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G = C24.40D6order 288 = 25·32

40th non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C24.40D6, (C3×SD16)⋊4S3, (C3×D4).19D6, C6.124(S3×D4), C242S310C2, C327D88C2, (C3×Q8).37D6, SD163(C3⋊S3), C3221(C4○D8), C3⋊Dic3.70D4, C327Q166C2, C35(Q8.7D6), C12.26D64C2, C12.D65C2, (C3×C12).97C23, C12.93(C22×S3), (C3×C24).39C22, (C32×SD16)⋊8C2, C12⋊S3.18C22, C324C8.27C22, (D4×C32).20C22, (Q8×C32).17C22, C324Q8.18C22, (C8×C3⋊S3)⋊9C2, C8.11(C2×C3⋊S3), D4.5(C2×C3⋊S3), C2.21(D4×C3⋊S3), Q8.7(C2×C3⋊S3), (C2×C3⋊S3).47D4, C4.7(C22×C3⋊S3), (C3×C6).245(C2×D4), (C4×C3⋊S3).73C22, SmallGroup(288,773)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C24.40D6
C1C3C32C3×C6C3×C12C4×C3⋊S3C12.D6 — C24.40D6
C32C3×C6C3×C12 — C24.40D6
C1C2C4SD16

Generators and relations for C24.40D6
 G = < a,b,c | a24=b6=1, c2=a12, bab-1=a19, cac-1=a11, cbc-1=a12b-1 >

Subgroups: 772 in 186 conjugacy classes, 55 normal (27 characteristic)
C1, C2, C2 [×3], C3 [×4], C4, C4 [×3], C22 [×3], S3 [×8], C6 [×4], C6 [×4], C8, C8, C2×C4 [×3], D4, D4 [×3], Q8, Q8, C32, Dic3 [×8], C12 [×4], C12 [×4], D6 [×8], C2×C6 [×4], C2×C8, D8, SD16, SD16, Q16, C4○D4 [×2], C3⋊S3 [×2], C3×C6, C3×C6, C3⋊C8 [×4], C24 [×4], Dic6 [×4], C4×S3 [×8], D12 [×8], C2×Dic3 [×4], C3⋊D4 [×4], C3×D4 [×4], C3×Q8 [×4], C4○D8, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C8 [×4], C24⋊C2 [×4], D4⋊S3 [×4], C3⋊Q16 [×4], C3×SD16 [×4], D42S3 [×4], Q83S3 [×4], C324C8, C3×C24, C324Q8, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C12⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, Q8×C32, Q8.7D6 [×4], C8×C3⋊S3, C242S3, C327D8, C327Q16, C32×SD16, C12.D6, C12.26D6, C24.40D6
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, Q8.7D6 [×4], D4×C3⋊S3, C24.40D6

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

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

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

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

42 conjugacy classes

class 1 2A2B2C2D3A3B3C3D4A4B4C4D4E6A6B6C6D6E6F6G6H8A8B8C8D12A12B12C12D12E12F12G12H24A···24H
order12222333344444666666668888121212121212121224···24
size1141836222224993622228888221818444488884···4

42 irreducible representations

dim11111111222222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6C4○D8S3×D4Q8.7D6
kernelC24.40D6C8×C3⋊S3C242S3C327D8C327Q16C32×SD16C12.D6C12.26D6C3×SD16C3⋊Dic3C2×C3⋊S3C24C3×D4C3×Q8C32C6C3
# reps11111111411444448

Matrix representation of C24.40D6 in GL6(𝔽73)

010000
72720000
000100
0072100
0000069
00001861
,
72720000
100000
0007200
0017200
00003238
00004841
,
100000
72720000
0072000
0072100
000004
0000180

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,0,18,0,0,0,0,69,61],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,32,48,0,0,0,0,38,41],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,4,0] >;

C24.40D6 in GAP, Magma, Sage, TeX

C_{24}._{40}D_6
% in TeX

G:=Group("C24.40D6");
// GroupNames label

G:=SmallGroup(288,773);
// by ID

G=gap.SmallGroup(288,773);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,135,100,346,185,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=b^6=1,c^2=a^12,b*a*b^-1=a^19,c*a*c^-1=a^11,c*b*c^-1=a^12*b^-1>;
// generators/relations

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