metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: 2+ 1+4.4S3, C4oD4.26D6, (C2xD4).83D6, (C3xD4).33D4, (C3xQ8).33D4, C6.81C22wrC2, Q8.14D6:5C2, C12.218(C2xD4), C3:5(D4.9D4), (C22xC6).25D4, C23.12D6:8C2, C12.D4:12C2, D4.15(C3:D4), (C2xC12).22C23, Q8.22(C3:D4), (C4xDic3):9C22, Q8:3Dic3:12C2, (C2xDic6):16C22, (C6xD4).108C22, C23.13(C3:D4), C4.Dic3:11C22, C2.15(C24:4S3), (C3x2+ 1+4).1C2, (C2xC6).43(C2xD4), C4.65(C2xC3:D4), (C2xC4).22(C22xS3), C22.15(C2xC3:D4), (C3xC4oD4).20C22, SmallGroup(192,801)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for 2+ 1+4.4S3
G = < a,b,c,d,e,f | a4=b2=d2=e3=1, c2=f2=a2, bab=faf-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf-1=ab, dcd=fcf-1=a2c, ce=ec, de=ed, fdf-1=cd, fef-1=e-1 >
Subgroups: 392 in 152 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, C2xC6, C2xC6, C42, C22:C4, M4(2), SD16, Q16, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C3:C8, Dic6, C2xDic3, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C22xC6, C22xC6, C4.D4, C4wrC2, C4.4D4, C8.C22, 2+ 1+4, C4.Dic3, C4xDic3, D4.S3, C3:Q16, C6.D4, C2xDic6, C6xD4, C6xD4, C3xC4oD4, C3xC4oD4, D4.9D4, C12.D4, Q8:3Dic3, C23.12D6, Q8.14D6, C3x2+ 1+4, 2+ 1+4.4S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C22wrC2, C2xC3:D4, D4.9D4, C24:4S3, 2+ 1+4.4S3
(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 45)(2 48)(3 47)(4 46)(5 19)(6 18)(7 17)(8 20)(9 15)(10 14)(11 13)(12 16)(21 34)(22 33)(23 36)(24 35)(25 43)(26 42)(27 41)(28 44)(29 39)(30 38)(31 37)(32 40)
(1 2 3 4)(5 8 7 6)(9 12 11 10)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 36 35 34)(37 40 39 38)(41 44 43 42)(45 48 47 46)
(1 35)(2 36)(3 33)(4 34)(5 28)(6 25)(7 26)(8 27)(9 32)(10 29)(11 30)(12 31)(13 38)(14 39)(15 40)(16 37)(17 42)(18 43)(19 44)(20 41)(21 46)(22 47)(23 48)(24 45)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 45)(6 11 46)(7 12 47)(8 9 48)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 42 37)(34 43 38)(35 44 39)(36 41 40)
(1 22 3 24)(2 21 4 23)(5 40 7 38)(6 39 8 37)(9 42 11 44)(10 41 12 43)(13 27 15 25)(14 26 16 28)(17 29 19 31)(18 32 20 30)(33 46 35 48)(34 45 36 47)
G:=sub<Sym(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), (1,45)(2,48)(3,47)(4,46)(5,19)(6,18)(7,17)(8,20)(9,15)(10,14)(11,13)(12,16)(21,34)(22,33)(23,36)(24,35)(25,43)(26,42)(27,41)(28,44)(29,39)(30,38)(31,37)(32,40), (1,2,3,4)(5,8,7,6)(9,12,11,10)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,36,35,34)(37,40,39,38)(41,44,43,42)(45,48,47,46), (1,35)(2,36)(3,33)(4,34)(5,28)(6,25)(7,26)(8,27)(9,32)(10,29)(11,30)(12,31)(13,38)(14,39)(15,40)(16,37)(17,42)(18,43)(19,44)(20,41)(21,46)(22,47)(23,48)(24,45), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,45)(6,11,46)(7,12,47)(8,9,48)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,42,37)(34,43,38)(35,44,39)(36,41,40), (1,22,3,24)(2,21,4,23)(5,40,7,38)(6,39,8,37)(9,42,11,44)(10,41,12,43)(13,27,15,25)(14,26,16,28)(17,29,19,31)(18,32,20,30)(33,46,35,48)(34,45,36,47)>;
G:=Group( (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), (1,45)(2,48)(3,47)(4,46)(5,19)(6,18)(7,17)(8,20)(9,15)(10,14)(11,13)(12,16)(21,34)(22,33)(23,36)(24,35)(25,43)(26,42)(27,41)(28,44)(29,39)(30,38)(31,37)(32,40), (1,2,3,4)(5,8,7,6)(9,12,11,10)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,36,35,34)(37,40,39,38)(41,44,43,42)(45,48,47,46), (1,35)(2,36)(3,33)(4,34)(5,28)(6,25)(7,26)(8,27)(9,32)(10,29)(11,30)(12,31)(13,38)(14,39)(15,40)(16,37)(17,42)(18,43)(19,44)(20,41)(21,46)(22,47)(23,48)(24,45), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,45)(6,11,46)(7,12,47)(8,9,48)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,42,37)(34,43,38)(35,44,39)(36,41,40), (1,22,3,24)(2,21,4,23)(5,40,7,38)(6,39,8,37)(9,42,11,44)(10,41,12,43)(13,27,15,25)(14,26,16,28)(17,29,19,31)(18,32,20,30)(33,46,35,48)(34,45,36,47) );
G=PermutationGroup([[(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)], [(1,45),(2,48),(3,47),(4,46),(5,19),(6,18),(7,17),(8,20),(9,15),(10,14),(11,13),(12,16),(21,34),(22,33),(23,36),(24,35),(25,43),(26,42),(27,41),(28,44),(29,39),(30,38),(31,37),(32,40)], [(1,2,3,4),(5,8,7,6),(9,12,11,10),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,36,35,34),(37,40,39,38),(41,44,43,42),(45,48,47,46)], [(1,35),(2,36),(3,33),(4,34),(5,28),(6,25),(7,26),(8,27),(9,32),(10,29),(11,30),(12,31),(13,38),(14,39),(15,40),(16,37),(17,42),(18,43),(19,44),(20,41),(21,46),(22,47),(23,48),(24,45)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,10,45),(6,11,46),(7,12,47),(8,9,48),(21,25,30),(22,26,31),(23,27,32),(24,28,29),(33,42,37),(34,43,38),(35,44,39),(36,41,40)], [(1,22,3,24),(2,21,4,23),(5,40,7,38),(6,39,8,37),(9,42,11,44),(10,41,12,43),(13,27,15,25),(14,26,16,28),(17,29,19,31),(18,32,20,30),(33,46,35,48),(34,45,36,47)]])
33 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | ··· | 6J | 8A | 8B | 12A | ··· | 12F |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 |
size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 24 | 2 | 4 | ··· | 4 | 24 | 24 | 4 | ··· | 4 |
33 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | C3:D4 | C3:D4 | C3:D4 | D4.9D4 | 2+ 1+4.4S3 |
kernel | 2+ 1+4.4S3 | C12.D4 | Q8:3Dic3 | C23.12D6 | Q8.14D6 | C3x2+ 1+4 | 2+ 1+4 | C3xD4 | C3xQ8 | C22xC6 | C2xD4 | C4oD4 | D4 | Q8 | C23 | C3 | C1 |
# reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 1 |
Matrix representation of 2+ 1+4.4S3 ►in GL6(F73)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 46 | 0 | 0 | 0 |
0 | 0 | 0 | 27 | 0 | 0 |
0 | 0 | 0 | 0 | 46 | 0 |
0 | 0 | 0 | 0 | 0 | 27 |
72 | 0 | 0 | 0 | 0 | 0 |
0 | 72 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 27 |
0 | 0 | 0 | 0 | 46 | 0 |
0 | 0 | 0 | 27 | 0 | 0 |
0 | 0 | 46 | 0 | 0 | 0 |
72 | 0 | 0 | 0 | 0 | 0 |
0 | 72 | 0 | 0 | 0 | 0 |
0 | 0 | 46 | 0 | 0 | 0 |
0 | 0 | 0 | 27 | 0 | 0 |
0 | 0 | 0 | 0 | 27 | 0 |
0 | 0 | 0 | 0 | 0 | 46 |
0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 27 | 0 |
0 | 0 | 0 | 0 | 0 | 46 |
0 | 0 | 46 | 0 | 0 | 0 |
0 | 0 | 0 | 27 | 0 | 0 |
36 | 45 | 0 | 0 | 0 | 0 |
45 | 36 | 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 | 1 |
36 | 45 | 0 | 0 | 0 | 0 |
28 | 37 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 72 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 46 |
0 | 0 | 0 | 0 | 46 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,46,0,0,0,0,0,0,27],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,46,0,0,0,0,27,0,0,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,46],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,27,0,0,0,0,0,0,46,0,0],[36,45,0,0,0,0,45,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,28,0,0,0,0,45,37,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,46,0,0,0,0,46,0] >;
2+ 1+4.4S3 in GAP, Magma, Sage, TeX
2_+^{1+4}._4S_3
% in TeX
G:=Group("ES+(2,2).4S3");
// GroupNames label
G:=SmallGroup(192,801);
// by ID
G=gap.SmallGroup(192,801);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,570,1684,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=b^2=d^2=e^3=1,c^2=f^2=a^2,b*a*b=f*a*f^-1=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,f*b*f^-1=a*b,d*c*d=f*c*f^-1=a^2*c,c*e=e*c,d*e=e*d,f*d*f^-1=c*d,f*e*f^-1=e^-1>;
// generators/relations