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G = C24.47D6order 288 = 25·32

47th non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C24.47D6, C12.40(C4×S3), C24⋊S310C2, C12⋊S3.3C4, C34(D12.C4), (C2×C12).146D6, C3212(C8○D4), (C3×M4(2))⋊6S3, C62.64(C2×C4), C327D4.3C4, M4(2)⋊5(C3⋊S3), C324Q8.3C4, (C3×C24).49C22, (C3×C12).179C23, (C6×C12).137C22, C12.210(C22×S3), C12.59D6.6C2, (C32×M4(2))⋊10C2, C324C8.40C22, C4.5(C4×C3⋊S3), C6.76(S3×C2×C4), (C8×C3⋊S3)⋊16C2, C8.12(C2×C3⋊S3), (C2×C6).23(C4×S3), C22.1(C4×C3⋊S3), (C3×C12).74(C2×C4), (C2×C324C8)⋊8C2, C4.39(C22×C3⋊S3), (C4×C3⋊S3).94C22, C3⋊Dic3.39(C2×C4), (C3×C6).107(C22×C4), C2.17(C2×C4×C3⋊S3), (C2×C4).45(C2×C3⋊S3), (C2×C3⋊S3).33(C2×C4), SmallGroup(288,764)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C24.47D6
C1C3C32C3×C6C3×C12C4×C3⋊S3C12.59D6 — C24.47D6
C32C3×C6 — C24.47D6
C1C4M4(2)

Generators and relations for C24.47D6
 G = < a,b,c | a24=b6=1, c2=a12, bab-1=a13, cac-1=a17, cbc-1=a12b-1 >

Subgroups: 572 in 186 conjugacy classes, 73 normal (21 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 [×3], M4(2), M4(2) [×2], 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], C2×C3⋊C8 [×4], C3×M4(2) [×4], C4○D12 [×4], C324C8 [×2], C3×C24 [×2], C324Q8, C4×C3⋊S3 [×2], C12⋊S3, C327D4 [×2], C6×C12, D12.C4 [×4], C8×C3⋊S3 [×2], C24⋊S3 [×2], C2×C324C8, C32×M4(2), C12.59D6, C24.47D6
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, D12.C4 [×4], C2×C4×C3⋊S3, C24.47D6

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

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

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

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

60 conjugacy classes

class 1 2A2B2C2D3A3B3C3D4A4B4C4D4E6A6B6C6D6E6F6G6H8A8B8C8D8E8F8G8H8I8J12A···12H12I12J12K12L24A···24P
order1222233334444466666666888888888812···121212121224···24
size112181822221121818222244442222999918182···244444···4

60 irreducible representations

dim1111111112222224
type+++++++++
imageC1C2C2C2C2C2C4C4C4S3D6D6C4×S3C4×S3C8○D4D12.C4
kernelC24.47D6C8×C3⋊S3C24⋊S3C2×C324C8C32×M4(2)C12.59D6C324Q8C12⋊S3C327D4C3×M4(2)C24C2×C12C12C2×C6C32C3
# reps1221112244848848

Matrix representation of C24.47D6 in GL6(𝔽73)

72720000
100000
00464600
0027000
0000630
00004310
,
7200000
0720000
0007200
001100
00005139
00005522
,
100000
72720000
0007200
0072000
0000270
0000846

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,46,27,0,0,0,0,46,0,0,0,0,0,0,0,63,43,0,0,0,0,0,10],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,1,0,0,0,0,0,0,51,55,0,0,0,0,39,22],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,27,8,0,0,0,0,0,46] >;

C24.47D6 in GAP, Magma, Sage, TeX

C_{24}._{47}D_6
% in TeX

G:=Group("C24.47D6");
// GroupNames label

G:=SmallGroup(288,764);
// by ID

G=gap.SmallGroup(288,764);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,219,58,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=b^6=1,c^2=a^12,b*a*b^-1=a^13,c*a*c^-1=a^17,c*b*c^-1=a^12*b^-1>;
// generators/relations

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