metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24:8D6, C6.262+ 1+4, C3:2D42, D6:6(C2xD4), C3:D4:4D4, (C2xD4):18D6, C22:4(S3xD4), C22wrC2:4S3, D6:D4:9C2, C23:2D6:4C2, C22:C4:24D6, Dic3:3(C2xD4), (C6xD4):7C22, Dic3:D4:13C2, C12:3D4:11C2, D6:C4:11C22, C6.56(C22xD4), C23.14D6:2C2, Dic3:4D4:2C2, (C2xD12):19C22, (C2xC6).134C24, (C2xC12).28C23, Dic3:C4:9C22, (S3xC23):7C22, (C23xC6):10C22, C2.28(D4:6D6), (C4xDic3):14C22, C6.D4:15C22, (C22xS3).53C23, C23.118(C22xS3), (C22xC6).181C23, C22.155(S3xC23), (C2xDic3).221C23, (C22xDic3):13C22, (C2xS3xD4):7C2, (C2xC6):6(C2xD4), C2.29(C2xS3xD4), (S3xC2xC4):7C22, (C3xC22wrC2):5C2, (C22xC3:D4):8C2, (C2xC3:D4):39C22, (C3xC22:C4):5C22, (C2xC4).28(C22xS3), SmallGroup(192,1149)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24:8D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, eae-1=faf=ac=ca, ad=da, fbf=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
Subgroups: 1344 in 428 conjugacy classes, 115 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, C2xC4, D4, C23, C23, C23, Dic3, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C22:C4, C4:C4, C22xC4, C2xD4, C2xD4, C2xD4, C24, C24, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C3:D4, C2xC12, C2xC12, C3xD4, C22xS3, C22xS3, C22xC6, C22xC6, C22xC6, C4xD4, C22wrC2, C22wrC2, C4:D4, C4:1D4, C22xD4, C4xDic3, Dic3:C4, D6:C4, C6.D4, C3xC22:C4, C3xC22:C4, S3xC2xC4, C2xD12, C2xD12, S3xD4, C22xDic3, C2xC3:D4, C2xC3:D4, C6xD4, C6xD4, S3xC23, S3xC23, C23xC6, D42, Dic3:4D4, D6:D4, Dic3:D4, C23:2D6, C23.14D6, C12:3D4, C3xC22wrC2, C2xS3xD4, C22xC3:D4, C24:8D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C24, C22xS3, C22xD4, 2+ 1+4, S3xD4, S3xC23, D42, C2xS3xD4, D4:6D6, C24:8D6
(1 25)(2 29)(3 27)(4 38)(5 42)(6 40)(7 28)(8 26)(9 30)(10 41)(11 39)(12 37)(13 32)(14 44)(15 34)(16 46)(17 36)(18 48)(19 43)(20 33)(21 45)(22 35)(23 47)(24 31)
(1 13)(2 17)(3 15)(4 16)(5 14)(6 18)(7 19)(8 23)(9 21)(10 22)(11 20)(12 24)(25 32)(26 47)(27 34)(28 43)(29 36)(30 45)(31 37)(33 39)(35 41)(38 46)(40 48)(42 44)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 28)(26 29)(27 30)(31 48)(32 43)(33 44)(34 45)(35 46)(36 47)(37 40)(38 41)(39 42)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 38)(26 39)(27 40)(28 41)(29 42)(30 37)(31 45)(32 46)(33 47)(34 48)(35 43)(36 44)
(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 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(25 37)(26 42)(27 41)(28 40)(29 39)(30 38)(31 43)(32 48)(33 47)(34 46)(35 45)(36 44)
G:=sub<Sym(48)| (1,25)(2,29)(3,27)(4,38)(5,42)(6,40)(7,28)(8,26)(9,30)(10,41)(11,39)(12,37)(13,32)(14,44)(15,34)(16,46)(17,36)(18,48)(19,43)(20,33)(21,45)(22,35)(23,47)(24,31), (1,13)(2,17)(3,15)(4,16)(5,14)(6,18)(7,19)(8,23)(9,21)(10,22)(11,20)(12,24)(25,32)(26,47)(27,34)(28,43)(29,36)(30,45)(31,37)(33,39)(35,41)(38,46)(40,48)(42,44), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,28)(26,29)(27,30)(31,48)(32,43)(33,44)(34,45)(35,46)(36,47)(37,40)(38,41)(39,42), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,38)(26,39)(27,40)(28,41)(29,42)(30,37)(31,45)(32,46)(33,47)(34,48)(35,43)(36,44), (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), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(25,37)(26,42)(27,41)(28,40)(29,39)(30,38)(31,43)(32,48)(33,47)(34,46)(35,45)(36,44)>;
G:=Group( (1,25)(2,29)(3,27)(4,38)(5,42)(6,40)(7,28)(8,26)(9,30)(10,41)(11,39)(12,37)(13,32)(14,44)(15,34)(16,46)(17,36)(18,48)(19,43)(20,33)(21,45)(22,35)(23,47)(24,31), (1,13)(2,17)(3,15)(4,16)(5,14)(6,18)(7,19)(8,23)(9,21)(10,22)(11,20)(12,24)(25,32)(26,47)(27,34)(28,43)(29,36)(30,45)(31,37)(33,39)(35,41)(38,46)(40,48)(42,44), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,28)(26,29)(27,30)(31,48)(32,43)(33,44)(34,45)(35,46)(36,47)(37,40)(38,41)(39,42), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,38)(26,39)(27,40)(28,41)(29,42)(30,37)(31,45)(32,46)(33,47)(34,48)(35,43)(36,44), (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), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(25,37)(26,42)(27,41)(28,40)(29,39)(30,38)(31,43)(32,48)(33,47)(34,46)(35,45)(36,44) );
G=PermutationGroup([[(1,25),(2,29),(3,27),(4,38),(5,42),(6,40),(7,28),(8,26),(9,30),(10,41),(11,39),(12,37),(13,32),(14,44),(15,34),(16,46),(17,36),(18,48),(19,43),(20,33),(21,45),(22,35),(23,47),(24,31)], [(1,13),(2,17),(3,15),(4,16),(5,14),(6,18),(7,19),(8,23),(9,21),(10,22),(11,20),(12,24),(25,32),(26,47),(27,34),(28,43),(29,36),(30,45),(31,37),(33,39),(35,41),(38,46),(40,48),(42,44)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,28),(26,29),(27,30),(31,48),(32,43),(33,44),(34,45),(35,46),(36,47),(37,40),(38,41),(39,42)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,38),(26,39),(27,40),(28,41),(29,42),(30,37),(31,45),(32,46),(33,47),(34,48),(35,43),(36,44)], [(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)], [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(25,37),(26,42),(27,41),(28,40),(29,39),(30,38),(31,43),(32,48),(33,47),(34,46),(35,45),(36,44)]])
39 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | ··· | 6I | 6J | 12A | 12B | 12C |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 12 | 12 | 12 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | 2+ 1+4 | S3xD4 | D4:6D6 |
kernel | C24:8D6 | Dic3:4D4 | D6:D4 | Dic3:D4 | C23:2D6 | C23.14D6 | C12:3D4 | C3xC22wrC2 | C2xS3xD4 | C22xC3:D4 | C22wrC2 | C3:D4 | C22:C4 | C2xD4 | C24 | C6 | C22 | C2 |
# reps | 1 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 8 | 3 | 3 | 1 | 1 | 4 | 2 |
Matrix representation of C24:8D6 ►in GL6(Z)
-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 | 1 | -2 |
0 | 0 | 0 | 0 | 0 | -1 |
1 | -2 | 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 | -2 |
0 | 0 | 0 | 0 | 0 | -1 |
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 | -1 | 0 |
0 | 0 | 0 | 0 | 0 | -1 |
-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 | -1 | 0 |
0 | 0 | 0 | 0 | 0 | -1 |
1 | 0 | 0 | 0 | 0 | 0 |
1 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | -1 | -1 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 | -1 |
-1 | 0 | 0 | 0 | 0 | 0 |
0 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | -1 | -1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | -1 | 0 |
0 | 0 | 0 | 0 | -1 | 1 |
G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,-2,-1],[1,0,0,0,0,0,-2,-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,-2,-1],[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,-1,0,0,0,0,0,0,-1],[-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,-1,0,0,0,0,0,0,-1],[1,1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1] >;
C24:8D6 in GAP, Magma, Sage, TeX
C_2^4\rtimes_8D_6
% in TeX
G:=Group("C2^4:8D6");
// GroupNames label
G:=SmallGroup(192,1149);
// by ID
G=gap.SmallGroup(192,1149);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,297,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=f^2=1,a*b=b*a,e*a*e^-1=f*a*f=a*c=c*a,a*d=d*a,f*b*f=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations