metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.76C24, C4oD4.61D6, (C6xD4).13C4, C3:C8.34C23, (C6xQ8).13C4, D4.Dic3:9C2, C3:3(Q8oM4(2)), C6.49(C23xC4), C4.75(S3xC23), C4oD4.5Dic3, D4.9(C2xDic3), C12.98(C22xC4), (C2xQ8).11Dic3, Q8.15(C2xDic3), (C2xD4).10Dic3, (C22xC4).296D6, (C2xC12).554C23, C4.Dic3:35C22, C2.11(C23xDic3), C23.16(C2xDic3), C4.20(C22xDic3), C22.2(C22xDic3), (C22xC12).289C22, (C2xC3:C8):21C22, (C3xC4oD4).4C4, (C2xC4oD4).15S3, (C6xC4oD4).10C2, (C3xD4).26(C2xC4), (C3xQ8).28(C2xC4), (C2xC12).136(C2xC4), (C2xC4.Dic3):29C2, (C22xC6).81(C2xC4), (C2xC6).29(C22xC4), (C2xC4).31(C2xDic3), (C2xC4).635(C22xS3), (C3xC4oD4).49C22, SmallGroup(192,1378)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.76C24
G = < a,b,c,d,e | a12=c2=d2=e2=1, b2=a9, bab-1=a5, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=a6b, dcd=a6c, ce=ec, de=ed >
Subgroups: 392 in 258 conjugacy classes, 187 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, C12, C12, C2xC6, C2xC6, C2xC6, C2xC8, M4(2), C22xC4, C2xD4, C2xQ8, C4oD4, C3:C8, C2xC12, C2xC12, C3xD4, C3xQ8, C22xC6, C2xM4(2), C8oD4, C2xC4oD4, C2xC3:C8, C4.Dic3, C22xC12, C6xD4, C6xQ8, C3xC4oD4, Q8oM4(2), C2xC4.Dic3, D4.Dic3, C6xC4oD4, C12.76C24
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, Dic3, D6, C22xC4, C24, C2xDic3, C22xS3, C23xC4, C22xDic3, S3xC23, Q8oM4(2), C23xDic3, C12.76C24
►On 48 points
(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 28 10 25 7 34 4 31)(2 33 11 30 8 27 5 36)(3 26 12 35 9 32 6 29)(13 46 22 43 19 40 16 37)(14 39 23 48 20 45 17 42)(15 44 24 41 21 38 18 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 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
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), (1,28,10,25,7,34,4,31)(2,33,11,30,8,27,5,36)(3,26,12,35,9,32,6,29)(13,46,22,43,19,40,16,37)(14,39,23,48,20,45,17,42)(15,44,24,41,21,38,18,47), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,37)(35,38)(36,39), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)>;
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), (1,28,10,25,7,34,4,31)(2,33,11,30,8,27,5,36)(3,26,12,35,9,32,6,29)(13,46,22,43,19,40,16,37)(14,39,23,48,20,45,17,42)(15,44,24,41,21,38,18,47), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,37)(35,38)(36,39), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48) );
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)], [(1,28,10,25,7,34,4,31),(2,33,11,30,8,27,5,36),(3,26,12,35,9,32,6,29),(13,46,22,43,19,40,16,37),(14,39,23,48,20,45,17,42),(15,44,24,41,21,38,18,47)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,37),(32,38),(33,39),(34,40),(35,41),(36,42)], [(1,22),(2,23),(3,24),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(25,40),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,37),(35,38),(36,39)], [(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)]])
54 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2H | 3 | 4A | 4B | 4C | ··· | 4I | 6A | 6B | 6C | 6D | ··· | 6I | 8A | ··· | 8P | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | ··· | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | Dic3 | Dic3 | Dic3 | D6 | Q8oM4(2) | C12.76C24 |
| kernel | C12.76C24 | C2xC4.Dic3 | D4.Dic3 | C6xC4oD4 | C6xD4 | C6xQ8 | C3xC4oD4 | C2xC4oD4 | C22xC4 | C2xD4 | C2xQ8 | C4oD4 | C4oD4 | C3 | C1 |
| # reps | 1 | 6 | 8 | 1 | 6 | 2 | 8 | 1 | 3 | 3 | 1 | 4 | 4 | 2 | 4 |
Matrix representation of C12.76C24 ►in GL4(F73) generated by
| 24 | 0 | 0 | 0 |
| 0 | 24 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 46 | 43 | 0 | 0 |
| 34 | 27 | 0 | 0 |
| 0 | 0 | 46 | 43 |
| 0 | 0 | 34 | 27 |
| 72 | 0 | 0 | 0 |
| 31 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 31 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [24,0,0,0,0,24,0,0,0,0,3,0,0,0,0,3],[0,0,46,0,0,0,0,46,1,0,0,0,0,1,0,0],[46,34,0,0,43,27,0,0,0,0,46,34,0,0,43,27],[72,31,0,0,0,1,0,0,0,0,72,31,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72] >;
C12.76C24 in GAP, Magma, Sage, TeX
C_{12}._{76}C_2^4 % in TeX
G:=Group("C12.76C2^4"); // GroupNames label
G:=SmallGroup(192,1378);
// by ID
G=gap.SmallGroup(192,1378);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,387,1123,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=c^2=d^2=e^2=1,b^2=a^9,b*a*b^-1=a^5,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=a^6*b,d*c*d=a^6*c,c*e=e*c,d*e=e*d>;
// generators/relations