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G = C3×C23.21D6order 288 = 25·32

Direct product of C3 and C23.21D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×C23.21D6, C62.62D4, C62.177C23, D6⋊C46C6, C6.6(C6×D4), C4⋊Dic35C6, C2.8(C6×D12), C6.94(C2×D12), (C2×C6).46D12, (C2×C12).232D6, C23.26(S3×C6), C22.4(C3×D12), (C22×Dic3)⋊5C6, (C22×C6).107D6, (C6×C12).191C22, (C2×C62).53C22, C6.115(D42S3), (C6×Dic3).123C22, C3216(C22.D4), (C2×C4).7(S3×C6), (Dic3×C2×C6)⋊6C2, (C2×C6).5(C3×D4), (C3×D6⋊C4)⋊18C2, C22⋊C46(C3×S3), (C3×C22⋊C4)⋊4C6, (C2×C12).3(C2×C6), C6.23(C3×C4○D4), (C2×C3⋊D4).5C6, C22.45(S3×C2×C6), (C3×C22⋊C4)⋊14S3, (C3×C4⋊Dic3)⋊29C2, (C3×C6).176(C2×D4), (C6×C3⋊D4).12C2, (S3×C2×C6).56C22, C2.10(C3×D42S3), (C22×S3).6(C2×C6), (C2×C6).32(C22×C6), (C22×C6).27(C2×C6), C32(C3×C22.D4), (C3×C6).129(C4○D4), (C32×C22⋊C4)⋊13C2, (C2×C6).310(C22×S3), (C2×Dic3).23(C2×C6), SmallGroup(288,657)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C3×C23.21D6
C1C3C6C2×C6C62S3×C2×C6C6×C3⋊D4 — C3×C23.21D6
C3C2×C6 — C3×C23.21D6
C1C2×C6C3×C22⋊C4

Generators and relations for C3×C23.21D6
 G = < a,b,c,d,e | a3=b2=c2=d12=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 418 in 173 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 [×10], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C32, Dic3 [×3], C12 [×9], D6 [×3], C2×C6 [×2], C2×C6 [×4], C2×C6 [×12], C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, C3×S3, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3, C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×2], C2×C12 [×4], C2×C12 [×7], 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], C4⋊Dic3 [×2], D6⋊C4 [×2], C3×C22⋊C4 [×2], C3×C22⋊C4 [×3], C3×C4⋊C4 [×2], C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×Dic3, C6×Dic3 [×2], C6×Dic3 [×2], C3×C3⋊D4 [×2], C6×C12 [×2], S3×C2×C6, C2×C62, C23.21D6, C3×C22.D4, C3×C4⋊Dic3 [×2], C3×D6⋊C4 [×2], C32×C22⋊C4, Dic3×C2×C6, C6×C3⋊D4, C3×C23.21D6
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, D12 [×2], C3×D4 [×2], C22×S3, C22×C6, C22.D4, S3×C6 [×3], C2×D12, D42S3 [×2], C6×D4, C3×C4○D4 [×2], C3×D12 [×2], S3×C2×C6, C23.21D6, C3×C22.D4, C6×D12, C3×D42S3 [×2], C3×C23.21D6

Smallest permutation representation of C3×C23.21D6
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 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(2 41)(4 43)(6 45)(8 47)(10 37)(12 39)(14 29)(16 31)(18 33)(20 35)(22 25)(24 27)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 37)(11 38)(12 39)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 25)(23 26)(24 27)
(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 21 40 36)(2 35 41 20)(3 19 42 34)(4 33 43 18)(5 17 44 32)(6 31 45 16)(7 15 46 30)(8 29 47 14)(9 13 48 28)(10 27 37 24)(11 23 38 26)(12 25 39 22)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C3D3E4A4B4C4D4E4F4G6A···6F6G···6S6T···6Y6Z6AA12A···12P12Q···12X12Y12Z
order12222223333344444446···66···66···66612···1212···121212
size1111221211222446666121···12···24···412124···46···61212

72 irreducible representations

dim11111111111122222222222244
type+++++++++++-
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D6D6C4○D4C3×S3D12C3×D4S3×C6S3×C6C3×C4○D4C3×D12D42S3C3×D42S3
kernelC3×C23.21D6C3×C4⋊Dic3C3×D6⋊C4C32×C22⋊C4Dic3×C2×C6C6×C3⋊D4C23.21D6C4⋊Dic3D6⋊C4C3×C22⋊C4C22×Dic3C2×C3⋊D4C3×C22⋊C4C62C2×C12C22×C6C3×C6C22⋊C4C2×C6C2×C6C2×C4C23C6C22C6C2
# reps12211124422212214244428824

Matrix representation of C3×C23.21D6 in GL4(𝔽13) generated by

1000
0100
0030
0003
,
11100
01200
0010
0001
,
12000
01200
0010
0001
,
8000
8500
0060
00011
,
5000
0500
00011
0060
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,11,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[8,8,0,0,0,5,0,0,0,0,6,0,0,0,0,11],[5,0,0,0,0,5,0,0,0,0,0,6,0,0,11,0] >;

C3×C23.21D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{21}D_6
% in TeX

G:=Group("C3xC2^3.21D6");
// GroupNames label

G:=SmallGroup(288,657);
// by ID

G=gap.SmallGroup(288,657);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,590,555,394,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^12=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations

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