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G = C6×D4⋊S3order 288 = 25·32

Direct product of C6 and D4⋊S3

direct product, metabelian, supersoluble, monomial

Aliases: C6×D4⋊S3, C62.121D4, C33(C6×D8), C62(C3×D8), (C3×C6)⋊6D8, (C6×D4)⋊1C6, (C6×D4)⋊8S3, D43(S3×C6), D125(C2×C6), (C3×D4)⋊21D6, (C2×D12)⋊8C6, C6.44(C6×D4), C3213(C2×D8), (C6×D12)⋊10C2, C12.14(C3×D4), (C3×C12).84D4, (C2×C12).323D6, (C3×D12)⋊23C22, C12.87(C3⋊D4), C12.11(C22×C6), (C3×C12).82C23, C12.162(C22×S3), (C6×C12).117C22, (D4×C32)⋊14C22, (C2×C3⋊C8)⋊4C6, C3⋊C87(C2×C6), (D4×C3×C6)⋊1C2, (C6×C3⋊C8)⋊13C2, C4.11(S3×C2×C6), (C3×D4)⋊3(C2×C6), (C2×D4)⋊1(C3×S3), C4.5(C3×C3⋊D4), C2.8(C6×C3⋊D4), (C3×C3⋊C8)⋊32C22, (C2×C4).47(S3×C6), (C2×C6).47(C3×D4), (C2×C12).28(C2×C6), (C3×C6).254(C2×D4), C6.145(C2×C3⋊D4), C22.21(C3×C3⋊D4), (C2×C6).114(C3⋊D4), SmallGroup(288,702)

Series: Derived Chief Lower central Upper central

C1C12 — C6×D4⋊S3
C1C3C6C12C3×C12C3×D12C6×D12 — C6×D4⋊S3
C3C6C12 — C6×D4⋊S3
C1C2×C6C2×C12C6×D4

Generators and relations for C6×D4⋊S3
 G = < a,b,c,d,e | a6=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Subgroups: 490 in 179 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C22, C22 [×8], S3 [×2], C6 [×2], C6 [×4], C6 [×13], C8 [×2], C2×C4, D4 [×2], D4 [×4], C23 [×2], C32, C12 [×4], C12 [×2], D6 [×4], C2×C6 [×2], C2×C6 [×21], C2×C8, D8 [×4], C2×D4, C2×D4, C3×S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C3⋊C8 [×2], C24 [×2], D12 [×2], D12, C2×C12 [×2], C2×C12, C3×D4 [×4], C3×D4 [×9], C22×S3, C22×C6 [×5], C2×D8, C3×C12 [×2], S3×C6 [×4], C62, C62 [×4], C2×C3⋊C8, D4⋊S3 [×4], C2×C24, C3×D8 [×4], C2×D12, C6×D4 [×2], C6×D4 [×2], C3×C3⋊C8 [×2], C3×D12 [×2], C3×D12, C6×C12, D4×C32 [×2], D4×C32, S3×C2×C6, C2×C62, C2×D4⋊S3, C6×D8, C6×C3⋊C8, C3×D4⋊S3 [×4], C6×D12, D4×C3×C6, C6×D4⋊S3
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], D8 [×2], C2×D4, C3×S3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C22×C6, C2×D8, S3×C6 [×3], D4⋊S3 [×2], C3×D8 [×2], C2×C3⋊D4, C6×D4, C3×C3⋊D4 [×2], S3×C2×C6, C2×D4⋊S3, C6×D8, C3×D4⋊S3 [×2], C6×C3⋊D4, C6×D4⋊S3

Smallest permutation representation of C6×D4⋊S3
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 11 19 29)(2 12 20 30)(3 7 21 25)(4 8 22 26)(5 9 23 27)(6 10 24 28)(13 47 33 41)(14 48 34 42)(15 43 35 37)(16 44 36 38)(17 45 31 39)(18 46 32 40)
(1 29)(2 30)(3 25)(4 26)(5 27)(6 28)(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 40)(2 41)(3 42)(4 37)(5 38)(6 39)(7 34)(8 35)(9 36)(10 31)(11 32)(12 33)(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), (1,11,19,29)(2,12,20,30)(3,7,21,25)(4,8,22,26)(5,9,23,27)(6,10,24,28)(13,47,33,41)(14,48,34,42)(15,43,35,37)(16,44,36,38)(17,45,31,39)(18,46,32,40), (1,29)(2,30)(3,25)(4,26)(5,27)(6,28)(7,21)(8,22)(9,23)(10,24)(11,19)(12,20)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46), (1,40)(2,41)(3,42)(4,37)(5,38)(6,39)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(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), (1,11,19,29)(2,12,20,30)(3,7,21,25)(4,8,22,26)(5,9,23,27)(6,10,24,28)(13,47,33,41)(14,48,34,42)(15,43,35,37)(16,44,36,38)(17,45,31,39)(18,46,32,40), (1,29)(2,30)(3,25)(4,26)(5,27)(6,28)(7,21)(8,22)(9,23)(10,24)(11,19)(12,20)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46), (1,40)(2,41)(3,42)(4,37)(5,38)(6,39)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(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)], [(1,11,19,29),(2,12,20,30),(3,7,21,25),(4,8,22,26),(5,9,23,27),(6,10,24,28),(13,47,33,41),(14,48,34,42),(15,43,35,37),(16,44,36,38),(17,45,31,39),(18,46,32,40)], [(1,29),(2,30),(3,25),(4,26),(5,27),(6,28),(7,21),(8,22),(9,23),(10,24),(11,19),(12,20),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,47,45),(44,48,46)], [(1,40),(2,41),(3,42),(4,37),(5,38),(6,39),(7,34),(8,35),(9,36),(10,31),(11,32),(12,33),(13,30),(14,25),(15,26),(16,27),(17,28),(18,29),(19,46),(20,47),(21,48),(22,43),(23,44),(24,45)])

72 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A4B6A···6F6G···6O6P···6AE6AF6AG6AH6AI8A8B8C8D12A12B12C12D12E···12J24A···24H
order1222222233333446···66···66···6666688881212121212···1224···24
size111144121211222221···12···24···412121212666622224···46···6

72 irreducible representations

dim1111111111222222222222222244
type++++++++++++
imageC1C2C2C2C2C3C6C6C6C6S3D4D4D6D6D8C3×S3C3⋊D4C3×D4C3⋊D4C3×D4S3×C6S3×C6C3×D8C3×C3⋊D4C3×C3⋊D4D4⋊S3C3×D4⋊S3
kernelC6×D4⋊S3C6×C3⋊C8C3×D4⋊S3C6×D12D4×C3×C6C2×D4⋊S3C2×C3⋊C8D4⋊S3C2×D12C6×D4C6×D4C3×C12C62C2×C12C3×D4C3×C6C2×D4C12C12C2×C6C2×C6C2×C4D4C6C4C22C6C2
# reps1141122822111124222222484424

Matrix representation of C6×D4⋊S3 in GL4(𝔽73) generated by

9000
0900
00720
00072
,
1000
0100
0001
00720
,
1000
0100
0001
0010
,
8000
06400
0010
0001
,
07200
72000
005716
001616
G:=sub<GL(4,GF(73))| [9,0,0,0,0,9,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[8,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,57,16,0,0,16,16] >;

C6×D4⋊S3 in GAP, Magma, Sage, TeX

C_6\times D_4\rtimes S_3
% in TeX

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

G:=SmallGroup(288,702);
// by ID

G=gap.SmallGroup(288,702);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,2524,648,102,9414]);
// 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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations

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