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G = C24.26D6order 288 = 25·32

26th non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C24.26D6, (C3×D8)⋊3S3, D83(C3⋊S3), (C3×D4).15D6, C6.120(S3×D4), (C32×D8)⋊6C2, C34(D83S3), C325Q168C2, C3219(C4○D8), C3⋊Dic3.69D4, C12.D64C2, C329SD166C2, (C3×C24).29C22, C12.89(C22×S3), (C3×C12).93C23, C324C8.26C22, (D4×C32).16C22, C324Q8.16C22, (C8×C3⋊S3)⋊4C2, C8.8(C2×C3⋊S3), D4.1(C2×C3⋊S3), C2.17(D4×C3⋊S3), (C2×C3⋊S3).46D4, C4.3(C22×C3⋊S3), (C3×C6).241(C2×D4), (C4×C3⋊S3).71C22, SmallGroup(288,769)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C24.26D6
C1C3C32C3×C6C3×C12C4×C3⋊S3C12.D6 — C24.26D6
C32C3×C6C3×C12 — C24.26D6
C1C2C4D8

Generators and relations for C24.26D6
 G = < a,b,c | a24=b6=1, c2=a12, bab-1=a7, cac-1=a-1, cbc-1=a12b-1 >

Subgroups: 708 in 186 conjugacy classes, 55 normal (17 characteristic)
C1, C2, C2 [×3], C3 [×4], C4, C4 [×3], C22 [×3], S3 [×4], C6 [×4], C6 [×8], C8, C8, C2×C4 [×3], D4 [×2], D4 [×2], Q8 [×2], C32, Dic3 [×12], C12 [×4], D6 [×4], C2×C6 [×8], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3⋊S3, C3×C6, C3×C6 [×2], C3⋊C8 [×4], C24 [×4], Dic6 [×8], C4×S3 [×4], C2×Dic3 [×8], C3⋊D4 [×8], C3×D4 [×8], C4○D8, C3⋊Dic3, C3⋊Dic3 [×2], C3×C12, C2×C3⋊S3, C62 [×2], S3×C8 [×4], Dic12 [×4], D4.S3 [×8], C3×D8 [×4], D42S3 [×8], C324C8, C3×C24, C324Q8 [×2], C4×C3⋊S3, C2×C3⋊Dic3 [×2], C327D4 [×2], D4×C32 [×2], D83S3 [×4], C8×C3⋊S3, C325Q16, C329SD16 [×2], C32×D8, C12.D6 [×2], C24.26D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C22×S3 [×4], C4○D8, C2×C3⋊S3 [×3], S3×D4 [×4], C22×C3⋊S3, D83S3 [×4], D4×C3⋊S3, C24.26D6

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

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

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

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

42 conjugacy classes

class 1 2A2B2C2D3A3B3C3D4A4B4C4D4E6A6B6C6D6E···6L8A8B8C8D12A12B12C12D24A···24H
order1222233334444466666···688881212121224···24
size1144182222299363622228···822181844444···4

42 irreducible representations

dim11111122222244
type++++++++++++-
imageC1C2C2C2C2C2S3D4D4D6D6C4○D8S3×D4D83S3
kernelC24.26D6C8×C3⋊S3C325Q16C329SD16C32×D8C12.D6C3×D8C3⋊Dic3C2×C3⋊S3C24C3×D4C32C6C3
# reps11121241148448

Matrix representation of C24.26D6 in GL6(𝔽73)

100000
010000
002300
00727200
0000100
00001222
,
110000
7200000
0072000
0007200
00002222
00006151
,
110000
0720000
0072000
001100
00005151
00003222

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,72,0,0,0,0,3,72,0,0,0,0,0,0,10,12,0,0,0,0,0,22],[1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,22,61,0,0,0,0,22,51],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,51,32,0,0,0,0,51,22] >;

C24.26D6 in GAP, Magma, Sage, TeX

C_{24}._{26}D_6
% in TeX

G:=Group("C24.26D6");
// GroupNames label

G:=SmallGroup(288,769);
// by ID

G=gap.SmallGroup(288,769);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,135,346,185,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=b^6=1,c^2=a^12,b*a*b^-1=a^7,c*a*c^-1=a^-1,c*b*c^-1=a^12*b^-1>;
// generators/relations

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