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G = C3xQ8.11D6order 288 = 25·32

Direct product of C3 and Q8.11D6

direct product, metabelian, supersoluble, monomial

Aliases: C3xQ8.11D6, C62.125D4, (C6xQ8):6C6, (C6xQ8):13S3, C3:Q16:5C6, C6.54(C6xD4), Q8:2S3:5C6, C4oD12.5C6, (C3xC12).89D4, C12.19(C3xD4), C4.Dic3:7C6, (C3xQ8).71D6, Q8.16(S3xC6), D12.10(C2xC6), (C2xC12).243D6, Dic6.9(C2xC6), C12.15(C22xC6), (C3xC12).86C23, C12.104(C3:D4), C12.166(C22xS3), (C6xC12).125C22, (C3xD12).39C22, C32:22(C8.C22), (C3xDic6).39C22, (Q8xC32).23C22, (Q8xC3xC6):2C2, C3:C8.3(C2xC6), C4.15(S3xC2xC6), (C2xQ8):6(C3xS3), (C2xC4).18(S3xC6), (C2xC6).51(C3xD4), C4.17(C3xC3:D4), C2.18(C6xC3:D4), C3:4(C3xC8.C22), (C2xC12).36(C2xC6), (C3xC3:Q16):13C2, (C3xC6).262(C2xD4), C6.155(C2xC3:D4), (C3xC3:C8).22C22, (C3xC4.Dic3):6C2, (C3xQ8).18(C2xC6), (C3xQ8:2S3):11C2, (C3xC4oD12).11C2, (C2xC6).64(C3:D4), C22.11(C3xC3:D4), SmallGroup(288,713)

Series: Derived Chief Lower central Upper central

C1C12 — C3xQ8.11D6
C1C3C6C12C3xC12C3xD12C3xC4oD12 — C3xQ8.11D6
C3C6C12 — C3xQ8.11D6
C1C6C2xC12C6xQ8

Generators and relations for C3xQ8.11D6
 G = < a,b,c,d,e | a3=b4=1, c2=d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d5 >

Subgroups: 298 in 139 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, M4(2), SD16, Q16, C2xQ8, C4oD4, C3xS3, C3xC6, C3xC6, C3:C8, C24, Dic6, C4xS3, D12, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C3xQ8, C8.C22, C3xDic3, C3xC12, C3xC12, S3xC6, C62, C4.Dic3, Q8:2S3, C3:Q16, C3xM4(2), C3xSD16, C3xQ16, C4oD12, C6xQ8, C6xQ8, C3xC4oD4, C3xC3:C8, C3xDic6, S3xC12, C3xD12, C3xC3:D4, C6xC12, C6xC12, Q8xC32, Q8xC32, Q8.11D6, C3xC8.C22, C3xC4.Dic3, C3xQ8:2S3, C3xC3:Q16, C3xC4oD12, Q8xC3xC6, C3xQ8.11D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2xC6, C2xD4, C3xS3, C3:D4, C3xD4, C22xS3, C22xC6, C8.C22, S3xC6, C2xC3:D4, C6xD4, C3xC3:D4, S3xC2xC6, Q8.11D6, C3xC8.C22, C6xC3:D4, C3xQ8.11D6

Smallest permutation representation of C3xQ8.11D6
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 4 7 10)(2 5 8 11)(3 6 9 12)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 46 43 40)(38 47 44 41)(39 48 45 42)
(1 17 7 23)(2 24 8 18)(3 19 9 13)(4 14 10 20)(5 21 11 15)(6 16 12 22)(25 37 31 43)(26 44 32 38)(27 39 33 45)(28 46 34 40)(29 41 35 47)(30 48 36 42)
(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 36 7 30)(2 29 8 35)(3 34 9 28)(4 27 10 33)(5 32 11 26)(6 25 12 31)(13 37 19 43)(14 42 20 48)(15 47 21 41)(16 40 22 46)(17 45 23 39)(18 38 24 44)

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

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

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

63 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E6A6B6C···6M6N6O8A8B12A12B12C12D12E···12Z12AA12AB24A24B24C24D
order12223333344444666···666881212121212···12121224242424
size1121211222224412112···21212121222224···4121212121212

63 irreducible representations

dim111111111111222222222222224444
type+++++++++++-
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D4D6D6C3xS3C3:D4C3xD4C3:D4C3xD4S3xC6S3xC6C3xC3:D4C3xC3:D4C8.C22Q8.11D6C3xC8.C22C3xQ8.11D6
kernelC3xQ8.11D6C3xC4.Dic3C3xQ8:2S3C3xC3:Q16C3xC4oD12Q8xC3xC6Q8.11D6C4.Dic3Q8:2S3C3:Q16C4oD12C6xQ8C6xQ8C3xC12C62C2xC12C3xQ8C2xQ8C12C12C2xC6C2xC6C2xC4Q8C4C22C32C3C3C1
# reps112211224422111122222224441224

Matrix representation of C3xQ8.11D6 in GL4(F7) generated by

2000
0200
0020
0002
,
1653
2022
5564
1630
,
0342
0636
2265
6112
,
4051
4412
5420
4664
,
5310
2301
3424
5554
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,2,5,1,6,0,5,6,5,2,6,3,3,2,4,0],[0,0,2,6,3,6,2,1,4,3,6,1,2,6,5,2],[4,4,5,4,0,4,4,6,5,1,2,6,1,2,0,4],[5,2,3,5,3,3,4,5,1,0,2,5,0,1,4,4] >;

C3xQ8.11D6 in GAP, Magma, Sage, TeX

C_3\times Q_8._{11}D_6
% in TeX

G:=Group("C3xQ8.11D6");
// GroupNames label

G:=SmallGroup(288,713);
// by ID

G=gap.SmallGroup(288,713);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,268,2524,648,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=1,c^2=d^6=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b*c,e*d*e^-1=d^5>;
// generators/relations

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