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G = C3×C23.28D6order 288 = 25·32

Direct product of C3 and C23.28D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×C23.28D6, C62.120D4, C62.197C23, D6⋊C42C6, C6.42(C6×D4), Dic3⋊C43C6, (C22×C12)⋊6C6, (C22×C12)⋊7S3, C6.D46C6, (C2×C12).353D6, C23.33(S3×C6), (C22×C6).128D6, C6.127(C4○D12), (C6×C12).283C22, (C2×C62).100C22, (C6×Dic3).98C22, C3221(C22.D4), (C2×C6×C12)⋊2C2, C2.6(C6×C3⋊D4), (C3×D6⋊C4)⋊33C2, (C2×C4).68(S3×C6), (C22×C4)⋊7(C3×S3), C6.17(C3×C4○D4), (C2×C6).46(C3×D4), (C2×C3⋊D4).6C6, C22.55(S3×C2×C6), (C2×C12).92(C2×C6), C2.18(C3×C4○D12), (C3×C6).253(C2×D4), (C6×C3⋊D4).13C2, C6.143(C2×C3⋊D4), (S3×C2×C6).59C22, C22.9(C3×C3⋊D4), (C3×Dic3⋊C4)⋊35C2, (C2×C6).62(C3⋊D4), (C22×S3).9(C2×C6), (C22×C6).64(C2×C6), (C2×C6).52(C22×C6), C34(C3×C22.D4), (C3×C6).105(C4○D4), (C3×C6.D4)⋊23C2, (C2×C6).330(C22×S3), (C2×Dic3).10(C2×C6), SmallGroup(288,700)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C3×C23.28D6
C1C3C6C2×C6C62S3×C2×C6C6×C3⋊D4 — C3×C23.28D6
C3C2×C6 — C3×C23.28D6
C1C2×C6C22×C12

Generators and relations for C3×C23.28D6
 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=ce5 >

Subgroups: 410 in 183 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×2], C22 [×5], S3, C6 [×2], C6 [×4], C6 [×12], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C32, Dic3 [×3], C12 [×11], D6 [×3], C2×C6 [×2], C2×C6 [×4], C2×C6 [×14], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3×S3, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12 [×4], C2×C12 [×13], C3×D4 [×2], C22×S3, C22×C6 [×2], C22×C6 [×2], C22.D4, C3×Dic3 [×3], C3×C12 [×2], S3×C6 [×3], C62, C62 [×2], C62 [×2], Dic3⋊C4 [×2], D6⋊C4 [×2], C6.D4, C3×C22⋊C4 [×3], C3×C4⋊C4 [×2], C2×C3⋊D4, C22×C12 [×2], C22×C12, C6×D4, C6×Dic3, C6×Dic3 [×2], C3×C3⋊D4 [×2], C6×C12 [×2], C6×C12 [×2], S3×C2×C6, C2×C62, C23.28D6, C3×C22.D4, C3×Dic3⋊C4 [×2], C3×D6⋊C4 [×2], C3×C6.D4, C6×C3⋊D4, C2×C6×C12, C3×C23.28D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C4○D4 [×2], C3×S3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C22×C6, C22.D4, S3×C6 [×3], C4○D12 [×2], C2×C3⋊D4, C6×D4, C3×C4○D4 [×2], C3×C3⋊D4 [×2], S3×C2×C6, C23.28D6, C3×C22.D4, C3×C4○D12 [×2], C6×C3⋊D4, C3×C23.28D6

Smallest permutation representation of C3×C23.28D6
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 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 25)(12 26)(13 46)(14 47)(15 48)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)
(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 18 33 45)(2 44 34 17)(3 16 35 43)(4 42 36 15)(5 14 25 41)(6 40 26 13)(7 24 27 39)(8 38 28 23)(9 22 29 37)(10 48 30 21)(11 20 31 47)(12 46 32 19)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C3D3E4A4B4C4D4E4F4G6A···6F6G···6AE6AF6AG12A···12AF12AG···12AL
order12222223333344444446···66···66612···1212···12
size111122121122222221212121···12···212122···212···12

90 irreducible representations

dim11111111111122222222222222
type++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D6D6C4○D4C3×S3C3⋊D4C3×D4S3×C6S3×C6C4○D12C3×C4○D4C3×C3⋊D4C3×C4○D12
kernelC3×C23.28D6C3×Dic3⋊C4C3×D6⋊C4C3×C6.D4C6×C3⋊D4C2×C6×C12C23.28D6Dic3⋊C4D6⋊C4C6.D4C2×C3⋊D4C22×C12C22×C12C62C2×C12C22×C6C3×C6C22×C4C2×C6C2×C6C2×C4C23C6C6C22C2
# reps122111244222122142444288816

Matrix representation of C3×C23.28D6 in GL4(𝔽13) generated by

1000
0100
0090
0009
,
1000
01200
0010
00012
,
1000
0100
00120
00012
,
12000
01200
00120
00012
,
8000
0800
0020
0007
,
0800
8000
0006
00110
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[8,0,0,0,0,8,0,0,0,0,2,0,0,0,0,7],[0,8,0,0,8,0,0,0,0,0,0,11,0,0,6,0] >;

C3×C23.28D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{28}D_6
% in TeX

G:=Group("C3xC2^3.28D6");
// GroupNames label

G:=SmallGroup(288,700);
// by ID

G=gap.SmallGroup(288,700);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,590,268,9414]);
// 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=c*e^5>;
// generators/relations

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