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G = C6.D24order 288 = 25·32

8th non-split extension by C6 of D24 acting via D24/C24=C2

metabelian, supersoluble, monomial

Aliases: C482S3, C6.8D24, C24.73D6, C326SD32, C12.44D12, (C3×C48)⋊3C2, C162(C3⋊S3), (C3×C6).24D8, C31(C48⋊C2), C325Q161C2, C325D8.1C2, (C3×C12).119D4, C4.2(C12⋊S3), C2.4(C325D8), (C3×C24).51C22, C8.14(C2×C3⋊S3), SmallGroup(288,275)

Series: Derived Chief Lower central Upper central

C1C3×C24 — C6.D24
C1C3C32C3×C6C3×C12C3×C24C325D8 — C6.D24
C32C3×C6C3×C12C3×C24 — C6.D24
C1C2C4C8C16

Generators and relations for C6.D24
 G = < a,b,c | a6=1, b24=c2=a3, ab=ba, cac-1=a-1, cbc-1=b23 >

Subgroups: 536 in 78 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2, C3 [×4], C4, C4, C22, S3 [×4], C6 [×4], C8, D4, Q8, C32, Dic3 [×4], C12 [×4], D6 [×4], C16, D8, Q16, C3⋊S3, C3×C6, C24 [×4], Dic6 [×4], D12 [×4], SD32, C3⋊Dic3, C3×C12, C2×C3⋊S3, C48 [×4], D24 [×4], Dic12 [×4], C3×C24, C324Q8, C12⋊S3, C48⋊C2 [×4], C3×C48, C325D8, C325Q16, C6.D24
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D8, C3⋊S3, D12 [×4], SD32, C2×C3⋊S3, D24 [×4], C12⋊S3, C48⋊C2 [×4], C325D8, C6.D24

Smallest permutation representation of C6.D24
On 144 points
Generators in S144
(1 89 99 25 65 123)(2 90 100 26 66 124)(3 91 101 27 67 125)(4 92 102 28 68 126)(5 93 103 29 69 127)(6 94 104 30 70 128)(7 95 105 31 71 129)(8 96 106 32 72 130)(9 49 107 33 73 131)(10 50 108 34 74 132)(11 51 109 35 75 133)(12 52 110 36 76 134)(13 53 111 37 77 135)(14 54 112 38 78 136)(15 55 113 39 79 137)(16 56 114 40 80 138)(17 57 115 41 81 139)(18 58 116 42 82 140)(19 59 117 43 83 141)(20 60 118 44 84 142)(21 61 119 45 85 143)(22 62 120 46 86 144)(23 63 121 47 87 97)(24 64 122 48 88 98)
(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 24 25 48)(2 47 26 23)(3 22 27 46)(4 45 28 21)(5 20 29 44)(6 43 30 19)(7 18 31 42)(8 41 32 17)(9 16 33 40)(10 39 34 15)(11 14 35 38)(12 37 36 13)(49 138 73 114)(50 113 74 137)(51 136 75 112)(52 111 76 135)(53 134 77 110)(54 109 78 133)(55 132 79 108)(56 107 80 131)(57 130 81 106)(58 105 82 129)(59 128 83 104)(60 103 84 127)(61 126 85 102)(62 101 86 125)(63 124 87 100)(64 99 88 123)(65 122 89 98)(66 97 90 121)(67 120 91 144)(68 143 92 119)(69 118 93 142)(70 141 94 117)(71 116 95 140)(72 139 96 115)

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

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

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

75 conjugacy classes

class 1 2A2B3A3B3C3D4A4B6A6B6C6D8A8B12A···12H16A16B16C16D24A···24P48A···48AF
order12233334466668812···121616161624···2448···48
size117222222722222222···222222···22···2

75 irreducible representations

dim111122222222
type++++++++++
imageC1C2C2C2S3D4D6D8D12SD32D24C48⋊C2
kernelC6.D24C3×C48C325D8C325Q16C48C3×C12C24C3×C6C12C32C6C3
# reps11114142841632

Matrix representation of C6.D24 in GL4(𝔽97) generated by

09600
19600
00960
00096
,
586800
292900
009552
004543
,
586800
293900
00245
004395
G:=sub<GL(4,GF(97))| [0,1,0,0,96,96,0,0,0,0,96,0,0,0,0,96],[58,29,0,0,68,29,0,0,0,0,95,45,0,0,52,43],[58,29,0,0,68,39,0,0,0,0,2,43,0,0,45,95] >;

C6.D24 in GAP, Magma, Sage, TeX

C_6.D_{24}
% in TeX

G:=Group("C6.D24");
// GroupNames label

G:=SmallGroup(288,275);
// by ID

G=gap.SmallGroup(288,275);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,590,58,675,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^6=1,b^24=c^2=a^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^23>;
// generators/relations

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