direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3xC22:Q8, C4:C4:25D6, D6:3(C2xQ8), (C2xQ8):17D6, C22:3(S3xQ8), D6.59(C2xD4), (C4xS3).41D4, C4.185(S3xD4), (C6xQ8):5C22, (C22xS3):6Q8, C4.D12:23C2, D6:Q8:17C2, D6:3Q8:12C2, C12.230(C2xD4), C22:C4.55D6, C6.72(C22xD4), D6.38(C4oD4), C6.34(C22xQ8), (C2xC6).170C24, (C2xC12).51C23, D6:C4.20C22, C4:Dic3:34C22, Dic3.47(C2xD4), (C22xC4).387D6, C12.48D4:35C2, Dic3:C4:30C22, (C2xDic6):27C22, Dic3.D4:22C2, (C22xC6).198C23, C22.191(S3xC23), C23.197(C22xS3), (C2xDic3).85C23, (S3xC23).108C22, (C22xS3).192C23, (C22xC12).250C22, C6.D4.32C22, (C22xDic3).224C22, (C2xS3xQ8):5C2, (S3xC4:C4):24C2, (C2xC6):2(C2xQ8), C2.45(C2xS3xD4), C3:4(C2xC22:Q8), C2.17(C2xS3xQ8), C2.47(S3xC4oD4), (S3xC22xC4).7C2, (C3xC22:Q8):6C2, (C3xC4:C4):17C22, C6.159(C2xC4oD4), (S3xC22:C4).1C2, (S3xC2xC4).92C22, (C2xC4).45(C22xS3), (C3xC22:C4).25C22, SmallGroup(192,1185)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3xC22:Q8
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=1, f2=e2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >
Subgroups: 800 in 322 conjugacy classes, 121 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, S3, C6, C6, C2xC4, C2xC4, C2xC4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC6, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xQ8, C2xQ8, C24, Dic6, C4xS3, C4xS3, C2xDic3, C2xDic3, C2xDic3, C2xC12, C2xC12, C2xC12, C3xQ8, C22xS3, C22xS3, C22xS3, C22xC6, C2xC22:C4, C2xC4:C4, C22:Q8, C22:Q8, C23xC4, C22xQ8, Dic3:C4, C4:Dic3, C4:Dic3, D6:C4, C6.D4, C3xC22:C4, C3xC4:C4, C3xC4:C4, C2xDic6, C2xDic6, S3xC2xC4, S3xC2xC4, S3xC2xC4, S3xQ8, C22xDic3, C22xC12, C6xQ8, S3xC23, C2xC22:Q8, Dic3.D4, S3xC22:C4, S3xC4:C4, S3xC4:C4, D6:Q8, C4.D12, C12.48D4, D6:3Q8, C3xC22:Q8, S3xC22xC4, C2xS3xQ8, S3xC22:Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2xD4, C2xQ8, C4oD4, C24, C22xS3, C22:Q8, C22xD4, C22xQ8, C2xC4oD4, S3xD4, S3xQ8, S3xC23, C2xC22:Q8, C2xS3xD4, C2xS3xQ8, S3xC4oD4, S3xC22:Q8
(1 32 47)(2 29 48)(3 30 45)(4 31 46)(5 44 14)(6 41 15)(7 42 16)(8 43 13)(9 19 39)(10 20 40)(11 17 37)(12 18 38)(21 33 25)(22 34 26)(23 35 27)(24 36 28)
(1 9)(2 10)(3 11)(4 12)(5 34)(6 35)(7 36)(8 33)(13 25)(14 26)(15 27)(16 28)(17 45)(18 46)(19 47)(20 48)(21 43)(22 44)(23 41)(24 42)(29 40)(30 37)(31 38)(32 39)
(1 11)(2 12)(3 9)(4 10)(5 7)(6 8)(13 15)(14 16)(17 32)(18 29)(19 30)(20 31)(21 23)(22 24)(25 27)(26 28)(33 35)(34 36)(37 47)(38 48)(39 45)(40 46)(41 43)(42 44)
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)(33 43)(34 44)(35 41)(36 42)(37 45)(38 46)(39 47)(40 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 26 3 28)(2 25 4 27)(5 17 7 19)(6 20 8 18)(9 14 11 16)(10 13 12 15)(21 31 23 29)(22 30 24 32)(33 46 35 48)(34 45 36 47)(37 42 39 44)(38 41 40 43)
G:=sub<Sym(48)| (1,32,47)(2,29,48)(3,30,45)(4,31,46)(5,44,14)(6,41,15)(7,42,16)(8,43,13)(9,19,39)(10,20,40)(11,17,37)(12,18,38)(21,33,25)(22,34,26)(23,35,27)(24,36,28), (1,9)(2,10)(3,11)(4,12)(5,34)(6,35)(7,36)(8,33)(13,25)(14,26)(15,27)(16,28)(17,45)(18,46)(19,47)(20,48)(21,43)(22,44)(23,41)(24,42)(29,40)(30,37)(31,38)(32,39), (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)(17,32)(18,29)(19,30)(20,31)(21,23)(22,24)(25,27)(26,28)(33,35)(34,36)(37,47)(38,48)(39,45)(40,46)(41,43)(42,44), (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,30)(18,31)(19,32)(20,29)(33,43)(34,44)(35,41)(36,42)(37,45)(38,46)(39,47)(40,48), (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), (1,26,3,28)(2,25,4,27)(5,17,7,19)(6,20,8,18)(9,14,11,16)(10,13,12,15)(21,31,23,29)(22,30,24,32)(33,46,35,48)(34,45,36,47)(37,42,39,44)(38,41,40,43)>;
G:=Group( (1,32,47)(2,29,48)(3,30,45)(4,31,46)(5,44,14)(6,41,15)(7,42,16)(8,43,13)(9,19,39)(10,20,40)(11,17,37)(12,18,38)(21,33,25)(22,34,26)(23,35,27)(24,36,28), (1,9)(2,10)(3,11)(4,12)(5,34)(6,35)(7,36)(8,33)(13,25)(14,26)(15,27)(16,28)(17,45)(18,46)(19,47)(20,48)(21,43)(22,44)(23,41)(24,42)(29,40)(30,37)(31,38)(32,39), (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)(17,32)(18,29)(19,30)(20,31)(21,23)(22,24)(25,27)(26,28)(33,35)(34,36)(37,47)(38,48)(39,45)(40,46)(41,43)(42,44), (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,30)(18,31)(19,32)(20,29)(33,43)(34,44)(35,41)(36,42)(37,45)(38,46)(39,47)(40,48), (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), (1,26,3,28)(2,25,4,27)(5,17,7,19)(6,20,8,18)(9,14,11,16)(10,13,12,15)(21,31,23,29)(22,30,24,32)(33,46,35,48)(34,45,36,47)(37,42,39,44)(38,41,40,43) );
G=PermutationGroup([[(1,32,47),(2,29,48),(3,30,45),(4,31,46),(5,44,14),(6,41,15),(7,42,16),(8,43,13),(9,19,39),(10,20,40),(11,17,37),(12,18,38),(21,33,25),(22,34,26),(23,35,27),(24,36,28)], [(1,9),(2,10),(3,11),(4,12),(5,34),(6,35),(7,36),(8,33),(13,25),(14,26),(15,27),(16,28),(17,45),(18,46),(19,47),(20,48),(21,43),(22,44),(23,41),(24,42),(29,40),(30,37),(31,38),(32,39)], [(1,11),(2,12),(3,9),(4,10),(5,7),(6,8),(13,15),(14,16),(17,32),(18,29),(19,30),(20,31),(21,23),(22,24),(25,27),(26,28),(33,35),(34,36),(37,47),(38,48),(39,45),(40,46),(41,43),(42,44)], [(1,9),(2,10),(3,11),(4,12),(5,22),(6,23),(7,24),(8,21),(13,25),(14,26),(15,27),(16,28),(17,30),(18,31),(19,32),(20,29),(33,43),(34,44),(35,41),(36,42),(37,45),(38,46),(39,47),(40,48)], [(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)], [(1,26,3,28),(2,25,4,27),(5,17,7,19),(6,20,8,18),(9,14,11,16),(10,13,12,15),(21,31,23,29),(22,30,24,32),(33,46,35,48),(34,45,36,47),(37,42,39,44),(38,41,40,43)]])
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 | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
42 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | ||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | D6 | D6 | C4oD4 | S3xD4 | S3xQ8 | S3xC4oD4 |
kernel | S3xC22:Q8 | Dic3.D4 | S3xC22:C4 | S3xC4:C4 | D6:Q8 | C4.D12 | C12.48D4 | D6:3Q8 | C3xC22:Q8 | S3xC22xC4 | C2xS3xQ8 | C22:Q8 | C4xS3 | C22xS3 | C22:C4 | C4:C4 | C22xC4 | C2xQ8 | D6 | C4 | C22 | C2 |
# reps | 1 | 2 | 2 | 3 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 3 | 1 | 1 | 4 | 2 | 2 | 2 |
Matrix representation of S3xC22:Q8 ►in GL6(F13)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 12 | 12 |
0 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 12 | 0 | 0 | 0 |
0 | 0 | 0 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 12 | 12 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 2 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 12 | 0 | 0 | 0 |
0 | 0 | 0 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
7 | 5 | 0 | 0 | 0 | 0 |
3 | 6 | 0 | 0 | 0 | 0 |
0 | 0 | 5 | 0 | 0 | 0 |
0 | 0 | 10 | 8 | 0 | 0 |
0 | 0 | 0 | 0 | 12 | 0 |
0 | 0 | 0 | 0 | 0 | 12 |
1 | 6 | 0 | 0 | 0 | 0 |
4 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 5 | 8 | 0 | 0 |
0 | 0 | 0 | 8 | 0 | 0 |
0 | 0 | 0 | 0 | 12 | 0 |
0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,3,0,0,0,0,5,6,0,0,0,0,0,0,5,10,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,4,0,0,0,0,6,12,0,0,0,0,0,0,5,0,0,0,0,0,8,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
S3xC22:Q8 in GAP, Magma, Sage, TeX
S_3\times C_2^2\rtimes Q_8
% in TeX
G:=Group("S3xC2^2:Q8");
// GroupNames label
G:=SmallGroup(192,1185);
// by ID
G=gap.SmallGroup(192,1185);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,794,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^4=1,f^2=e^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations