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G = C3×Q8.11D6order 288 = 25·32

Direct product of C3 and Q8.11D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×Q8.11D6, C62.125D4, (C6×Q8)⋊6C6, (C6×Q8)⋊13S3, C3⋊Q165C6, C6.54(C6×D4), Q82S35C6, C4○D12.5C6, (C3×C12).89D4, C12.19(C3×D4), C4.Dic37C6, (C3×Q8).71D6, Q8.16(S3×C6), D12.10(C2×C6), (C2×C12).243D6, Dic6.9(C2×C6), C12.15(C22×C6), (C3×C12).86C23, C12.104(C3⋊D4), C12.166(C22×S3), (C6×C12).125C22, (C3×D12).39C22, C3222(C8.C22), (C3×Dic6).39C22, (Q8×C32).23C22, (Q8×C3×C6)⋊2C2, C3⋊C8.3(C2×C6), C4.15(S3×C2×C6), (C2×Q8)⋊6(C3×S3), (C2×C4).18(S3×C6), (C2×C6).51(C3×D4), C4.17(C3×C3⋊D4), C2.18(C6×C3⋊D4), C34(C3×C8.C22), (C2×C12).36(C2×C6), (C3×C3⋊Q16)⋊13C2, (C3×C6).262(C2×D4), C6.155(C2×C3⋊D4), (C3×C3⋊C8).22C22, (C3×C4.Dic3)⋊6C2, (C3×Q8).18(C2×C6), (C3×Q82S3)⋊11C2, (C3×C4○D12).11C2, (C2×C6).64(C3⋊D4), C22.11(C3×C3⋊D4), SmallGroup(288,713)

Series: Derived Chief Lower central Upper central

C1C12 — C3×Q8.11D6
C1C3C6C12C3×C12C3×D12C3×C4○D12 — C3×Q8.11D6
C3C6C12 — C3×Q8.11D6
C1C6C2×C12C6×Q8

Generators and relations for C3×Q8.11D6
 G = < a,b,c,d,e | a3=b4=1, c2=d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d5 >

Subgroups: 298 in 139 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6 [×2], C6 [×6], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×2], Q8 [×2], C32, Dic3, C12 [×4], C12 [×11], D6, C2×C6 [×2], C2×C6 [×2], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12 [×2], C2×C12 [×6], C3×D4 [×2], C3×Q8 [×4], C3×Q8 [×7], C8.C22, C3×Dic3, C3×C12 [×2], C3×C12 [×2], S3×C6, C62, C4.Dic3, Q82S3 [×2], C3⋊Q16 [×2], C3×M4(2), C3×SD16 [×2], C3×Q16 [×2], C4○D12, C6×Q8 [×2], C6×Q8, C3×C4○D4, C3×C3⋊C8 [×2], C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C6×C12, Q8×C32 [×2], Q8×C32, Q8.11D6, C3×C8.C22, C3×C4.Dic3, C3×Q82S3 [×2], C3×C3⋊Q16 [×2], C3×C4○D12, Q8×C3×C6, C3×Q8.11D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C22×C6, C8.C22, S3×C6 [×3], C2×C3⋊D4, C6×D4, C3×C3⋊D4 [×2], S3×C2×C6, Q8.11D6, C3×C8.C22, C6×C3⋊D4, C3×Q8.11D6

Smallest permutation representation of C3×Q8.11D6
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 4 7 10)(2 5 8 11)(3 6 9 12)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 46 43 40)(38 47 44 41)(39 48 45 42)
(1 17 7 23)(2 24 8 18)(3 19 9 13)(4 14 10 20)(5 21 11 15)(6 16 12 22)(25 46 31 40)(26 41 32 47)(27 48 33 42)(28 43 34 37)(29 38 35 44)(30 45 36 39)
(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 33 7 27)(2 26 8 32)(3 31 9 25)(4 36 10 30)(5 29 11 35)(6 34 12 28)(13 37 19 43)(14 42 20 48)(15 47 21 41)(16 40 22 46)(17 45 23 39)(18 38 24 44)

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

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

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

63 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E6A6B6C···6M6N6O8A8B12A12B12C12D12E···12Z12AA12AB24A24B24C24D
order12223333344444666···666881212121212···12121224242424
size1121211222224412112···21212121222224···4121212121212

63 irreducible representations

dim111111111111222222222222224444
type+++++++++++-
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D4D6D6C3×S3C3⋊D4C3×D4C3⋊D4C3×D4S3×C6S3×C6C3×C3⋊D4C3×C3⋊D4C8.C22Q8.11D6C3×C8.C22C3×Q8.11D6
kernelC3×Q8.11D6C3×C4.Dic3C3×Q82S3C3×C3⋊Q16C3×C4○D12Q8×C3×C6Q8.11D6C4.Dic3Q82S3C3⋊Q16C4○D12C6×Q8C6×Q8C3×C12C62C2×C12C3×Q8C2×Q8C12C12C2×C6C2×C6C2×C4Q8C4C22C32C3C3C1
# reps112211224422111122222224441224

Matrix representation of C3×Q8.11D6 in GL4(𝔽7) generated by

2000
0200
0020
0002
,
1653
2022
5564
1630
,
0342
0636
2265
6112
,
4051
4412
5420
4664
,
5310
2301
3424
5554
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,2,5,1,6,0,5,6,5,2,6,3,3,2,4,0],[0,0,2,6,3,6,2,1,4,3,6,1,2,6,5,2],[4,4,5,4,0,4,4,6,5,1,2,6,1,2,0,4],[5,2,3,5,3,3,4,5,1,0,2,5,0,1,4,4] >;

C3×Q8.11D6 in GAP, Magma, Sage, TeX

C_3\times Q_8._{11}D_6
% in TeX

G:=Group("C3xQ8.11D6");
// GroupNames label

G:=SmallGroup(288,713);
// by ID

G=gap.SmallGroup(288,713);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,268,2524,648,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=1,c^2=d^6=e^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^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b*c,e*d*e^-1=d^5>;
// generators/relations

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