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G = C3×C8⋊S3order 144 = 24·32

Direct product of C3 and C8⋊S3

direct product, metacyclic, supersoluble, monomial

Aliases: C3×C8⋊S3, C245C6, C247S3, D6.C12, C12.65D6, Dic3.C12, C325M4(2), C3⋊C84C6, C83(C3×S3), (C3×C24)⋊9C2, (C4×S3).2C6, (S3×C6).3C4, C2.3(S3×C12), C4.13(S3×C6), C6.22(C4×S3), C6.2(C2×C12), (S3×C12).5C2, C12.14(C2×C6), C31(C3×M4(2)), (C3×Dic3).3C4, (C3×C12).43C22, (C3×C3⋊C8)⋊11C2, (C3×C6).18(C2×C4), SmallGroup(144,70)

Series: Derived Chief Lower central Upper central

C1C6 — C3×C8⋊S3
C1C3C6C12C3×C12S3×C12 — C3×C8⋊S3
C3C6 — C3×C8⋊S3
C1C12C24

Generators and relations for C3×C8⋊S3
 G = < a,b,c,d | a3=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

6C2
2C3
3C22
3C4
2S3
2C6
6C6
3C8
3C2×C4
2C12
3C12
3C2×C6
2C3×S3
3M4(2)
2C24
3C2×C12
3C24
3C3×M4(2)

Smallest permutation representation of C3×C8⋊S3
On 48 points
Generators in S48
(1 30 40)(2 31 33)(3 32 34)(4 25 35)(5 26 36)(6 27 37)(7 28 38)(8 29 39)(9 48 23)(10 41 24)(11 42 17)(12 43 18)(13 44 19)(14 45 20)(15 46 21)(16 47 22)
(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 40 30)(2 33 31)(3 34 32)(4 35 25)(5 36 26)(6 37 27)(7 38 28)(8 39 29)(9 48 23)(10 41 24)(11 42 17)(12 43 18)(13 44 19)(14 45 20)(15 46 21)(16 47 22)
(1 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)(17 40)(18 37)(19 34)(20 39)(21 36)(22 33)(23 38)(24 35)(25 41)(26 46)(27 43)(28 48)(29 45)(30 42)(31 47)(32 44)

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

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

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

C3×C8⋊S3 is a maximal subgroup of
C24⋊D6  C241D6  D24⋊S3  C24.3D6  Dic12⋊S3  C24.64D6  C24.D6  C3×S3×M4(2)  He35M4(2)  C72⋊C6  He36M4(2)
C3×C8⋊S3 is a maximal quotient of
He35M4(2)  C72⋊C6

54 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B4C6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E···12J12K12L24A···24P24Q24R24S24T
order12233333444666666688881212121212···12121224···2424242424
size116112221161122266226611112···2662···26666

54 irreducible representations

dim1111111111112222222222
type++++++
imageC1C2C2C2C3C4C4C6C6C6C12C12S3D6M4(2)C3×S3C4×S3S3×C6C8⋊S3C3×M4(2)S3×C12C3×C8⋊S3
kernelC3×C8⋊S3C3×C3⋊C8C3×C24S3×C12C8⋊S3C3×Dic3S3×C6C3⋊C8C24C4×S3Dic3D6C24C12C32C8C6C4C3C3C2C1
# reps1111222222441122224448

Matrix representation of C3×C8⋊S3 in GL4(𝔽5) generated by

0002
0320
0110
2004
,
4001
0240
0230
1001
,
0002
0130
0430
2004
,
0320
4003
2003
0340
G:=sub<GL(4,GF(5))| [0,0,0,2,0,3,1,0,0,2,1,0,2,0,0,4],[4,0,0,1,0,2,2,0,0,4,3,0,1,0,0,1],[0,0,0,2,0,1,4,0,0,3,3,0,2,0,0,4],[0,4,2,0,3,0,0,3,2,0,0,4,0,3,3,0] >;

C3×C8⋊S3 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes S_3
% in TeX

G:=Group("C3xC8:S3");
// GroupNames label

G:=SmallGroup(144,70);
// by ID

G=gap.SmallGroup(144,70);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,313,79,69,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of C3×C8⋊S3 in TeX

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