direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3xD4:C4, D6.10D8, D6.5SD16, C4:C4:18D6, (S3xD4):1C4, D4:5(C4xS3), (C2xC8):25D6, C2.2(S3xD8), D12:2(C2xC4), C6.21(C2xD8), (C4xS3).37D4, C6.D8:4C2, C4.154(S3xD4), C2.2(S3xSD16), (C2xC24):27C22, (C2xD4).130D6, C2.D24:22C2, C12.103(C2xD4), D4:Dic3:3C2, C12.4(C22xC4), C6.22(C2xSD16), C22.68(S3xD4), C4:Dic3:17C22, (C2xDic3).88D4, (C6xD4).30C22, (C2xC12).209C23, D6.18(C22:C4), (C2xD12).47C22, (C22xS3).104D4, Dic3.6(C22:C4), C4.4(S3xC2xC4), (S3xC4:C4):1C2, (S3xC2xC8):16C2, (C2xS3xD4).3C2, (C3xD4):2(C2xC4), C3:1(C2xD4:C4), (C3xC4:C4):1C22, (C2xC3:C8):30C22, (C4xS3).12(C2xC4), (C2xC6).222(C2xD4), C6.16(C2xC22:C4), C2.17(S3xC22:C4), (C3xD4:C4):23C2, (S3xC2xC4).222C22, (C2xC4).316(C22xS3), SmallGroup(192,328)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3xD4:C4
G = < a,b,c,d,e | a3=b2=c4=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=ece-1=c-1, ede-1=cd >
Subgroups: 712 in 202 conjugacy classes, 63 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C2xC4, C2xC4, D4, D4, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C4:C4, C4:C4, C2xC8, C2xC8, C22xC4, C2xD4, C2xD4, C24, C3:C8, C24, C4xS3, C4xS3, D12, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xD4, C22xS3, C22xS3, C22xC6, D4:C4, D4:C4, C2xC4:C4, C22xC8, C22xD4, S3xC8, C2xC3:C8, Dic3:C4, C4:Dic3, C3xC4:C4, C2xC24, S3xC2xC4, S3xC2xC4, C2xD12, S3xD4, S3xD4, C2xC3:D4, C6xD4, S3xC23, C2xD4:C4, C6.D8, C2.D24, D4:Dic3, C3xD4:C4, S3xC4:C4, S3xC2xC8, C2xS3xD4, S3xD4:C4
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, C22:C4, D8, SD16, C22xC4, C2xD4, C4xS3, C22xS3, D4:C4, C2xC22:C4, C2xD8, C2xSD16, S3xC2xC4, S3xD4, C2xD4:C4, S3xC22:C4, S3xD8, S3xSD16, S3xD4:C4
►On 48 points
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 45 10)(6 46 11)(7 47 12)(8 48 9)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 39 44)(34 40 41)(35 37 42)(36 38 43)
(1 22)(2 23)(3 24)(4 21)(5 39)(6 40)(7 37)(8 38)(9 43)(10 44)(11 41)(12 42)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 45)(34 46)(35 47)(36 48)
(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 21)(2 24)(3 23)(4 22)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 29)(14 32)(15 31)(16 30)(17 25)(18 28)(19 27)(20 26)(33 45)(34 48)(35 47)(36 46)
(1 46 22 34)(2 45 23 33)(3 48 24 36)(4 47 21 35)(5 32 44 13)(6 31 41 16)(7 30 42 15)(8 29 43 14)(9 28 38 19)(10 27 39 18)(11 26 40 17)(12 25 37 20)
G:=sub<Sym(48)| (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,45,10)(6,46,11)(7,47,12)(8,48,9)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,39,44)(34,40,41)(35,37,42)(36,38,43), (1,22)(2,23)(3,24)(4,21)(5,39)(6,40)(7,37)(8,38)(9,43)(10,44)(11,41)(12,42)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30)(33,45)(34,46)(35,47)(36,48), (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), (1,21)(2,24)(3,23)(4,22)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,29)(14,32)(15,31)(16,30)(17,25)(18,28)(19,27)(20,26)(33,45)(34,48)(35,47)(36,46), (1,46,22,34)(2,45,23,33)(3,48,24,36)(4,47,21,35)(5,32,44,13)(6,31,41,16)(7,30,42,15)(8,29,43,14)(9,28,38,19)(10,27,39,18)(11,26,40,17)(12,25,37,20)>;
G:=Group( (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,45,10)(6,46,11)(7,47,12)(8,48,9)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,39,44)(34,40,41)(35,37,42)(36,38,43), (1,22)(2,23)(3,24)(4,21)(5,39)(6,40)(7,37)(8,38)(9,43)(10,44)(11,41)(12,42)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30)(33,45)(34,46)(35,47)(36,48), (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), (1,21)(2,24)(3,23)(4,22)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,29)(14,32)(15,31)(16,30)(17,25)(18,28)(19,27)(20,26)(33,45)(34,48)(35,47)(36,46), (1,46,22,34)(2,45,23,33)(3,48,24,36)(4,47,21,35)(5,32,44,13)(6,31,41,16)(7,30,42,15)(8,29,43,14)(9,28,38,19)(10,27,39,18)(11,26,40,17)(12,25,37,20) );
G=PermutationGroup([[(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,45,10),(6,46,11),(7,47,12),(8,48,9),(21,25,30),(22,26,31),(23,27,32),(24,28,29),(33,39,44),(34,40,41),(35,37,42),(36,38,43)], [(1,22),(2,23),(3,24),(4,21),(5,39),(6,40),(7,37),(8,38),(9,43),(10,44),(11,41),(12,42),(13,27),(14,28),(15,25),(16,26),(17,31),(18,32),(19,29),(20,30),(33,45),(34,46),(35,47),(36,48)], [(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)], [(1,21),(2,24),(3,23),(4,22),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,29),(14,32),(15,31),(16,30),(17,25),(18,28),(19,27),(20,26),(33,45),(34,48),(35,47),(36,46)], [(1,46,22,34),(2,45,23,33),(3,48,24,36),(4,47,21,35),(5,32,44,13),(6,31,41,16),(7,30,42,15),(8,29,43,14),(9,28,38,19),(10,27,39,18),(11,26,40,17),(12,25,37,20)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | D8 | SD16 | C4xS3 | S3xD4 | S3xD4 | S3xD8 | S3xSD16 |
| kernel | S3xD4:C4 | C6.D8 | C2.D24 | D4:Dic3 | C3xD4:C4 | S3xC4:C4 | S3xC2xC8 | C2xS3xD4 | S3xD4 | D4:C4 | C4xS3 | C2xDic3 | C22xS3 | C4:C4 | C2xC8 | C2xD4 | D6 | D6 | D4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of S3xD4:C4 ►in GL4(F73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 16 | 16 | 0 | 0 |
| 16 | 57 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,72,0],[0,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,72,0,0,0,0,72],[16,16,0,0,16,57,0,0,0,0,27,0,0,0,0,27] >;
S3xD4:C4 in GAP, Magma, Sage, TeX
S_3\times D_4\rtimes C_4
% in TeX
G:=Group("S3xD4:C4"); // GroupNames label
G:=SmallGroup(192,328);
// by ID
G=gap.SmallGroup(192,328);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,58,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=d^2=e^4=1,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,d*c*d=e*c*e^-1=c^-1,e*d*e^-1=c*d>;
// generators/relations