33,131,84,65,109)(9,34,144,85,54,110,21,46,132,73,66,98)(10,47,121,74,55,99,22,35,133,86,67,111)(11,36,122,87,56,112,23,48,134,75,68,100)(12,25,123,76,57,101,24,37,135,88,69,113), (1,102,13,114)(2,101,14,113)(3,100,15,112)(4,99,16,111)(5,98,17,110)(6,97,18,109)(7,120,19,108)(8,119,20,107)(9,118,21,106)(10,117,22,105)(11,116,23,104)(12,115,24,103)(25,59,37,71)(26,58,38,70)(27,57,39,69)(28,56,40,68)(29,55,41,67)(30,54,42,66)(31,53,43,65)(32,52,44,64)(33,51,45,63)(34,50,46,62)(35,49,47,61)(36,72,48,60)(73,128,85,140)(74,127,86,139)(75,126,87,138)(76,125,88,137)(77,124,89,136)(78,123,90,135)(79,122,91,134)(80,121,92,133)(81,144,93,132)(82,143,94,131)(83,142,95,130)(84,141,96,129)>;

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), (1,26,136,77,70,102,13,38,124,89,58,114)(2,39,137,90,71,115,14,27,125,78,59,103)(3,28,138,79,72,104,15,40,126,91,60,116)(4,41,139,92,49,117,16,29,127,80,61,105)(5,30,140,81,50,106,17,42,128,93,62,118)(6,43,141,94,51,119,18,31,129,82,63,107)(7,32,142,83,52,108,19,44,130,95,64,120)(8,45,143,96,53,97,20,33,131,84,65,109)(9,34,144,85,54,110,21,46,132,73,66,98)(10,47,121,74,55,99,22,35,133,86,67,111)(11,36,122,87,56,112,23,48,134,75,68,100)(12,25,123,76,57,101,24,37,135,88,69,113), (1,102,13,114)(2,101,14,113)(3,100,15,112)(4,99,16,111)(5,98,17,110)(6,97,18,109)(7,120,19,108)(8,119,20,107)(9,118,21,106)(10,117,22,105)(11,116,23,104)(12,115,24,103)(25,59,37,71)(26,58,38,70)(27,57,39,69)(28,56,40,68)(29,55,41,67)(30,54,42,66)(31,53,43,65)(32,52,44,64)(33,51,45,63)(34,50,46,62)(35,49,47,61)(36,72,48,60)(73,128,85,140)(74,127,86,139)(75,126,87,138)(76,125,88,137)(77,124,89,136)(78,123,90,135)(79,122,91,134)(80,121,92,133)(81,144,93,132)(82,143,94,131)(83,142,95,130)(84,141,96,129) );

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)], [(1,26,136,77,70,102,13,38,124,89,58,114),(2,39,137,90,71,115,14,27,125,78,59,103),(3,28,138,79,72,104,15,40,126,91,60,116),(4,41,139,92,49,117,16,29,127,80,61,105),(5,30,140,81,50,106,17,42,128,93,62,118),(6,43,141,94,51,119,18,31,129,82,63,107),(7,32,142,83,52,108,19,44,130,95,64,120),(8,45,143,96,53,97,20,33,131,84,65,109),(9,34,144,85,54,110,21,46,132,73,66,98),(10,47,121,74,55,99,22,35,133,86,67,111),(11,36,122,87,56,112,23,48,134,75,68,100),(12,25,123,76,57,101,24,37,135,88,69,113)], [(1,102,13,114),(2,101,14,113),(3,100,15,112),(4,99,16,111),(5,98,17,110),(6,97,18,109),(7,120,19,108),(8,119,20,107),(9,118,21,106),(10,117,22,105),(11,116,23,104),(12,115,24,103),(25,59,37,71),(26,58,38,70),(27,57,39,69),(28,56,40,68),(29,55,41,67),(30,54,42,66),(31,53,43,65),(32,52,44,64),(33,51,45,63),(34,50,46,62),(35,49,47,61),(36,72,48,60),(73,128,85,140),(74,127,86,139),(75,126,87,138),(76,125,88,137),(77,124,89,136),(78,123,90,135),(79,122,91,134),(80,121,92,133),(81,144,93,132),(82,143,94,131),(83,142,95,130),(84,141,96,129)]])

51 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E6A6B6C6D6E6F6G6H8A8B12A···12H12I12J12K12L24A···24P
order1222333344444666666668812···121212121224···24
size1123622222236363622224444442···244444···4

51 irreducible representations

dim111111222222244
type+++++++++++++--
imageC1C2C2C2C2C2S3D4D4D6D6D12D12C8.C22C8.D6
kernelC24.5D6C242S3C325Q16C32×M4(2)C2×C324Q8C12.59D6C3×M4(2)C3×C12C62C24C2×C12C12C2×C6C32C3
# reps122111411848818

Matrix representation of C24.5D6 in GL6(𝔽73)

72720000
100000
007138026
0035334747
005637235
0036193840
,
7200000
0720000
007454340
002835333
0057216628
0052364538
,
7200000
110000
007454340
0038667030
00345223
001839171

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,71,35,56,36,0,0,38,33,37,19,0,0,0,47,2,38,0,0,26,47,35,40],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,7,28,57,52,0,0,45,35,21,36,0,0,43,33,66,45,0,0,40,3,28,38],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,7,38,34,18,0,0,45,66,52,39,0,0,43,70,2,1,0,0,40,30,3,71] >;

C24.5D6 in GAP, Magma, Sage, TeX

C_{24}._5D_6
% in TeX

G:=Group("C24.5D6");
// GroupNames label

G:=SmallGroup(288,766);
// by ID

G=gap.SmallGroup(288,766);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,58,675,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=1,b^6=c^2=a^12,b*a*b^-1=a^13,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations

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