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G = C24.94D6order 288 = 25·32

28th non-split extension by C24 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: C24.94D6, C62.9C8, C24.12Dic3, C3210M5(2), C12.3(C3⋊C8), (C3×C12).7C8, (C3×C24).12C4, (C6×C12).24C4, (C6×C24).21C2, (C2×C24).31S3, C4.(C324C8), C24.S39C2, C8.2(C3⋊Dic3), C32(C12.C8), (C3×C24).71C22, (C2×C12).19Dic3, C12.57(C2×Dic3), C22.(C324C8), C6.15(C2×C3⋊C8), C8.21(C2×C3⋊S3), (C2×C6).8(C3⋊C8), (C2×C8).7(C3⋊S3), (C3×C6).45(C2×C8), C4.10(C2×C3⋊Dic3), C2.4(C2×C324C8), (C3×C12).133(C2×C4), (C2×C4).5(C3⋊Dic3), SmallGroup(288,287)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C24.94D6
C1C3C32C3×C6C3×C12C3×C24C24.S3 — C24.94D6
C32C3×C6 — C24.94D6
C1C8C2×C8

Generators and relations for C24.94D6
 G = < a,b,c | a24=b6=1, c2=a21, ab=ba, cac-1=a17, cbc-1=a12b-1 >

Subgroups: 132 in 78 conjugacy classes, 57 normal (19 characteristic)
C1, C2, C2, C3 [×4], C4 [×2], C22, C6 [×4], C6 [×4], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], C16 [×2], C2×C8, C3×C6, C3×C6, C24 [×8], C2×C12 [×4], M5(2), C3×C12 [×2], C62, C3⋊C16 [×8], C2×C24 [×4], C3×C24 [×2], C6×C12, C12.C8 [×4], C24.S3 [×2], C6×C24, C24.94D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C8 [×2], C2×C4, Dic3 [×8], D6 [×4], C2×C8, C3⋊S3, C3⋊C8 [×8], C2×Dic3 [×4], M5(2), C3⋊Dic3 [×2], C2×C3⋊S3, C2×C3⋊C8 [×4], C324C8 [×2], C2×C3⋊Dic3, C12.C8 [×4], C2×C324C8, C24.94D6

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

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

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

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

84 conjugacy classes

class 1 2A2B3A3B3C3D4A4B4C6A···6L8A8B8C8D8E8F12A···12P16A···16H24A···24AF
order12233334446···688888812···1216···1624···24
size11222221122···21111222···218···182···2

84 irreducible representations

dim111111122222222
type++++-+-
imageC1C2C2C4C4C8C8S3Dic3D6Dic3C3⋊C8C3⋊C8M5(2)C12.C8
kernelC24.94D6C24.S3C6×C24C3×C24C6×C12C3×C12C62C2×C24C24C24C2×C12C12C2×C6C32C3
# reps1212244444488432

Matrix representation of C24.94D6 in GL4(𝔽97) generated by

22000
02200
00730
00088
,
35000
26100
00610
00062
,
773100
412000
00035
00240
G:=sub<GL(4,GF(97))| [22,0,0,0,0,22,0,0,0,0,73,0,0,0,0,88],[35,2,0,0,0,61,0,0,0,0,61,0,0,0,0,62],[77,41,0,0,31,20,0,0,0,0,0,24,0,0,35,0] >;

C24.94D6 in GAP, Magma, Sage, TeX

C_{24}._{94}D_6
% in TeX

G:=Group("C24.94D6");
// GroupNames label

G:=SmallGroup(288,287);
// by ID

G=gap.SmallGroup(288,287);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,58,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=b^6=1,c^2=a^21,a*b=b*a,c*a*c^-1=a^17,c*b*c^-1=a^12*b^-1>;
// generators/relations

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