Copied to
clipboard

G = C3×S3×D8order 288 = 25·32

Direct product of C3, S3 and D8

direct product, metabelian, supersoluble, monomial

Aliases: C3×S3×D8, D244C6, C2418D6, C84(S3×C6), C32(C6×D8), (S3×C8)⋊1C6, C242(C2×C6), D4⋊S31C6, (C3×D8)⋊2C6, (S3×D4)⋊1C6, D41(S3×C6), (S3×C24)⋊5C2, (C3×D4)⋊12D6, D121(C2×C6), C6.27(C6×D4), C3211(C2×D8), (C3×D24)⋊12C2, (C3×C24)⋊8C22, D6.12(C3×D4), (S3×C6).48D4, C6.187(S3×D4), (C32×D8)⋊3C2, C12.1(C22×C6), Dic3.3(C3×D4), (C3×D12)⋊10C22, (C3×C12).72C23, (C3×Dic3).30D4, (D4×C32)⋊5C22, (S3×C12).47C22, C12.152(C22×S3), C3⋊C85(C2×C6), (C3×S3×D4)⋊4C2, C4.1(S3×C2×C6), C2.15(C3×S3×D4), (C3×D4⋊S3)⋊9C2, (C3×D4)⋊1(C2×C6), (C3×C3⋊C8)⋊31C22, (C4×S3).7(C2×C6), (C3×C6).215(C2×D4), SmallGroup(288,681)

Series: Derived Chief Lower central Upper central

C1C12 — C3×S3×D8
C1C3C6C12C3×C12S3×C12C3×S3×D4 — C3×S3×D8
C3C6C12 — C3×S3×D8
C1C6C12C3×D8

Generators and relations for C3×S3×D8
 G = < a,b,c,d,e | a3=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 522 in 163 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4, C4, C22 [×9], S3 [×2], S3 [×2], C6 [×2], C6 [×11], C8, C8, C2×C4, D4 [×2], D4 [×4], C23 [×2], C32, Dic3, C12 [×2], C12 [×2], D6, D6 [×6], C2×C6 [×13], C2×C8, D8, D8 [×3], C2×D4 [×2], C3×S3 [×2], C3×S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8, C24 [×2], C24 [×2], C4×S3, D12 [×2], C3⋊D4 [×2], C2×C12, C3×D4 [×4], C3×D4 [×6], C22×S3 [×2], C22×C6 [×2], C2×D8, C3×Dic3, C3×C12, S3×C6, S3×C6 [×6], C62 [×2], S3×C8, D24, D4⋊S3 [×2], C2×C24, C3×D8 [×2], C3×D8 [×4], S3×D4 [×2], C6×D4 [×2], C3×C3⋊C8, C3×C24, S3×C12, C3×D12 [×2], C3×C3⋊D4 [×2], D4×C32 [×2], S3×C2×C6 [×2], S3×D8, C6×D8, S3×C24, C3×D24, C3×D4⋊S3 [×2], C32×D8, C3×S3×D4 [×2], C3×S3×D8
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], D8 [×2], C2×D4, C3×S3, C3×D4 [×2], C22×S3, C22×C6, C2×D8, S3×C6 [×3], C3×D8 [×2], S3×D4, C6×D4, S3×C2×C6, S3×D8, C6×D8, C3×S3×D4, C3×S3×D8

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

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

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

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

63 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M6N···6S6T6U6V6W8A8B8C8D12A12B12C12D12E12F12G24A24B24C24D24E···24J24K24L24M24N
order12222222333334466666666666666···666668888121212121212122424242424···2424242424
size1133441212112222611222333344448···8121212122266224446622224···46666

63 irreducible representations

dim1111111111112222222222224444
type++++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D4D4D6D6D8C3×S3C3×D4C3×D4S3×C6S3×C6C3×D8S3×D4S3×D8C3×S3×D4C3×S3×D8
kernelC3×S3×D8S3×C24C3×D24C3×D4⋊S3C32×D8C3×S3×D4S3×D8S3×C8D24D4⋊S3C3×D8S3×D4C3×D8C3×Dic3S3×C6C24C3×D4C3×S3D8Dic3D6C8D4S3C6C3C2C1
# reps1112122224241111242222481224

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

4000
0400
0040
0004
,
2331
6114
2206
4322
,
4253
6214
0654
5033
,
3302
1311
2263
3423
,
2061
6466
2412
6430
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,6,2,4,3,1,2,3,3,1,0,2,1,4,6,2],[4,6,0,5,2,2,6,0,5,1,5,3,3,4,4,3],[3,1,2,3,3,3,2,4,0,1,6,2,2,1,3,3],[2,6,2,6,0,4,4,4,6,6,1,3,1,6,2,0] >;

C3×S3×D8 in GAP, Magma, Sage, TeX

C_3\times S_3\times D_8
% in TeX

G:=Group("C3xS3xD8");
// GroupNames label

G:=SmallGroup(288,681);
// by ID

G=gap.SmallGroup(288,681);
# by ID

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

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

׿
×
𝔽