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G = C3×Q82S3order 144 = 24·32

Direct product of C3 and Q82S3

direct product, metabelian, supersoluble, monomial

Aliases: C3×Q82S3, D12.2C6, C12.35D6, C3210SD16, C3⋊C83C6, C4.3(S3×C6), Q83(C3×S3), (C3×Q8)⋊6S3, (C3×Q8)⋊3C6, C6.9(C3×D4), C12.3(C2×C6), C33(C3×SD16), (C3×C6).30D4, (C3×D12).3C2, (Q8×C32)⋊1C2, C6.31(C3⋊D4), (C3×C12).10C22, (C3×C3⋊C8)⋊10C2, C2.6(C3×C3⋊D4), SmallGroup(144,82)

Series: Derived Chief Lower central Upper central

C1C12 — C3×Q82S3
C1C3C6C12C3×C12C3×D12 — C3×Q82S3
C3C6C12 — C3×Q82S3
C1C6C12C3×Q8

Generators and relations for C3×Q82S3
 G = < a,b,c,d,e | a3=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

12C2
2C3
2C4
6C22
2C6
4S3
12C6
3D4
3C8
2C12
2C12
2C12
2C12
2D6
2C12
6C2×C6
4C3×S3
3SD16
2C3×Q8
3C24
3C3×D4
2C3×C12
2S3×C6
3C3×SD16

Smallest permutation representation of C3×Q82S3
On 48 points
Generators in S48
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(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 24 3 22)(2 23 4 21)(5 40 7 38)(6 39 8 37)(9 42 11 44)(10 41 12 43)(13 27 15 25)(14 26 16 28)(17 29 19 31)(18 32 20 30)(33 48 35 46)(34 47 36 45)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 27 32)(22 28 29)(23 25 30)(24 26 31)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 34)(2 33)(3 36)(4 35)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 47)(22 46)(23 45)(24 48)

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

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

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

C3×Q82S3 is a maximal subgroup of
D126D6  D12.9D6  D12.10D6  D12.24D6  D12.12D6  D12.14D6  D12.15D6  C3×S3×SD16  He310SD16  D36.C6  He311SD16  C32.GL2(𝔽3)  C322GL2(𝔽3)  C323GL2(𝔽3)
C3×Q82S3 is a maximal quotient of
He310SD16  D36.C6

36 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B6A6B6C6D6E6F6G8A8B12A12B12C···12M24A24B24C24D
order1223333344666666688121212···1224242424
size1112112222411222121266224···46666

36 irreducible representations

dim11111111222222222244
type++++++++
imageC1C2C2C2C3C6C6C6S3D4D6SD16C3×S3C3⋊D4C3×D4S3×C6C3×SD16C3×C3⋊D4Q82S3C3×Q82S3
kernelC3×Q82S3C3×C3⋊C8C3×D12Q8×C32Q82S3C3⋊C8D12C3×Q8C3×Q8C3×C6C12C32Q8C6C6C4C3C2C3C1
# reps11112222111222224412

Matrix representation of C3×Q82S3 in GL4(𝔽7) generated by

2000
0200
0020
0002
,
3555
2312
4156
6053
,
2333
2604
5135
3523
,
4000
1166
0242
6113
,
6513
5262
0503
1626
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[3,2,4,6,5,3,1,0,5,1,5,5,5,2,6,3],[2,2,5,3,3,6,1,5,3,0,3,2,3,4,5,3],[4,1,0,6,0,1,2,1,0,6,4,1,0,6,2,3],[6,5,0,1,5,2,5,6,1,6,0,2,3,2,3,6] >;

C3×Q82S3 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes_2S_3
% in TeX

G:=Group("C3xQ8:2S3");
// GroupNames label

G:=SmallGroup(144,82);
// by ID

G=gap.SmallGroup(144,82);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,169,151,867,441,69,3461]);
// Polycyclic

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

Export

Subgroup lattice of C3×Q82S3 in TeX

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