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G = C6×D42S3order 288 = 25·32

Direct product of C6 and D42S3

direct product, metabelian, supersoluble, monomial

Aliases: C6×D42S3, C62.269C23, D45(S3×C6), (C6×D4)⋊6C6, (C6×D4)⋊17S3, (C3×D4)⋊23D6, Dic67(C2×C6), C6.6(C23×C6), (C6×Dic6)⋊22C2, (C2×Dic6)⋊12C6, (C2×C12).334D6, C23.29(S3×C6), C6.74(S3×C23), (C3×C6).43C24, D6.2(C22×C6), (S3×C12)⋊20C22, (S3×C6).29C23, C12.20(C22×C6), (C22×C6).109D6, C12.171(C22×S3), (C6×C12).163C22, (C3×C12).121C23, (C3×Dic6)⋊32C22, (C6×Dic3)⋊34C22, (C22×Dic3)⋊11C6, (D4×C32)⋊19C22, (C2×C62).83C22, Dic3.3(C22×C6), (C3×Dic3).30C23, (S3×C2×C4)⋊4C6, (D4×C3×C6)⋊10C2, C32(C6×C4○D4), C62(C3×C4○D4), C4.20(S3×C2×C6), (S3×C2×C12)⋊12C2, (C4×S3)⋊4(C2×C6), (C2×D4)⋊8(C3×S3), (C3×D4)⋊6(C2×C6), C3⋊D42(C2×C6), C2.7(S3×C22×C6), C22.1(S3×C2×C6), (C3×C6)⋊9(C4○D4), (C6×C3⋊D4)⋊24C2, (C2×C3⋊D4)⋊10C6, (C2×C4).60(S3×C6), (Dic3×C2×C6)⋊19C2, C3215(C2×C4○D4), (C2×C12).45(C2×C6), (C2×Dic3)⋊9(C2×C6), (C2×C6).1(C22×C6), (C3×C3⋊D4)⋊16C22, (S3×C2×C6).110C22, (C22×C6).34(C2×C6), (C2×C6).20(C22×S3), (C22×S3).30(C2×C6), SmallGroup(288,993)

Series: Derived Chief Lower central Upper central

C1C6 — C6×D42S3
C1C3C6C3×C6S3×C6S3×C2×C6S3×C2×C12 — C6×D42S3
C3C6 — C6×D42S3
C1C2×C6C6×D4

Generators and relations for C6×D42S3
 G = < a,b,c,d,e | a6=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 682 in 355 conjugacy classes, 178 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], C22 [×8], S3 [×2], C6 [×2], C6 [×4], C6 [×17], C2×C4, C2×C4 [×15], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, C32, Dic3 [×6], C12 [×4], C12 [×8], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×8], C2×C6 [×21], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], C3×S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×4], Dic6 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×10], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×16], C3×D4 [×8], C3×D4 [×12], C3×Q8 [×4], C22×S3, C22×C6 [×4], C22×C6 [×3], C2×C4○D4, C3×Dic3 [×6], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, C62 [×4], C62 [×4], C2×Dic6, S3×C2×C4, D42S3 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C22×C12 [×3], C6×D4 [×2], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], C3×Dic6 [×4], S3×C12 [×4], C6×Dic3, C6×Dic3 [×10], C3×C3⋊D4 [×8], C6×C12, D4×C32 [×4], S3×C2×C6, C2×C62 [×2], C2×D42S3, C6×C4○D4, C6×Dic6, S3×C2×C12, C3×D42S3 [×8], Dic3×C2×C6 [×2], C6×C3⋊D4 [×2], D4×C3×C6, C6×D42S3
Quotients: C1, C2 [×15], C3, C22 [×35], S3, C6 [×15], C23 [×15], D6 [×7], C2×C6 [×35], C4○D4 [×2], C24, C3×S3, C22×S3 [×7], C22×C6 [×15], C2×C4○D4, S3×C6 [×7], D42S3 [×2], C3×C4○D4 [×2], S3×C23, C23×C6, S3×C2×C6 [×7], C2×D42S3, C6×C4○D4, C3×D42S3 [×2], S3×C22×C6, C6×D42S3

Smallest permutation representation of C6×D42S3
On 48 points
Generators in S48
(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 8 23 28)(2 9 24 29)(3 10 19 30)(4 11 20 25)(5 12 21 26)(6 7 22 27)(13 31 41 46)(14 32 42 47)(15 33 37 48)(16 34 38 43)(17 35 39 44)(18 36 40 45)
(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 35)(2 36)(3 31)(4 32)(5 33)(6 34)(7 38)(8 39)(9 40)(10 41)(11 42)(12 37)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I3A3B3C3D3E4A4B4C4D4E4F4G4H4I4J6A···6F6G···6W6X···6AI6AJ6AK6AL6AM12A12B12C12D12E···12L12M···12R12S···12Z
order12222222223333344444444446···66···66···666661212121212···1212···1212···12
size11112222661122222333366661···12···24···4666622223···34···46···6

90 irreducible representations

dim11111111111111222222222244
type+++++++++++-
imageC1C2C2C2C2C2C2C3C6C6C6C6C6C6S3D6D6D6C4○D4C3×S3S3×C6S3×C6S3×C6C3×C4○D4D42S3C3×D42S3
kernelC6×D42S3C6×Dic6S3×C2×C12C3×D42S3Dic3×C2×C6C6×C3⋊D4D4×C3×C6C2×D42S3C2×Dic6S3×C2×C4D42S3C22×Dic3C2×C3⋊D4C6×D4C6×D4C2×C12C3×D4C22×C6C3×C6C2×D4C2×C4D4C23C6C6C2
# reps111822122216442114242284824

Matrix representation of C6×D42S3 in GL4(𝔽13) generated by

10000
01000
0010
0001
,
12000
01200
00012
0010
,
1000
0100
0010
00012
,
3000
0900
0010
0001
,
01200
12000
0005
0080
G:=sub<GL(4,GF(13))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12],[3,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,12,0,0,0,0,0,0,8,0,0,5,0] >;

C6×D42S3 in GAP, Magma, Sage, TeX

C_6\times D_4\rtimes_2S_3
% in TeX

G:=Group("C6xD4:2S3");
// GroupNames label

G:=SmallGroup(288,993);
// by ID

G=gap.SmallGroup(288,993);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,268,1571,409,9414]);
// Polycyclic

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

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