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G = C3×Q83S3order 144 = 24·32

Direct product of C3 and Q83S3

direct product, metabelian, supersoluble, monomial

Aliases: C3×Q83S3, D124C6, C12.39D6, (C4×S3)⋊3C6, C4.7(S3×C6), (C3×Q8)⋊5C6, (C3×Q8)⋊7S3, Q84(C3×S3), (S3×C12)⋊7C2, (C3×D12)⋊9C2, C12.7(C2×C6), D6.3(C2×C6), C6.8(C22×C6), (Q8×C32)⋊4C2, C3211(C4○D4), C6.47(C22×S3), (C3×C6).26C23, Dic3.5(C2×C6), (S3×C6).12C22, (C3×C12).23C22, (C3×Dic3).14C22, C2.9(S3×C2×C6), C33(C3×C4○D4), SmallGroup(144,165)

Series: Derived Chief Lower central Upper central

C1C6 — C3×Q83S3
C1C3C6C3×C6S3×C6S3×C12 — C3×Q83S3
C3C6 — C3×Q83S3
C1C6C3×Q8

Generators and relations for C3×Q83S3
 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, ce=ec, ede=d-1 >

Subgroups: 172 in 86 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×3], C4, C22 [×3], S3 [×3], C6 [×2], C6 [×4], C2×C4 [×3], D4 [×3], Q8, C32, Dic3, C12 [×6], C12 [×4], D6 [×3], C2×C6 [×3], C4○D4, C3×S3 [×3], C3×C6, C4×S3 [×3], D12 [×3], C2×C12 [×3], C3×D4 [×3], C3×Q8 [×2], C3×Q8, C3×Dic3, C3×C12 [×3], S3×C6 [×3], Q83S3, C3×C4○D4, S3×C12 [×3], C3×D12 [×3], Q8×C32, C3×Q83S3
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], C23, D6 [×3], C2×C6 [×7], C4○D4, C3×S3, C22×S3, C22×C6, S3×C6 [×3], Q83S3, C3×C4○D4, S3×C2×C6, C3×Q83S3

Smallest permutation representation of C3×Q83S3
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 37 7 39)(6 40 8 38)(9 43 11 41)(10 42 12 44)(13 27 15 25)(14 26 16 28)(17 29 19 31)(18 32 20 30)(33 45 35 47)(34 48 36 46)
(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)(5,37,7,39)(6,40,8,38)(9,43,11,41)(10,42,12,44)(13,27,15,25)(14,26,16,28)(17,29,19,31)(18,32,20,30)(33,45,35,47)(34,48,36,46), (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)(5,37,7,39)(6,40,8,38)(9,43,11,41)(10,42,12,44)(13,27,15,25)(14,26,16,28)(17,29,19,31)(18,32,20,30)(33,45,35,47)(34,48,36,46), (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),(5,37,7,39),(6,40,8,38),(9,43,11,41),(10,42,12,44),(13,27,15,25),(14,26,16,28),(17,29,19,31),(18,32,20,30),(33,45,35,47),(34,48,36,46)], [(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×Q83S3 is a maximal subgroup of
D126D6  D12.11D6  D12.12D6  D12.13D6  D12.25D6  D1215D6  D1216D6  C3×S3×C4○D4  Q8⋊C93S3  (Q8×He3)⋊C2  D363C6  He35D4⋊C2
C3×Q83S3 is a maximal quotient of
C3×Q8×Dic3  (Q8×He3)⋊C2  D363C6

45 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E4A4B4C4D4E6A6B6C6D6E6F···6K12A···12F12G12H12I12J12K···12S
order122223333344444666666···612···121212121212···12
size116661122222233112226···62···233334···4

45 irreducible representations

dim1111111122222244
type+++++++
imageC1C2C2C2C3C6C6C6S3D6C4○D4C3×S3S3×C6C3×C4○D4Q83S3C3×Q83S3
kernelC3×Q83S3S3×C12C3×D12Q8×C32Q83S3C4×S3D12C3×Q8C3×Q8C12C32Q8C4C3C3C1
# reps1331266213226412

Matrix representation of C3×Q83S3 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
3333
1442
2413
3366
,
4111
3036
6512
2562
,
4000
1166
0242
6113
,
3645
3353
5120
2646
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,1,2,3,3,4,4,3,3,4,1,6,3,2,3,6],[4,3,6,2,1,0,5,5,1,3,1,6,1,6,2,2],[4,1,0,6,0,1,2,1,0,6,4,1,0,6,2,3],[3,3,5,2,6,3,1,6,4,5,2,4,5,3,0,6] >;

C3×Q83S3 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes_3S_3
% in TeX

G:=Group("C3xQ8:3S3");
// GroupNames label

G:=SmallGroup(144,165);
// by ID

G=gap.SmallGroup(144,165);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,151,506,260,122,3461]);
// 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,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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