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G = C24.78D6order 288 = 25·32

12nd non-split extension by C24 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: C24.78D6, C12.61D12, C62.92D4, (C6×C24)⋊8C2, (C2×C24)⋊6S3, C34(C4○D24), (C2×C6).40D12, C6.58(C2×D12), C242S311C2, C325D811C2, (C3×C12).141D4, (C2×C12).399D6, C3217(C4○D8), C325Q1611C2, C12.59D61C2, (C3×C24).59C22, C4.20(C12⋊S3), C12.191(C22×S3), (C6×C12).309C22, (C3×C12).153C23, C12⋊S3.23C22, C22.1(C12⋊S3), C324Q8.23C22, (C2×C8)⋊4(C3⋊S3), C8.16(C2×C3⋊S3), (C3×C6).198(C2×D4), C4.28(C22×C3⋊S3), C2.13(C2×C12⋊S3), (C2×C4).81(C2×C3⋊S3), SmallGroup(288,761)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C24.78D6
C1C3C32C3×C6C3×C12C12⋊S3C12.59D6 — C24.78D6
C32C3×C6C3×C12 — C24.78D6
C1C4C2×C4C2×C8

Generators and relations for C24.78D6
 G = < a,b,c | a24=1, b6=c2=a12, ab=ba, cac-1=a11, cbc-1=b5 >

Subgroups: 836 in 186 conjugacy classes, 65 normal (21 characteristic)
C1, C2, C2 [×3], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×2], S3 [×8], C6 [×4], C6 [×4], C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], C32, Dic3 [×8], C12 [×8], D6 [×8], C2×C6 [×4], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3⋊S3 [×2], C3×C6, C3×C6, C24 [×8], Dic6 [×8], C4×S3 [×8], D12 [×8], C3⋊D4 [×8], C2×C12 [×4], C4○D8, C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C62, C24⋊C2 [×8], D24 [×4], Dic12 [×4], C2×C24 [×4], C4○D12 [×8], C3×C24 [×2], C324Q8 [×2], C4×C3⋊S3 [×2], C12⋊S3 [×2], C327D4 [×2], C6×C12, C4○D24 [×4], C242S3 [×2], C325D8, C325Q16, C6×C24, C12.59D6 [×2], C24.78D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, D12 [×8], C22×S3 [×4], C4○D8, C2×C3⋊S3 [×3], C2×D12 [×4], C12⋊S3 [×2], C22×C3⋊S3, C4○D24 [×4], C2×C12⋊S3, C24.78D6

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

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

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

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

78 conjugacy classes

class 1 2A2B2C2D3A3B3C3D4A4B4C4D4E6A···6L8A8B8C8D12A···12P24A···24AF
order122223333444446···6888812···1224···24
size1123636222211236362···222222···22···2

78 irreducible representations

dim111111222222222
type+++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D12D12C4○D8C4○D24
kernelC24.78D6C242S3C325D8C325Q16C6×C24C12.59D6C2×C24C3×C12C62C24C2×C12C12C2×C6C32C3
# reps1211124118488432

Matrix representation of C24.78D6 in GL4(𝔽73) generated by

0100
727200
002536
003762
,
07200
1100
00270
00027
,
0100
1000
002536
001148
G:=sub<GL(4,GF(73))| [0,72,0,0,1,72,0,0,0,0,25,37,0,0,36,62],[0,1,0,0,72,1,0,0,0,0,27,0,0,0,0,27],[0,1,0,0,1,0,0,0,0,0,25,11,0,0,36,48] >;

C24.78D6 in GAP, Magma, Sage, TeX

C_{24}._{78}D_6
% in TeX

G:=Group("C24.78D6");
// GroupNames label

G:=SmallGroup(288,761);
// by ID

G=gap.SmallGroup(288,761);
# by ID

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

G:=Group<a,b,c|a^24=1,b^6=c^2=a^12,a*b=b*a,c*a*c^-1=a^11,c*b*c^-1=b^5>;
// generators/relations

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