Copied to
clipboard

G = C24.95D6order 288 = 25·32

29th non-split extension by C24 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: C24.95D6, (C6×C24)⋊18C2, (C2×C24)⋊12S3, C34(C8○D12), C12.64(C4×S3), C24⋊S311C2, C12⋊S3.4C4, (C2×C12).398D6, C3210(C8○D4), C62.84(C2×C4), C327D4.4C4, C324Q8.4C4, (C3×C24).72C22, C12.58D617C2, C12.208(C22×S3), (C6×C12).308C22, (C3×C12).177C23, C12.59D6.9C2, C324C8.39C22, C6.74(S3×C2×C4), (C8×C3⋊S3)⋊12C2, (C2×C8)⋊7(C3⋊S3), C4.10(C4×C3⋊S3), C8.22(C2×C3⋊S3), (C2×C6).54(C4×S3), C22.2(C4×C3⋊S3), (C3×C12).95(C2×C4), C4.37(C22×C3⋊S3), (C4×C3⋊S3).92C22, C3⋊Dic3.38(C2×C4), (C3×C6).105(C22×C4), C2.15(C2×C4×C3⋊S3), (C2×C4).78(C2×C3⋊S3), (C2×C3⋊S3).32(C2×C4), SmallGroup(288,758)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C24.95D6
C1C3C32C3×C6C3×C12C4×C3⋊S3C12.59D6 — C24.95D6
C32C3×C6 — C24.95D6
C1C8C2×C8

Generators and relations for C24.95D6
 G = < a,b,c | a24=b6=1, c2=a12, ab=ba, cac-1=a5, cbc-1=a12b-1 >

Subgroups: 572 in 186 conjugacy classes, 73 normal (23 characteristic)
C1, C2, C2 [×3], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×2], S3 [×8], C6 [×4], C6 [×4], C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, C32, Dic3 [×8], C12 [×8], D6 [×8], C2×C6 [×4], C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, C3⋊S3 [×2], C3×C6, C3×C6, C3⋊C8 [×8], C24 [×8], Dic6 [×4], C4×S3 [×8], D12 [×4], C3⋊D4 [×8], C2×C12 [×4], C8○D4, C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C62, S3×C8 [×8], C8⋊S3 [×8], C4.Dic3 [×4], C2×C24 [×4], C4○D12 [×4], C324C8 [×2], C3×C24 [×2], C324Q8, C4×C3⋊S3 [×2], C12⋊S3, C327D4 [×2], C6×C12, C8○D12 [×4], C8×C3⋊S3 [×2], C24⋊S3 [×2], C12.58D6, C6×C24, C12.59D6, C24.95D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], C23, D6 [×12], C22×C4, C3⋊S3, C4×S3 [×8], C22×S3 [×4], C8○D4, C2×C3⋊S3 [×3], S3×C2×C4 [×4], C4×C3⋊S3 [×2], C22×C3⋊S3, C8○D12 [×4], C2×C4×C3⋊S3, C24.95D6

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

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

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

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

84 conjugacy classes

class 1 2A2B2C2D3A3B3C3D4A4B4C4D4E6A···6L8A8B8C8D8E8F8G8H8I8J12A···12P24A···24AF
order122223333444446···6888888888812···1224···24
size1121818222211218182···2111122181818182···22···2

84 irreducible representations

dim1111111112222222
type+++++++++
imageC1C2C2C2C2C2C4C4C4S3D6D6C4×S3C4×S3C8○D4C8○D12
kernelC24.95D6C8×C3⋊S3C24⋊S3C12.58D6C6×C24C12.59D6C324Q8C12⋊S3C327D4C2×C24C24C2×C12C12C2×C6C32C3
# reps12211122448488432

Matrix representation of C24.95D6 in GL4(𝔽73) generated by

02700
464600
00816
005765
,
0100
727200
003043
003060
,
07200
72000
002727
00046
G:=sub<GL(4,GF(73))| [0,46,0,0,27,46,0,0,0,0,8,57,0,0,16,65],[0,72,0,0,1,72,0,0,0,0,30,30,0,0,43,60],[0,72,0,0,72,0,0,0,0,0,27,0,0,0,27,46] >;

C24.95D6 in GAP, Magma, Sage, TeX

C_{24}._{95}D_6
% in TeX

G:=Group("C24.95D6");
// GroupNames label

G:=SmallGroup(288,758);
// by ID

G=gap.SmallGroup(288,758);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,58,80,2693,9414]);
// Polycyclic

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

׿
×
𝔽