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G = C3xC23.26D6order 288 = 25·32

Direct product of C3 and C23.26D6

direct product, metabelian, supersoluble, monomial

Aliases: C3xC23.26D6, C62.194C23, (C6xC12):14C4, (C2xC12):6C12, C4:Dic3:17C6, (C2xC12):9Dic3, C12.44(C2xC12), (C4xDic3):15C6, (C2xC12).447D6, C23.31(S3xC6), C4.15(C6xDic3), (Dic3xC12):31C2, (C22xC12).42S3, (C22xC12).23C6, C62.110(C2xC4), C6.24(C22xC12), C12.72(C2xDic3), C6.D4.5C6, (C22xC6).126D6, C6.125(C4oD12), C22.6(C6xDic3), (C6xC12).326C22, (C2xC62).97C22, C6.44(C22xDic3), C32:17(C42:C2), (C6xDic3).134C22, (C2xC6xC12).14C2, C2.5(Dic3xC2xC6), (C2xC4):4(C3xDic3), C6.15(C3xC4oD4), C2.4(C3xC4oD12), C22.22(S3xC2xC6), (C2xC4).102(S3xC6), (C2xC6).44(C2xC12), (C3xC4:Dic3):35C2, C3:4(C3xC42:C2), (C2xC12).110(C2xC6), (C3xC12).137(C2xC4), (C2xC6).49(C22xC6), (C22xC6).61(C2xC6), (C22xC4).11(C3xS3), (C2xC6).27(C2xDic3), (C3xC6).103(C4oD4), (C3xC6).115(C22xC4), (C2xC6).327(C22xS3), (C2xDic3).34(C2xC6), (C3xC6.D4).10C2, SmallGroup(288,697)

Series: Derived Chief Lower central Upper central

C1C6 — C3xC23.26D6
C1C3C6C2xC6C62C6xDic3Dic3xC12 — C3xC23.26D6
C3C6 — C3xC23.26D6
C1C2xC12C22xC12

Generators and relations for C3xC23.26D6
 G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e6=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 314 in 179 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2xC4, C2xC4, C2xC4, C23, C32, Dic3, C12, C12, C2xC6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C22xC4, C3xC6, C3xC6, C3xC6, C2xDic3, C2xC12, C2xC12, C2xC12, C22xC6, C22xC6, C42:C2, C3xDic3, C3xC12, C62, C62, C62, C4xDic3, C4:Dic3, C6.D4, C4xC12, C3xC22:C4, C3xC4:C4, C22xC12, C22xC12, C6xDic3, C6xC12, C6xC12, C2xC62, C23.26D6, C3xC42:C2, Dic3xC12, C3xC4:Dic3, C3xC6.D4, C2xC6xC12, C3xC23.26D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, C23, Dic3, C12, D6, C2xC6, C22xC4, C4oD4, C3xS3, C2xDic3, C2xC12, C22xS3, C22xC6, C42:C2, C3xDic3, S3xC6, C4oD12, C22xDic3, C22xC12, C3xC4oD4, C6xDic3, S3xC2xC6, C23.26D6, C3xC42:C2, C3xC4oD12, Dic3xC2xC6, C3xC23.26D6

Smallest permutation representation of C3xC23.26D6
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(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 41 45)(38 42 46)(39 43 47)(40 44 48)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 25)(9 26)(10 27)(11 28)(12 29)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 37)(22 38)(23 39)(24 40)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 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 38 36 16)(2 43 25 21)(3 48 26 14)(4 41 27 19)(5 46 28 24)(6 39 29 17)(7 44 30 22)(8 37 31 15)(9 42 32 20)(10 47 33 13)(11 40 34 18)(12 45 35 23)

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

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

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

108 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C4D4E4F4G···4N6A···6F6G···6AE12A···12H12I···12AJ12AK···12AZ
order122222333334444444···46···66···612···1212···1212···12
size111122112221111226···61···12···21···12···26···6

108 irreducible representations

dim111111111111222222222222
type++++++-++
imageC1C2C2C2C2C3C4C6C6C6C6C12S3Dic3D6D6C4oD4C3xS3C3xDic3S3xC6S3xC6C4oD12C3xC4oD4C3xC4oD12
kernelC3xC23.26D6Dic3xC12C3xC4:Dic3C3xC6.D4C2xC6xC12C23.26D6C6xC12C4xDic3C4:Dic3C6.D4C22xC12C2xC12C22xC12C2xC12C2xC12C22xC6C3xC6C22xC4C2xC4C2xC4C23C6C6C2
# reps12221284442161421428428816

Matrix representation of C3xC23.26D6 in GL4(F13) generated by

3000
0300
0010
0001
,
1000
0100
0010
001112
,
12000
01200
0010
0001
,
1000
0100
00120
00012
,
4000
01000
0080
0008
,
01000
9000
001212
0021
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,11,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[4,0,0,0,0,10,0,0,0,0,8,0,0,0,0,8],[0,9,0,0,10,0,0,0,0,0,12,2,0,0,12,1] >;

C3xC23.26D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{26}D_6
% in TeX

G:=Group("C3xC2^3.26D6");
// GroupNames label

G:=SmallGroup(288,697);
// by ID

G=gap.SmallGroup(288,697);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,344,1094,9414]);
// Polycyclic

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

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