metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:23D6, C6.732+ 1+4, (C2xQ8):12D6, C22:C4:35D6, (C4xC12):33C22, D6:3Q8:31C2, (C2xD4).111D6, C23:2D6.6C2, C4.4D4:13S3, (C6xQ8):15C22, C42:2S3:36C2, D6.23(C4oD4), C23.9D6:44C2, (C2xC6).223C24, C4:Dic3:42C22, Dic3:4D4:32C2, C2.76(D4:6D6), (C2xC12).632C23, Dic3:C4:67C22, D6:C4.136C22, (C4xDic3):57C22, (C6xD4).211C22, C23.8D6:40C2, (C22xC6).53C23, C23.55(C22xS3), C3:8(C22.45C24), C6.D4:34C22, C23.23D6:25C2, (S3xC23).66C22, C22.244(S3xC23), (C22xS3).217C23, (C2xDic3).255C23, (C22xDic3):28C22, C2.79(S3xC4oD4), (S3xC22:C4):19C2, C6.190(C2xC4oD4), (C3xC4.4D4):15C2, (S3xC2xC4).215C22, (C2xC4).74(C22xS3), (C3xC22:C4):31C22, (C2xC3:D4).61C22, SmallGroup(192,1238)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:23D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 672 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2xC6, C2xC6, C42, C42, C22:C4, C22:C4, C4:C4, C22xC4, C2xD4, C2xD4, C2xQ8, C24, C4xS3, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C22xS3, C22xC6, C2xC22:C4, C42:C2, C4xD4, C22wrC2, C22:Q8, C22.D4, C4.4D4, C42:2C2, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C6.D4, C4xC12, C3xC22:C4, S3xC2xC4, C22xDic3, C2xC3:D4, C6xD4, C6xQ8, S3xC23, C22.45C24, C42:2S3, C23.8D6, S3xC22:C4, Dic3:4D4, C23.9D6, C23.23D6, C23:2D6, D6:3Q8, C3xC4.4D4, C42:23D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C24, C22xS3, C2xC4oD4, 2+ 1+4, S3xC23, C22.45C24, D4:6D6, S3xC4oD4, C42:23D6
(1 31 7 21)(2 35 8 19)(3 33 9 23)(4 34 10 24)(5 32 11 22)(6 36 12 20)(13 38 16 43)(14 47 17 42)(15 40 18 45)(25 46 28 41)(26 39 29 44)(27 48 30 37)
(1 25 4 13)(2 29 5 17)(3 27 6 15)(7 28 10 16)(8 26 11 14)(9 30 12 18)(19 39 22 47)(20 45 23 37)(21 41 24 43)(31 46 34 38)(32 42 35 44)(33 48 36 40)
(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 9)(2 8)(3 7)(4 12)(5 11)(6 10)(13 15)(16 18)(19 35)(20 34)(21 33)(22 32)(23 31)(24 36)(25 27)(28 30)(37 41)(38 40)(43 45)(46 48)
G:=sub<Sym(48)| (1,31,7,21)(2,35,8,19)(3,33,9,23)(4,34,10,24)(5,32,11,22)(6,36,12,20)(13,38,16,43)(14,47,17,42)(15,40,18,45)(25,46,28,41)(26,39,29,44)(27,48,30,37), (1,25,4,13)(2,29,5,17)(3,27,6,15)(7,28,10,16)(8,26,11,14)(9,30,12,18)(19,39,22,47)(20,45,23,37)(21,41,24,43)(31,46,34,38)(32,42,35,44)(33,48,36,40), (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), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,15)(16,18)(19,35)(20,34)(21,33)(22,32)(23,31)(24,36)(25,27)(28,30)(37,41)(38,40)(43,45)(46,48)>;
G:=Group( (1,31,7,21)(2,35,8,19)(3,33,9,23)(4,34,10,24)(5,32,11,22)(6,36,12,20)(13,38,16,43)(14,47,17,42)(15,40,18,45)(25,46,28,41)(26,39,29,44)(27,48,30,37), (1,25,4,13)(2,29,5,17)(3,27,6,15)(7,28,10,16)(8,26,11,14)(9,30,12,18)(19,39,22,47)(20,45,23,37)(21,41,24,43)(31,46,34,38)(32,42,35,44)(33,48,36,40), (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), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,15)(16,18)(19,35)(20,34)(21,33)(22,32)(23,31)(24,36)(25,27)(28,30)(37,41)(38,40)(43,45)(46,48) );
G=PermutationGroup([[(1,31,7,21),(2,35,8,19),(3,33,9,23),(4,34,10,24),(5,32,11,22),(6,36,12,20),(13,38,16,43),(14,47,17,42),(15,40,18,45),(25,46,28,41),(26,39,29,44),(27,48,30,37)], [(1,25,4,13),(2,29,5,17),(3,27,6,15),(7,28,10,16),(8,26,11,14),(9,30,12,18),(19,39,22,47),(20,45,23,37),(21,41,24,43),(31,46,34,38),(32,42,35,44),(33,48,36,40)], [(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)], [(1,9),(2,8),(3,7),(4,12),(5,11),(6,10),(13,15),(16,18),(19,35),(20,34),(21,33),(22,32),(23,31),(24,36),(25,27),(28,30),(37,41),(38,40),(43,45),(46,48)]])
39 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
order | 1 | 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 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
39 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4oD4 | 2+ 1+4 | D4:6D6 | S3xC4oD4 |
kernel | C42:23D6 | C42:2S3 | C23.8D6 | S3xC22:C4 | Dic3:4D4 | C23.9D6 | C23.23D6 | C23:2D6 | D6:3Q8 | C3xC4.4D4 | C4.4D4 | C42 | C22:C4 | C2xD4 | C2xQ8 | D6 | C6 | C2 | C2 |
# reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 8 | 1 | 2 | 4 |
Matrix representation of C42:23D6 ►in GL6(F13)
0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 5 | 0 |
0 | 0 | 0 | 0 | 0 | 5 |
8 | 0 | 0 | 0 | 0 | 0 |
0 | 8 | 0 | 0 | 0 | 0 |
0 | 0 | 12 | 0 | 0 | 0 |
0 | 0 | 0 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 11 |
0 | 0 | 0 | 0 | 0 | 12 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 12 | 0 | 0 |
0 | 0 | 1 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 | 12 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 12 | 0 | 0 |
0 | 0 | 0 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 12 | 0 |
0 | 0 | 0 | 0 | 12 | 1 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,11,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;
C42:23D6 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{23}D_6
% in TeX
G:=Group("C4^2:23D6");
// GroupNames label
G:=SmallGroup(192,1238);
// by ID
G=gap.SmallGroup(192,1238);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,387,100,346,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations