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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 [×2], C3 [×4], C4 [×2], C4 [×3], C22, C22, S3 [×4], C6 [×4], C6 [×4], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], C32, Dic3 [×12], C12 [×8], D6 [×4], C2×C6 [×4], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C24 [×8], Dic6 [×16], C4×S3 [×4], D12 [×4], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12 [×4], C8.C22, C3⋊Dic3 [×3], C3×C12 [×2], C2×C3⋊S3, C62, C24⋊C2 [×8], Dic12 [×8], C3×M4(2) [×4], C2×Dic6 [×4], C4○D12 [×4], C3×C24 [×2], C324Q8, C324Q8 [×2], C324Q8, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C327D4, C6×C12, C8.D6 [×4], C242S3 [×2], C325Q16 [×2], C32×M4(2), C2×C324Q8, C12.59D6, C24.5D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, D12 [×8], C22×S3 [×4], C8.C22, C2×C3⋊S3 [×3], C2×D12 [×4], C12⋊S3 [×2], C22×C3⋊S3, C8.D6 [×4], 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)
(1 144 80 32 109 53 13 132 92 44 97 65)(2 133 81 45 110 66 14 121 93 33 98 54)(3 122 82 34 111 55 15 134 94 46 99 67)(4 135 83 47 112 68 16 123 95 35 100 56)(5 124 84 36 113 57 17 136 96 48 101 69)(6 137 85 25 114 70 18 125 73 37 102 58)(7 126 86 38 115 59 19 138 74 26 103 71)(8 139 87 27 116 72 20 127 75 39 104 60)(9 128 88 40 117 61 21 140 76 28 105 49)(10 141 89 29 118 50 22 129 77 41 106 62)(11 130 90 42 119 63 23 142 78 30 107 51)(12 143 91 31 120 52 24 131 79 43 108 64)
(1 53 13 65)(2 52 14 64)(3 51 15 63)(4 50 16 62)(5 49 17 61)(6 72 18 60)(7 71 19 59)(8 70 20 58)(9 69 21 57)(10 68 22 56)(11 67 23 55)(12 66 24 54)(25 75 37 87)(26 74 38 86)(27 73 39 85)(28 96 40 84)(29 95 41 83)(30 94 42 82)(31 93 43 81)(32 92 44 80)(33 91 45 79)(34 90 46 78)(35 89 47 77)(36 88 48 76)(97 132 109 144)(98 131 110 143)(99 130 111 142)(100 129 112 141)(101 128 113 140)(102 127 114 139)(103 126 115 138)(104 125 116 137)(105 124 117 136)(106 123 118 135)(107 122 119 134)(108 121 120 133)

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

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

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

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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