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G = C24.35D6order 288 = 25·32

35th non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C24.35D6, (C3×Q16)⋊4S3, C242S36C2, C24⋊S36C2, Q162(C3⋊S3), C6.126(S3×D4), (C3×Q8).39D6, C35(Q16⋊S3), C3⋊Dic3.68D4, (C32×Q16)⋊7C2, C327Q168C2, C3211SD167C2, (C3×C12).99C23, (C3×C24).34C22, C12.95(C22×S3), C12.26D6.3C2, C3221(C8.C22), C12⋊S3.19C22, C324C8.16C22, (Q8×C32).19C22, C324Q8.20C22, C8.3(C2×C3⋊S3), (Q8×C3⋊S3)⋊5C2, C2.23(D4×C3⋊S3), Q8.9(C2×C3⋊S3), (C2×C3⋊S3).68D4, C4.9(C22×C3⋊S3), (C3×C6).247(C2×D4), (C4×C3⋊S3).26C22, SmallGroup(288,775)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C24.35D6
C1C3C32C3×C6C3×C12C4×C3⋊S3Q8×C3⋊S3 — C24.35D6
C32C3×C6C3×C12 — C24.35D6
C1C2C4Q16

Generators and relations for C24.35D6
 G = < a,b,c | a24=1, b6=c2=a12, bab-1=a7, cac-1=a11, cbc-1=b5 >

Subgroups: 748 in 180 conjugacy classes, 55 normal (27 characteristic)
C1, C2, C2 [×2], C3 [×4], C4, C4 [×4], C22 [×2], S3 [×8], C6 [×4], C8, C8, C2×C4 [×3], D4 [×2], Q8 [×2], Q8 [×2], C32, Dic3 [×8], C12 [×4], C12 [×8], D6 [×8], M4(2), SD16 [×2], Q16, Q16, C2×Q8, C4○D4, C3⋊S3 [×2], C3×C6, C3⋊C8 [×4], C24 [×4], Dic6 [×8], C4×S3 [×12], D12 [×8], C3×Q8 [×8], C8.C22, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12 [×2], C2×C3⋊S3, C2×C3⋊S3, C8⋊S3 [×4], C24⋊C2 [×4], Q82S3 [×4], C3⋊Q16 [×4], C3×Q16 [×4], S3×Q8 [×4], Q83S3 [×4], C324C8, C3×C24, C324Q8, C324Q8, C4×C3⋊S3, C4×C3⋊S3 [×2], C12⋊S3, C12⋊S3, Q8×C32 [×2], Q16⋊S3 [×4], C24⋊S3, C242S3, C3211SD16, C327Q16, C32×Q16, Q8×C3⋊S3, C12.26D6, C24.35D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C22×S3 [×4], C8.C22, C2×C3⋊S3 [×3], S3×D4 [×4], C22×C3⋊S3, Q16⋊S3 [×4], D4×C3⋊S3, C24.35D6

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

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

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

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

39 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E6A6B6C6D8A8B12A12B12C12D12E···12L24A···24H
order12223333444446666881212121212···1224···24
size11183622222441836222243644448···84···4

39 irreducible representations

dim1111111122222444
type+++++++++++++-+
imageC1C2C2C2C2C2C2C2S3D4D4D6D6C8.C22S3×D4Q16⋊S3
kernelC24.35D6C24⋊S3C242S3C3211SD16C327Q16C32×Q16Q8×C3⋊S3C12.26D6C3×Q16C3⋊Dic3C2×C3⋊S3C24C3×Q8C32C6C3
# reps1111111141148148

Matrix representation of C24.35D6 in GL8(𝔽73)

13730150000
363758150000
435872360000
155837360000
00001122020
000051625353
0000111100
000062000
,
155837360000
15303710000
363758150000
367258430000
000037375736
00003603721
000049353636
00003814370
,
363758150000
13730150000
155837360000
435872360000
00006251053
00006211530
00000626211
00006202211

G:=sub<GL(8,GF(73))| [1,36,43,15,0,0,0,0,37,37,58,58,0,0,0,0,30,58,72,37,0,0,0,0,15,15,36,36,0,0,0,0,0,0,0,0,11,51,11,62,0,0,0,0,22,62,11,0,0,0,0,0,0,53,0,0,0,0,0,0,20,53,0,0],[15,15,36,36,0,0,0,0,58,30,37,72,0,0,0,0,37,37,58,58,0,0,0,0,36,1,15,43,0,0,0,0,0,0,0,0,37,36,49,38,0,0,0,0,37,0,35,14,0,0,0,0,57,37,36,37,0,0,0,0,36,21,36,0],[36,1,15,43,0,0,0,0,37,37,58,58,0,0,0,0,58,30,37,72,0,0,0,0,15,15,36,36,0,0,0,0,0,0,0,0,62,62,0,62,0,0,0,0,51,11,62,0,0,0,0,0,0,53,62,22,0,0,0,0,53,0,11,11] >;

C24.35D6 in GAP, Magma, Sage, TeX

C_{24}._{35}D_6
% in TeX

G:=Group("C24.35D6");
// GroupNames label

G:=SmallGroup(288,775);
// by ID

G=gap.SmallGroup(288,775);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,100,346,185,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=1,b^6=c^2=a^12,b*a*b^-1=a^7,c*a*c^-1=a^11,c*b*c^-1=b^5>;
// generators/relations

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