metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24:7D6, C6.252+ 1+4, (C2xD4):5D6, C22:C4:5D6, D6.2(C2xD4), C22wrC2:3S3, C23:2D6:3C2, (C6xD4):6C22, (C22xS3):6D4, Dic3:D4:12C2, D6:3D4:12C2, C24:4S3:6C2, D6:C4:10C22, C3:2(C23:3D4), (C23xC6):9C22, C22.40(S3xD4), C6.55(C22xD4), C23.9D6:12C2, (C2xD12):18C22, (C2xC6).133C24, (C2xC12).27C23, Dic3:C4:8C22, (S3xC23):6C22, C4:Dic3:25C22, (C22xC6).8C23, C2.27(D4:6D6), C23.21D6:9C2, C23.23D6:4C2, C6.D4:14C22, C23.117(C22xS3), C22.154(S3xC23), (C2xDic3).60C23, (C22xS3).182C23, (C22xDic3):12C22, (C2xS3xD4):6C2, C2.28(C2xS3xD4), (S3xC2xC4):6C22, (S3xC22:C4):2C2, (C2xC6).53(C2xD4), (C3xC22wrC2):4C2, (C22xC3:D4):7C2, (C2xC3:D4):38C22, (C3xC22:C4):4C22, (C2xC4).27(C22xS3), SmallGroup(192,1148)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24:7D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
Subgroups: 1072 in 346 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, C2xC4, D4, C23, C23, C23, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C22:C4, C22:C4, C22:C4, C4:C4, C22xC4, C2xD4, C2xD4, C2xD4, C24, C24, C4xS3, D12, C2xDic3, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C22xS3, C22xS3, C22xS3, C22xC6, C22xC6, C22xC6, C2xC22:C4, C22wrC2, C22wrC2, C4:D4, C22.D4, C22xD4, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C6.D4, C3xC22:C4, C3xC22:C4, S3xC2xC4, S3xC2xC4, C2xD12, S3xD4, C22xDic3, C2xC3:D4, C2xC3:D4, C2xC3:D4, C6xD4, C6xD4, S3xC23, C23xC6, C23:3D4, S3xC22:C4, C23.9D6, Dic3:D4, C23.21D6, C23.23D6, C23:2D6, D6:3D4, C24:4S3, C3xC22wrC2, C2xS3xD4, C22xC3:D4, C24:7D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C24, C22xS3, C22xD4, 2+ 1+4, S3xD4, S3xC23, C23:3D4, C2xS3xD4, D4:6D6, C24:7D6
►On 48 points
(1 37)(2 24)(3 39)(4 20)(5 41)(6 22)(7 27)(8 44)(9 29)(10 46)(11 25)(12 48)(13 43)(14 28)(15 45)(16 30)(17 47)(18 26)(19 35)(21 31)(23 33)(32 42)(34 38)(36 40)
(1 13)(2 44)(3 15)(4 46)(5 17)(6 48)(7 33)(8 24)(9 35)(10 20)(11 31)(12 22)(14 38)(16 40)(18 42)(19 29)(21 25)(23 27)(26 32)(28 34)(30 36)(37 43)(39 45)(41 47)
(1 33)(2 34)(3 35)(4 36)(5 31)(6 32)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(19 39)(20 40)(21 41)(22 42)(23 37)(24 38)(25 47)(26 48)(27 43)(28 44)(29 45)(30 46)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 35)(20 36)(21 31)(22 32)(23 33)(24 34)
(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 18)(8 17)(9 16)(10 15)(11 14)(12 13)(19 20)(21 24)(22 23)(25 44)(26 43)(27 48)(28 47)(29 46)(30 45)(31 34)(32 33)(35 36)(37 42)(38 41)(39 40)
G:=sub<Sym(48)| (1,37)(2,24)(3,39)(4,20)(5,41)(6,22)(7,27)(8,44)(9,29)(10,46)(11,25)(12,48)(13,43)(14,28)(15,45)(16,30)(17,47)(18,26)(19,35)(21,31)(23,33)(32,42)(34,38)(36,40), (1,13)(2,44)(3,15)(4,46)(5,17)(6,48)(7,33)(8,24)(9,35)(10,20)(11,31)(12,22)(14,38)(16,40)(18,42)(19,29)(21,25)(23,27)(26,32)(28,34)(30,36)(37,43)(39,45)(41,47), (1,33)(2,34)(3,35)(4,36)(5,31)(6,32)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(19,39)(20,40)(21,41)(22,42)(23,37)(24,38)(25,47)(26,48)(27,43)(28,44)(29,45)(30,46), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,35)(20,36)(21,31)(22,32)(23,33)(24,34), (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)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(19,20)(21,24)(22,23)(25,44)(26,43)(27,48)(28,47)(29,46)(30,45)(31,34)(32,33)(35,36)(37,42)(38,41)(39,40)>;
G:=Group( (1,37)(2,24)(3,39)(4,20)(5,41)(6,22)(7,27)(8,44)(9,29)(10,46)(11,25)(12,48)(13,43)(14,28)(15,45)(16,30)(17,47)(18,26)(19,35)(21,31)(23,33)(32,42)(34,38)(36,40), (1,13)(2,44)(3,15)(4,46)(5,17)(6,48)(7,33)(8,24)(9,35)(10,20)(11,31)(12,22)(14,38)(16,40)(18,42)(19,29)(21,25)(23,27)(26,32)(28,34)(30,36)(37,43)(39,45)(41,47), (1,33)(2,34)(3,35)(4,36)(5,31)(6,32)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(19,39)(20,40)(21,41)(22,42)(23,37)(24,38)(25,47)(26,48)(27,43)(28,44)(29,45)(30,46), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,35)(20,36)(21,31)(22,32)(23,33)(24,34), (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)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(19,20)(21,24)(22,23)(25,44)(26,43)(27,48)(28,47)(29,46)(30,45)(31,34)(32,33)(35,36)(37,42)(38,41)(39,40) );
G=PermutationGroup([[(1,37),(2,24),(3,39),(4,20),(5,41),(6,22),(7,27),(8,44),(9,29),(10,46),(11,25),(12,48),(13,43),(14,28),(15,45),(16,30),(17,47),(18,26),(19,35),(21,31),(23,33),(32,42),(34,38),(36,40)], [(1,13),(2,44),(3,15),(4,46),(5,17),(6,48),(7,33),(8,24),(9,35),(10,20),(11,31),(12,22),(14,38),(16,40),(18,42),(19,29),(21,25),(23,27),(26,32),(28,34),(30,36),(37,43),(39,45),(41,47)], [(1,33),(2,34),(3,35),(4,36),(5,31),(6,32),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(19,39),(20,40),(21,41),(22,42),(23,37),(24,38),(25,47),(26,48),(27,43),(28,44),(29,45),(30,46)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,27),(8,28),(9,29),(10,30),(11,25),(12,26),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,35),(20,36),(21,31),(22,32),(23,33),(24,34)], [(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),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13),(19,20),(21,24),(22,23),(25,44),(26,43),(27,48),(28,47),(29,46),(30,45),(31,34),(32,33),(35,36),(37,42),(38,41),(39,40)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3 | 4A | 4B | 4C | 4D | ··· | 4H | 6A | 6B | 6C | 6D | ··· | 6I | 6J | 12A | 12B | 12C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | 2+ 1+4 | S3xD4 | D4:6D6 |
| kernel | C24:7D6 | S3xC22:C4 | C23.9D6 | Dic3:D4 | C23.21D6 | C23.23D6 | C23:2D6 | D6:3D4 | C24:4S3 | C3xC22wrC2 | C2xS3xD4 | C22xC3:D4 | C22wrC2 | C22xS3 | C22:C4 | C2xD4 | C24 | C6 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 1 | 2 | 2 | 4 |
Matrix representation of C24:7D6 ►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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 6 | 0 | 0 | 0 | 0 |
| 9 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 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 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,9,0,0,0,0,6,5,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,0,0,2,9,0,0,0,0,4,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,12,1,0,0,0,0,12,0,0,0],[12,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,12,0,0,0,0,0,12,1,0,0] >;
C24:7D6 in GAP, Magma, Sage, TeX
C_2^4\rtimes_7D_6
% in TeX
G:=Group("C2^4:7D6"); // GroupNames label
G:=SmallGroup(192,1148);
// by ID
G=gap.SmallGroup(192,1148);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,675,297,6278]);
// 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,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f=b*c*d,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