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G = C3×D83S3order 288 = 25·32

Direct product of C3 and D83S3

direct product, metabelian, supersoluble, monomial

Aliases: C3×D83S3, C24.56D6, Dic124C6, (S3×C8)⋊2C6, (C3×D8)⋊3C6, D83(C3×S3), (C3×D8)⋊7S3, C8.8(S3×C6), (S3×C24)⋊6C2, C24.6(C2×C6), D4.S32C6, D6.1(C3×D4), D4.1(S3×C6), C6.29(C6×D4), D42S32C6, (C3×D4).24D6, (S3×C6).25D4, C6.189(S3×D4), (C32×D8)⋊4C2, C3218(C4○D8), C12.3(C22×C6), Dic6.1(C2×C6), (C3×Dic12)⋊12C2, (C3×C24).26C22, (C3×C12).74C23, Dic3.12(C3×D4), (C3×Dic3).49D4, (S3×C12).48C22, C12.154(C22×S3), (C3×Dic6).24C22, (D4×C32).11C22, C4.3(S3×C2×C6), C32(C3×C4○D8), C3⋊C8.5(C2×C6), C2.17(C3×S3×D4), (C4×S3).8(C2×C6), (C3×D4).1(C2×C6), (C3×D42S3)⋊5C2, (C3×D4.S3)⋊13C2, (C3×C6).217(C2×D4), (C3×C3⋊C8).39C22, SmallGroup(288,683)

Series: Derived Chief Lower central Upper central

C1C12 — C3×D83S3
C1C3C6C12C3×C12S3×C12C3×D42S3 — C3×D83S3
C3C6C12 — C3×D83S3
C1C6C12C3×D8

Generators and relations for C3×D83S3
 G = < a,b,c,d,e | a3=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b4c, ede=d-1 >

Subgroups: 330 in 135 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3, C6 [×2], C6 [×8], C8, C8, C2×C4 [×3], D4 [×2], D4 [×2], Q8 [×2], C32, Dic3, Dic3 [×2], C12 [×2], C12 [×4], D6, C2×C6 [×7], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3, C3×C6, C3×C6 [×2], C3⋊C8, C24 [×2], C24 [×2], Dic6 [×2], C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12 [×3], C3×D4 [×4], C3×D4 [×4], C3×Q8 [×2], C4○D8, C3×Dic3, C3×Dic3 [×2], C3×C12, S3×C6, C62 [×2], S3×C8, Dic12, D4.S3 [×2], C2×C24, C3×D8 [×2], C3×D8, C3×SD16 [×2], C3×Q16, D42S3 [×2], C3×C4○D4 [×2], C3×C3⋊C8, C3×C24, C3×Dic6 [×2], S3×C12, C6×Dic3 [×2], C3×C3⋊D4 [×2], D4×C32 [×2], D83S3, C3×C4○D8, S3×C24, C3×Dic12, C3×D4.S3 [×2], C32×D8, C3×D42S3 [×2], C3×D83S3
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], C22×S3, C22×C6, C4○D8, S3×C6 [×3], S3×D4, C6×D4, S3×C2×C6, D83S3, C3×C4○D8, C3×S3×D4, C3×D83S3

Smallest permutation representation of C3×D83S3
On 48 points
Generators in S48
(1 25 40)(2 26 33)(3 27 34)(4 28 35)(5 29 36)(6 30 37)(7 31 38)(8 32 39)(9 42 17)(10 43 18)(11 44 19)(12 45 20)(13 46 21)(14 47 22)(15 48 23)(16 41 24)
(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 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)(25 32)(26 31)(27 30)(28 29)(33 38)(34 37)(35 36)(39 40)(41 46)(42 45)(43 44)(47 48)
(1 40 25)(2 33 26)(3 34 27)(4 35 28)(5 36 29)(6 37 30)(7 38 31)(8 39 32)(9 42 17)(10 43 18)(11 44 19)(12 45 20)(13 46 21)(14 47 22)(15 48 23)(16 41 24)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)

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

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

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

63 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J6K6L···6Q8A8B8C8D12A12B12C12D12E12F12G12H12I12J12K12L12M24A24B24C24D24E···24J24K24L24M24N
order122223333344444666666666666···68888121212121212121212121212122424242424···2424242424
size11446112222331212112224444668···822662233334441212121222224···46666

63 irreducible representations

dim1111111111112222222222224444
type++++++++++++-
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D4D6D6C3×S3C3×D4C3×D4C4○D8S3×C6S3×C6C3×C4○D8S3×D4D83S3C3×S3×D4C3×D83S3
kernelC3×D83S3S3×C24C3×Dic12C3×D4.S3C32×D8C3×D42S3D83S3S3×C8Dic12D4.S3C3×D8D42S3C3×D8C3×Dic3S3×C6C24C3×D4D8Dic3D6C32C8D4C3C6C3C2C1
# reps1112122224241111222242481224

Matrix representation of C3×D83S3 in GL4(𝔽7) generated by

2000
0200
0020
0002
,
4536
4244
3301
2562
,
4214
6032
0010
2562
,
2511
3336
4346
2263
,
0301
1506
0566
5603
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,4,3,2,5,2,3,5,3,4,0,6,6,4,1,2],[4,6,0,2,2,0,0,5,1,3,1,6,4,2,0,2],[2,3,4,2,5,3,3,2,1,3,4,6,1,6,6,3],[0,1,0,5,3,5,5,6,0,0,6,0,1,6,6,3] >;

C3×D83S3 in GAP, Magma, Sage, TeX

C_3\times D_8\rtimes_3S_3
% in TeX

G:=Group("C3xD8:3S3");
// GroupNames label

G:=SmallGroup(288,683);
// by ID

G=gap.SmallGroup(288,683);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,1094,303,1271,648,102,9414]);
// Polycyclic

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

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