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G = Dic6.24D6order 288 = 25·32

11st non-split extension by Dic6 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: Dic6.24D6, C62.1C23, C32:32- 1+4, D4.11S32, C3:D4.1D6, D4:2S3:4S3, C3:3(Q8oD12), (S3xDic6):9C2, (C4xS3).15D6, (C3xD4).23D6, D6.4D6:4C2, D6.D6:5C2, D6.3D6:2C2, (S3xC6).8C23, (C3xC6).16C24, C6.16(S3xC23), D6.9(C22xS3), C12.D6:6C2, C32:7D4.C22, (C3xC12).29C23, C12.29(C22xS3), (C2xDic3).50D6, Dic3.D6:11C2, (S3xC12).32C22, C3:D12.2C22, D6:S3.8C22, C3:Dic3.19C23, (S3xDic3).3C22, C6.D6.6C22, C32:2Q8.9C22, Dic3.21(C22xS3), (C3xDic6).29C22, (C6xDic3).48C22, (C3xDic3).11C23, (D4xC32).21C22, C32:4Q8.21C22, C4.29(C2xS32), C22.1(C2xS32), C2.18(C22xS32), (C3xC3:D4).C22, (C3xD4:2S3):7C2, (C2xC6).2(C22xS3), (C2xC32:2Q8):16C2, (C2xC3:S3).22C23, (C4xC3:S3).42C22, (C2xC3:Dic3).104C22, SmallGroup(288,957)

Series: Derived Chief Lower central Upper central

C1C3xC6 — Dic6.24D6
C1C3C32C3xC6S3xC6S3xDic3S3xDic6 — Dic6.24D6
C32C3xC6 — Dic6.24D6
C1C2D4

Generators and relations for Dic6.24D6
 G = < a,b,c,d | a12=c6=1, b2=d2=a6, bab-1=a-1, cac-1=dad-1=a7, bc=cb, bd=db, dcd-1=a6c-1 >

Subgroups: 1018 in 312 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, D4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xQ8, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, Dic6, Dic6, C4xS3, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C3:D4, C2xC12, C3xD4, C3xD4, C3xQ8, 2- 1+4, C3xDic3, C3:Dic3, C3:Dic3, C3xC12, S3xC6, C2xC3:S3, C62, C2xDic6, C4oD12, D4:2S3, D4:2S3, S3xQ8, C3xC4oD4, S3xDic3, C6.D6, D6:S3, C3:D12, C32:2Q8, C32:2Q8, C3xDic6, S3xC12, C6xDic3, C3xC3:D4, C32:4Q8, C4xC3:S3, C2xC3:Dic3, C32:7D4, D4xC32, Q8oD12, S3xDic6, Dic3.D6, D6.D6, D6.3D6, D6.4D6, C2xC32:2Q8, C3xD4:2S3, C12.D6, Dic6.24D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22xS3, 2- 1+4, S32, S3xC23, C2xS32, Q8oD12, C22xS32, Dic6.24D6

Smallest permutation representation of Dic6.24D6
On 48 points
Generators in S48
(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)
(1 28 7 34)(2 27 8 33)(3 26 9 32)(4 25 10 31)(5 36 11 30)(6 35 12 29)(13 46 19 40)(14 45 20 39)(15 44 21 38)(16 43 22 37)(17 42 23 48)(18 41 24 47)
(1 18 9 14 5 22)(2 13 10 21 6 17)(3 20 11 16 7 24)(4 15 12 23 8 19)(25 44 29 48 33 40)(26 39 30 43 34 47)(27 46 31 38 35 42)(28 41 32 45 36 37)
(1 26 7 32)(2 33 8 27)(3 28 9 34)(4 35 10 29)(5 30 11 36)(6 25 12 31)(13 42 19 48)(14 37 20 43)(15 44 21 38)(16 39 22 45)(17 46 23 40)(18 41 24 47)

G:=sub<Sym(48)| (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), (1,28,7,34)(2,27,8,33)(3,26,9,32)(4,25,10,31)(5,36,11,30)(6,35,12,29)(13,46,19,40)(14,45,20,39)(15,44,21,38)(16,43,22,37)(17,42,23,48)(18,41,24,47), (1,18,9,14,5,22)(2,13,10,21,6,17)(3,20,11,16,7,24)(4,15,12,23,8,19)(25,44,29,48,33,40)(26,39,30,43,34,47)(27,46,31,38,35,42)(28,41,32,45,36,37), (1,26,7,32)(2,33,8,27)(3,28,9,34)(4,35,10,29)(5,30,11,36)(6,25,12,31)(13,42,19,48)(14,37,20,43)(15,44,21,38)(16,39,22,45)(17,46,23,40)(18,41,24,47)>;

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), (1,28,7,34)(2,27,8,33)(3,26,9,32)(4,25,10,31)(5,36,11,30)(6,35,12,29)(13,46,19,40)(14,45,20,39)(15,44,21,38)(16,43,22,37)(17,42,23,48)(18,41,24,47), (1,18,9,14,5,22)(2,13,10,21,6,17)(3,20,11,16,7,24)(4,15,12,23,8,19)(25,44,29,48,33,40)(26,39,30,43,34,47)(27,46,31,38,35,42)(28,41,32,45,36,37), (1,26,7,32)(2,33,8,27)(3,28,9,34)(4,35,10,29)(5,30,11,36)(6,25,12,31)(13,42,19,48)(14,37,20,43)(15,44,21,38)(16,39,22,45)(17,46,23,40)(18,41,24,47) );

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)], [(1,28,7,34),(2,27,8,33),(3,26,9,32),(4,25,10,31),(5,36,11,30),(6,35,12,29),(13,46,19,40),(14,45,20,39),(15,44,21,38),(16,43,22,37),(17,42,23,48),(18,41,24,47)], [(1,18,9,14,5,22),(2,13,10,21,6,17),(3,20,11,16,7,24),(4,15,12,23,8,19),(25,44,29,48,33,40),(26,39,30,43,34,47),(27,46,31,38,35,42),(28,41,32,45,36,37)], [(1,26,7,32),(2,33,8,27),(3,28,9,34),(4,35,10,29),(5,30,11,36),(6,25,12,31),(13,42,19,48),(14,37,20,43),(15,44,21,38),(16,39,22,45),(17,46,23,40),(18,41,24,47)]])

42 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B···4G4H4I4J6A6B6C···6G6H6I6J6K12A12B12C12D12E12F12G12H12I12J12K
order122222233344···4444666···666661212121212121212121212
size1122661822426···6181818224···4881212446666812121212

42 irreducible representations

dim111111111222222444448
type+++++++++++++++-+++--
imageC1C2C2C2C2C2C2C2C2S3D6D6D6D6D62- 1+4S32C2xS32C2xS32Q8oD12Dic6.24D6
kernelDic6.24D6S3xDic6Dic3.D6D6.D6D6.3D6D6.4D6C2xC32:2Q8C3xD4:2S3C12.D6D4:2S3Dic6C4xS3C2xDic3C3:D4C3xD4C32D4C4C22C3C1
# reps121142221222442111241

Matrix representation of Dic6.24D6 in GL6(F13)

110000
1200000
0012270
0011107
0040111
0004212
,
100000
12120000
005000
000500
0071280
001608
,
100000
010000
002200
0011400
001121111
0011029
,
100000
010000
008000
008500
000050
000058

G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,11,4,0,0,0,2,1,0,4,0,0,7,0,1,2,0,0,0,7,11,12],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,5,0,7,1,0,0,0,5,12,6,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,11,11,11,0,0,2,4,2,0,0,0,0,0,11,2,0,0,0,0,11,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,8,0,0,0,0,0,5,0,0,0,0,0,0,5,5,0,0,0,0,0,8] >;

Dic6.24D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6._{24}D_6
% in TeX

G:=Group("Dic6.24D6");
// GroupNames label

G:=SmallGroup(288,957);
// by ID

G=gap.SmallGroup(288,957);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,185,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^6=1,b^2=d^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^7,b*c=c*b,b*d=d*b,d*c*d^-1=a^6*c^-1>;
// generators/relations

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