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G = C3×D8.S3order 288 = 25·32

Direct product of C3 and D8.S3

direct product, metabelian, supersoluble, monomial

Aliases: C3×D8.S3, C24.53D6, C327SD32, Dic123C6, C3⋊C162C6, D8.(C3×S3), C8.5(S3×C6), C6.9(C3×D8), C24.3(C2×C6), (C3×D8).1C6, (C3×D8).7S3, C32(C3×SD32), C12.4(C3×D4), (C3×C6).31D8, (C3×C12).42D4, C6.31(D4⋊S3), (C3×Dic12)⋊7C2, (C32×D8).1C2, C12.84(C3⋊D4), (C3×C24).14C22, (C3×C3⋊C16)⋊5C2, C2.5(C3×D4⋊S3), C4.2(C3×C3⋊D4), SmallGroup(288,261)

Series: Derived Chief Lower central Upper central

C1C24 — C3×D8.S3
C1C3C6C12C24C3×C24C3×Dic12 — C3×D8.S3
C3C6C12C24 — C3×D8.S3
C1C6C12C24C3×D8

Generators and relations for C3×D8.S3
 G = < a,b,c,d,e | a3=b8=c2=d3=1, e2=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=b5c, ede-1=d-1 >

Subgroups: 186 in 63 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, C6 [×2], C6 [×5], C8, D4, Q8, C32, Dic3, C12 [×2], C12 [×2], C2×C6 [×4], C16, D8, Q16, C3×C6, C3×C6, C24 [×2], C24, Dic6, C3×D4 [×4], C3×Q8, SD32, C3×Dic3, C3×C12, C62, C3⋊C16, C48, Dic12, C3×D8 [×2], C3×D8, C3×Q16, C3×C24, C3×Dic6, D4×C32, D8.S3, C3×SD32, C3×C3⋊C16, C3×Dic12, C32×D8, C3×D8.S3
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, D8, C3×S3, C3⋊D4, C3×D4, SD32, S3×C6, D4⋊S3, C3×D8, C3×C3⋊D4, D8.S3, C3×SD32, C3×D4⋊S3, C3×D8.S3

Smallest permutation representation of C3×D8.S3
On 48 points
Generators in S48
(1 34 25)(2 35 26)(3 36 27)(4 37 28)(5 38 29)(6 39 30)(7 40 31)(8 33 32)(9 48 22)(10 41 23)(11 42 24)(12 43 17)(13 44 18)(14 45 19)(15 46 20)(16 47 21)
(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 8)(2 7)(3 6)(4 5)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)(25 32)(26 31)(27 30)(28 29)(33 34)(35 40)(36 39)(37 38)(42 48)(43 47)(44 46)
(1 25 34)(2 26 35)(3 27 36)(4 28 37)(5 29 38)(6 30 39)(7 31 40)(8 32 33)(9 48 22)(10 41 23)(11 42 24)(12 43 17)(13 44 18)(14 45 19)(15 46 20)(16 47 21)
(1 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(33 47 37 43)(34 46 38 42)(35 45 39 41)(36 44 40 48)

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

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

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

54 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B6A6B6C6D6E6F···6M8A8B12A12B12C12D12E12F12G16A16B16C16D24A24B24C24D24E···24J48A···48H
order1223333344666666···68812121212121212161616162424242424···2448···48
size11811222224112228···822224442424666622224···46···6

54 irreducible representations

dim111111112222222222224444
type+++++++++-
imageC1C2C2C2C3C6C6C6S3D4D6D8C3×S3C3⋊D4C3×D4SD32S3×C6C3×D8C3×C3⋊D4C3×SD32D4⋊S3D8.S3C3×D4⋊S3C3×D8.S3
kernelC3×D8.S3C3×C3⋊C16C3×Dic12C32×D8D8.S3C3⋊C16Dic12C3×D8C3×D8C3×C12C24C3×C6D8C12C12C32C8C6C4C3C6C3C2C1
# reps111122221112222424481224

Matrix representation of C3×D8.S3 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
3302
1311
2263
3423
,
0643
4444
3004
0663
,
4446
1563
5561
3454
,
6331
6366
5464
0326
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,1,2,3,3,3,2,4,0,1,6,2,2,1,3,3],[0,4,3,0,6,4,0,6,4,4,0,6,3,4,4,3],[4,1,5,3,4,5,5,4,4,6,6,5,6,3,1,4],[6,6,5,0,3,3,4,3,3,6,6,2,1,6,4,6] >;

C3×D8.S3 in GAP, Magma, Sage, TeX

C_3\times D_8.S_3
% in TeX

G:=Group("C3xD8.S3");
// GroupNames label

G:=SmallGroup(288,261);
// by ID

G=gap.SmallGroup(288,261);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,336,197,1011,514,192,2524,1271,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^3=1,e^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^5*c,e*d*e^-1=d^-1>;
// generators/relations

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