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G = C3xQ8.15D6order 288 = 25·32

Direct product of C3 and Q8.15D6

direct product, metabelian, supersoluble, monomial

Aliases: C3xQ8.15D6, C32:62- 1+4, C62.272C23, (S3xQ8):7C6, C4oD12:6C6, (C6xQ8):16S3, (C6xQ8):11C6, Q8:3S3:7C6, (C3xQ8).75D6, Q8.20(S3xC6), D12.13(C2xC6), (C2xC12).255D6, C6.10(C23xC6), C6.78(S3xC23), (C3xC6).47C24, D6.5(C22xC6), (S3xC6).32C23, C12.24(C22xC6), Dic6.13(C2xC6), C3:1(C3x2- 1+4), (S3xC12).36C22, C12.175(C22xS3), (C6xC12).166C22, (C3xC12).125C23, (C3xD12).44C22, Dic3.6(C22xC6), (C3xDic6).45C22, (C3xDic3).33C23, (Q8xC32).31C22, C4.24(S3xC2xC6), (Q8xC3xC6):11C2, (C3xS3xQ8):11C2, (C2xQ8):9(C3xS3), C22.7(S3xC2xC6), (C4xS3).5(C2xC6), (C2xC4).22(S3xC6), C3:D4.2(C2xC6), C2.11(S3xC22xC6), (C3xC4oD12):16C2, (C2xC12).48(C2xC6), (C3xQ8).22(C2xC6), (C3xQ8:3S3):11C2, (C2xC6).73(C22xC6), (C3xC3:D4).5C22, (C2xC6).165(C22xS3), SmallGroup(288,997)

Series: Derived Chief Lower central Upper central

C1C6 — C3xQ8.15D6
C1C3C6C3xC6S3xC6S3xC12C3xS3xQ8 — C3xQ8.15D6
C3C6 — C3xQ8.15D6
C1C6C6xQ8

Generators and relations for C3xQ8.15D6
 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=ece-1=b2c, ede-1=d5 >

Subgroups: 586 in 311 conjugacy classes, 170 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C2xQ8, C2xQ8, C4oD4, C3xS3, C3xC6, C3xC6, Dic6, C4xS3, D12, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C3xQ8, 2- 1+4, C3xDic3, C3xC12, S3xC6, C62, C4oD12, S3xQ8, Q8:3S3, C6xQ8, C6xQ8, C3xC4oD4, C3xDic6, S3xC12, C3xD12, C3xC3:D4, C6xC12, Q8xC32, Q8.15D6, C3x2- 1+4, C3xC4oD12, C3xS3xQ8, C3xQ8:3S3, Q8xC3xC6, C3xQ8.15D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2xC6, C24, C3xS3, C22xS3, C22xC6, 2- 1+4, S3xC6, S3xC23, C23xC6, S3xC2xC6, Q8.15D6, C3x2- 1+4, S3xC22xC6, C3xQ8.15D6

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

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

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

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

81 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C3D3E4A···4F4G4H4I4J6A6B6C···6M6N···6U12A···12L12M···12AD12AE···12AL
order1222222333334···44444666···66···612···1212···1212···12
size1126666112222···26666112···26···62···24···46···6

81 irreducible representations

dim11111111112222224444
type++++++++-
imageC1C2C2C2C2C3C6C6C6C6S3D6D6C3xS3S3xC6S3xC62- 1+4Q8.15D6C3x2- 1+4C3xQ8.15D6
kernelC3xQ8.15D6C3xC4oD12C3xS3xQ8C3xQ8:3S3Q8xC3xC6Q8.15D6C4oD12S3xQ8Q8:3S3C6xQ8C6xQ8C2xC12C3xQ8C2xQ8C2xC4Q8C32C3C3C1
# reps164412128821342681224

Matrix representation of C3xQ8.15D6 in GL4(F7) generated by

2000
0200
0020
0002
,
4341
0160
0260
4553
,
5111
0252
1311
1436
,
0323
4145
3430
2563
,
4501
0661
4164
4655
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,0,0,4,3,1,2,5,4,6,6,5,1,0,0,3],[5,0,1,1,1,2,3,4,1,5,1,3,1,2,1,6],[0,4,3,2,3,1,4,5,2,4,3,6,3,5,0,3],[4,0,4,4,5,6,1,6,0,6,6,5,1,1,4,5] >;

C3xQ8.15D6 in GAP, Magma, Sage, TeX

C_3\times Q_8._{15}D_6
% in TeX

G:=Group("C3xQ8.15D6");
// GroupNames label

G:=SmallGroup(288,997);
// by ID

G=gap.SmallGroup(288,997);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,344,555,268,1571,9414]);
// 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=e*c*e^-1=b^2*c,e*d*e^-1=d^5>;
// generators/relations

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