direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C8○D4, M4(2)⋊27D6, C12.71C24, C24.53C23, (C2×C8)⋊30D6, (S3×D4).2C4, (S3×Q8).2C4, C8○D12⋊16C2, C4○D4.58D6, D4.12(C4×S3), C3⋊C8.36C23, Q8.18(C4×S3), (S3×C8)⋊20C22, D12.C4⋊14C2, (C2×C24)⋊31C22, D12.20(C2×C4), D4⋊2S3.2C4, C6.34(C23×C4), C4.70(S3×C23), C8.66(C22×S3), Q8⋊3S3.2C4, C8⋊S3⋊20C22, (S3×M4(2))⋊12C2, D4.Dic3⋊14C2, (C4×S3).41C23, C12.38(C22×C4), Dic6.21(C2×C4), D6.15(C22×C4), (C2×C12).513C23, C4○D12.51C22, C4.Dic3⋊26C22, (C3×M4(2))⋊32C22, Dic3.15(C22×C4), C3⋊3(C2×C8○D4), (S3×C2×C8)⋊30C2, C4.38(S3×C2×C4), C22.4(S3×C2×C4), (C3×C8○D4)⋊13C2, (C2×C3⋊C8)⋊34C22, (S3×C4○D4).5C2, C3⋊D4.1(C2×C4), C2.35(S3×C22×C4), (C4×S3).18(C2×C4), (C3×D4).16(C2×C4), (C2×C6).4(C22×C4), (C3×Q8).17(C2×C4), (S3×C2×C4).254C22, (C22×S3).47(C2×C4), (C2×C4).606(C22×S3), (C2×Dic3).73(C2×C4), (C3×C4○D4).43C22, SmallGroup(192,1308)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C8○D4
G = < a,b,c,d,e | a3=b2=c8=e2=1, d2=c4, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c4d >
Subgroups: 512 in 266 conjugacy classes, 149 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, S3, C6, C6, C8, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C8, C2×M4(2), C8○D4, C8○D4, C2×C4○D4, S3×C8, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C4○D4, C2×C8○D4, S3×C2×C8, C8○D12, S3×M4(2), D12.C4, D4.Dic3, C3×C8○D4, S3×C4○D4, S3×C8○D4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C8○D4, C23×C4, S3×C2×C4, S3×C23, C2×C8○D4, S3×C22×C4, S3×C8○D4
►On 48 points
(1 26 35)(2 27 36)(3 28 37)(4 29 38)(5 30 39)(6 31 40)(7 32 33)(8 25 34)(9 44 23)(10 45 24)(11 46 17)(12 47 18)(13 48 19)(14 41 20)(15 42 21)(16 43 22)
(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)
(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 15 5 11)(2 16 6 12)(3 9 7 13)(4 10 8 14)(17 35 21 39)(18 36 22 40)(19 37 23 33)(20 38 24 34)(25 41 29 45)(26 42 30 46)(27 43 31 47)(28 44 32 48)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(41 45)(42 46)(43 47)(44 48)
G:=sub<Sym(48)| (1,26,35)(2,27,36)(3,28,37)(4,29,38)(5,30,39)(6,31,40)(7,32,33)(8,25,34)(9,44,23)(10,45,24)(11,46,17)(12,47,18)(13,48,19)(14,41,20)(15,42,21)(16,43,22), (17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33), (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), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,35,21,39)(18,36,22,40)(19,37,23,33)(20,38,24,34)(25,41,29,45)(26,42,30,46)(27,43,31,47)(28,44,32,48), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(41,45)(42,46)(43,47)(44,48)>;
G:=Group( (1,26,35)(2,27,36)(3,28,37)(4,29,38)(5,30,39)(6,31,40)(7,32,33)(8,25,34)(9,44,23)(10,45,24)(11,46,17)(12,47,18)(13,48,19)(14,41,20)(15,42,21)(16,43,22), (17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33), (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), (1,15,5,11)(2,16,6,12)(3,9,7,13)(4,10,8,14)(17,35,21,39)(18,36,22,40)(19,37,23,33)(20,38,24,34)(25,41,29,45)(26,42,30,46)(27,43,31,47)(28,44,32,48), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(41,45)(42,46)(43,47)(44,48) );
G=PermutationGroup([[(1,26,35),(2,27,36),(3,28,37),(4,29,38),(5,30,39),(6,31,40),(7,32,33),(8,25,34),(9,44,23),(10,45,24),(11,46,17),(12,47,18),(13,48,19),(14,41,20),(15,42,21),(16,43,22)], [(17,46),(18,47),(19,48),(20,41),(21,42),(22,43),(23,44),(24,45),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(31,40),(32,33)], [(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)], [(1,15,5,11),(2,16,6,12),(3,9,7,13),(4,10,8,14),(17,35,21,39),(18,36,22,40),(19,37,23,33),(20,38,24,34),(25,41,29,45),(26,42,30,46),(27,43,31,47),(28,44,32,48)], [(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(41,45),(42,46),(43,47),(44,48)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N | 8O | ··· | 8T | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D | 24E | ··· | 24J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 2 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D6 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | S3×C8○D4 |
| kernel | S3×C8○D4 | S3×C2×C8 | C8○D12 | S3×M4(2) | D12.C4 | D4.Dic3 | C3×C8○D4 | S3×C4○D4 | S3×D4 | D4⋊2S3 | S3×Q8 | Q8⋊3S3 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | S3 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 1 | 3 | 3 | 1 | 6 | 2 | 8 | 4 |
Matrix representation of S3×C8○D4 ►in GL4(𝔽73) generated by
| 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 72 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 22 | 0 |
| 0 | 0 | 0 | 22 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [0,1,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,27,0,0,0,0,22,0,0,0,0,22],[72,0,0,0,0,72,0,0,0,0,0,1,0,0,72,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,72] >;
S3×C8○D4 in GAP, Magma, Sage, TeX
S_3\times C_8\circ D_4
% in TeX
G:=Group("S3xC8oD4"); // GroupNames label
G:=SmallGroup(192,1308);
// by ID
G=gap.SmallGroup(192,1308);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,80,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^8=e^2=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^4*d>;
// generators/relations