metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.38D6, C6.32+ 1+4, C3:D4:8D4, Dic3:D4:1C2, D6:D4:2C2, C3:1(D4:5D4), C22:C4:40D6, D6.12(C2xD4), (C22xC4):11D6, C12:7D4:17C2, C24:4S3:2C2, D6:C4:46C22, C23.9D6:1C2, (C2xD12):2C22, (C2xC6).34C24, C4:Dic3:4C22, C6.37(C22xD4), C22.18(S3xD4), C2.7(D4:6D6), C22:5(C4oD12), Dic3.14(C2xD4), Dic3:4D4:40C2, (C2xC12).128C23, Dic3:C4:49C22, (C22xC12):14C22, Dic3.D4:2C2, C23.11D6:1C2, (C4xDic3):47C22, (C2xDic6):48C22, C23.28D6:9C2, C6.D4:8C22, (C23xC6).60C22, C22.73(S3xC23), (S3xC23).31C22, (C22xC6).387C23, C23.231(C22xS3), (C22xS3).152C23, (C2xDic3).180C23, (C22xDic3).78C22, C2.11(C2xS3xD4), (C4xC3:D4):1C2, (C2xC4oD12):3C2, (C2xC6):8(C4oD4), (S3xC2xC4):40C22, C6.14(C2xC4oD4), (C6xC22:C4):18C2, (C2xC22:C4):13S3, (S3xC22:C4):24C2, C2.16(C2xC4oD12), (C2xC6).383(C2xD4), (C22xC3:D4):5C2, (C2xC3:D4):1C22, (C3xC22:C4):53C22, (C2xC4).259(C22xS3), SmallGroup(192,1049)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.38D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
Subgroups: 952 in 334 conjugacy classes, 107 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C24, C24, Dic6, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C3:D4, C2xC12, C2xC12, C22xS3, C22xS3, C22xC6, C22xC6, C2xC22:C4, C2xC22:C4, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C4.4D4, C22xD4, C2xC4oD4, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C3xC22:C4, C2xDic6, S3xC2xC4, C2xD12, C4oD12, C22xDic3, C2xC3:D4, C2xC3:D4, C22xC12, S3xC23, C23xC6, D4:5D4, Dic3.D4, S3xC22:C4, Dic3:4D4, D6:D4, C23.9D6, Dic3:D4, C23.11D6, C4xC3:D4, C23.28D6, C12:7D4, C24:4S3, C6xC22:C4, C2xC4oD12, C22xC3:D4, C24.38D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C24, C22xS3, C22xD4, C2xC4oD4, 2+ 1+4, C4oD12, S3xD4, S3xC23, D4:5D4, C2xC4oD12, C2xS3xD4, D4:6D6, C24.38D6
►On 48 points
(1 7)(2 38)(3 9)(4 40)(5 11)(6 42)(8 44)(10 46)(12 48)(13 19)(14 34)(15 21)(16 36)(17 23)(18 26)(20 28)(22 30)(24 32)(25 31)(27 33)(29 35)(37 43)(39 45)(41 47)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 37)(34 38)(35 39)(36 40)
(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 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 25)(24 26)
(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 7 12)(2 11 8 5)(3 4 9 10)(13 24 19 18)(14 17 20 23)(15 22 21 16)(25 28 31 34)(26 33 32 27)(29 36 35 30)(37 42 43 48)(38 47 44 41)(39 40 45 46)
G:=sub<Sym(48)| (1,7)(2,38)(3,9)(4,40)(5,11)(6,42)(8,44)(10,46)(12,48)(13,19)(14,34)(15,21)(16,36)(17,23)(18,26)(20,28)(22,30)(24,32)(25,31)(27,33)(29,35)(37,43)(39,45)(41,47), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,25)(24,26), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,28,31,34)(26,33,32,27)(29,36,35,30)(37,42,43,48)(38,47,44,41)(39,40,45,46)>;
G:=Group( (1,7)(2,38)(3,9)(4,40)(5,11)(6,42)(8,44)(10,46)(12,48)(13,19)(14,34)(15,21)(16,36)(17,23)(18,26)(20,28)(22,30)(24,32)(25,31)(27,33)(29,35)(37,43)(39,45)(41,47), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,25)(24,26), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,28,31,34)(26,33,32,27)(29,36,35,30)(37,42,43,48)(38,47,44,41)(39,40,45,46) );
G=PermutationGroup([[(1,7),(2,38),(3,9),(4,40),(5,11),(6,42),(8,44),(10,46),(12,48),(13,19),(14,34),(15,21),(16,36),(17,23),(18,26),(20,28),(22,30),(24,32),(25,31),(27,33),(29,35),(37,43),(39,45),(41,47)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,37),(34,38),(35,39),(36,40)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,25),(24,26)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,24,19,18),(14,17,20,23),(15,22,21,16),(25,28,31,34),(26,33,32,27),(29,36,35,30),(37,42,43,48),(38,47,44,41),(39,40,45,46)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4oD4 | C4oD12 | 2+ 1+4 | S3xD4 | D4:6D6 |
| kernel | C24.38D6 | Dic3.D4 | S3xC22:C4 | Dic3:4D4 | D6:D4 | C23.9D6 | Dic3:D4 | C23.11D6 | C4xC3:D4 | C23.28D6 | C12:7D4 | C24:4S3 | C6xC22:C4 | C2xC4oD12 | C22xC3:D4 | C2xC22:C4 | C3:D4 | C22:C4 | C22xC4 | C24 | C2xC6 | C22 | C6 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 1 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C24.38D6 ►in GL4(F13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 5 | 12 |
| 2 | 9 | 0 | 0 |
| 4 | 11 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 5 | 0 | 0 |
| 8 | 5 | 0 | 0 |
| 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 |
| 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,5,0,0,0,12],[2,4,0,0,9,11,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[0,8,0,0,5,5,0,0,0,0,12,0,0,0,3,1],[5,5,0,0,0,8,0,0,0,0,12,0,0,0,3,1] >;
C24.38D6 in GAP, Magma, Sage, TeX
C_2^4._{38}D_6 % in TeX
G:=Group("C2^4.38D6"); // GroupNames label
G:=SmallGroup(192,1049);
// by ID
G=gap.SmallGroup(192,1049);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,100,675,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
// generators/relations