metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:25D6, C6.1372+ 1+4, C4:C4:33D6, (C4xD12):13C2, (C4xC12):7C22, D6:C4:7C22, Dic3:D4:44C2, D6:D4:27C2, C12:D4:36C2, C42:7S3:8C2, C42:2C2:2S3, D6:Q8:40C2, C22:C4.40D6, Dic3:5D4:39C2, D6.12(C4oD4), C23.9D6:48C2, D6.D4:38C2, C2.62(D4oD12), (C2xD12):29C22, (C2xC6).248C24, C4:Dic3:61C22, (C2xC12).193C23, Dic3:C4:27C22, C3:9(C22.32C24), (C4xDic3):38C22, (C2xDic6):11C22, (C22xC6).62C23, C23.64(C22xS3), C23.11D6:44C2, (S3xC23).68C22, C22.269(S3xC23), (C22xS3).111C23, (C2xDic3).264C23, C6.D4.64C22, (S3xC2xC4):27C22, C4:C4:S3:41C2, C2.95(S3xC4oD4), (S3xC22:C4):20C2, (C3xC4:C4):32C22, C6.206(C2xC4oD4), (C3xC42:2C2):3C2, (C2xC4).85(C22xS3), (C2xC3:D4).68C22, (C3xC22:C4).73C22, SmallGroup(192,1263)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:25D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=b-1, dcd=c-1 >
Subgroups: 784 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2xC6, C2xC6, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C2xD4, C2xQ8, C24, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C22xS3, C22xS3, C22xC6, C2xC22:C4, C4xD4, C22wrC2, C4:D4, C22:Q8, C22.D4, C4.4D4, C42:2C2, C42:2C2, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C4xC12, C3xC22:C4, C3xC4:C4, C2xDic6, S3xC2xC4, C2xD12, C2xC3:D4, S3xC23, C22.32C24, C4xD12, C42:7S3, S3xC22:C4, D6:D4, C23.9D6, Dic3:D4, C23.11D6, Dic3:5D4, D6.D4, C12:D4, D6:Q8, C4:C4:S3, C3xC42:2C2, C42:25D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C24, C22xS3, C2xC4oD4, 2+ 1+4, S3xC23, C22.32C24, S3xC4oD4, D4oD12, C42:25D6
►On 48 points
(1 31 4 28)(2 35 5 26)(3 33 6 30)(7 34 10 25)(8 32 11 29)(9 36 12 27)(13 41 16 43)(14 47 17 39)(15 37 18 45)(19 46 22 38)(20 42 23 44)(21 48 24 40)
(1 22 10 13)(2 20 11 17)(3 24 12 15)(4 19 7 16)(5 23 8 14)(6 21 9 18)(25 41 31 38)(26 44 32 47)(27 37 33 40)(28 46 34 43)(29 39 35 42)(30 48 36 45)
(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 21)(14 20)(15 19)(16 24)(17 23)(18 22)(25 33)(26 32)(27 31)(28 36)(29 35)(30 34)(37 41)(38 40)(43 45)(46 48)
G:=sub<Sym(48)| (1,31,4,28)(2,35,5,26)(3,33,6,30)(7,34,10,25)(8,32,11,29)(9,36,12,27)(13,41,16,43)(14,47,17,39)(15,37,18,45)(19,46,22,38)(20,42,23,44)(21,48,24,40), (1,22,10,13)(2,20,11,17)(3,24,12,15)(4,19,7,16)(5,23,8,14)(6,21,9,18)(25,41,31,38)(26,44,32,47)(27,37,33,40)(28,46,34,43)(29,39,35,42)(30,48,36,45), (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)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34)(37,41)(38,40)(43,45)(46,48)>;
G:=Group( (1,31,4,28)(2,35,5,26)(3,33,6,30)(7,34,10,25)(8,32,11,29)(9,36,12,27)(13,41,16,43)(14,47,17,39)(15,37,18,45)(19,46,22,38)(20,42,23,44)(21,48,24,40), (1,22,10,13)(2,20,11,17)(3,24,12,15)(4,19,7,16)(5,23,8,14)(6,21,9,18)(25,41,31,38)(26,44,32,47)(27,37,33,40)(28,46,34,43)(29,39,35,42)(30,48,36,45), (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)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34)(37,41)(38,40)(43,45)(46,48) );
G=PermutationGroup([[(1,31,4,28),(2,35,5,26),(3,33,6,30),(7,34,10,25),(8,32,11,29),(9,36,12,27),(13,41,16,43),(14,47,17,39),(15,37,18,45),(19,46,22,38),(20,42,23,44),(21,48,24,40)], [(1,22,10,13),(2,20,11,17),(3,24,12,15),(4,19,7,16),(5,23,8,14),(6,21,9,18),(25,41,31,38),(26,44,32,47),(27,37,33,40),(28,46,34,43),(29,39,35,42),(30,48,36,45)], [(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),(13,21),(14,20),(15,19),(16,24),(17,23),(18,22),(25,33),(26,32),(27,31),(28,36),(29,35),(30,34),(37,41),(38,40),(43,45),(46,48)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | S3 | D6 | D6 | D6 | C4oD4 | 2+ 1+4 | S3xC4oD4 | D4oD12 |
| kernel | C42:25D6 | C4xD12 | C42:7S3 | S3xC22:C4 | D6:D4 | C23.9D6 | Dic3:D4 | C23.11D6 | Dic3:5D4 | D6.D4 | C12:D4 | D6:Q8 | C4:C4:S3 | C3xC42:2C2 | C42:2C2 | C42 | C22:C4 | C4:C4 | D6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 4 | 2 | 2 | 4 |
Matrix representation of C42:25D6 ►in GL6(F13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 7 | 1 |
| 0 | 0 | 0 | 12 | 12 | 6 |
| 0 | 0 | 10 | 7 | 1 | 0 |
| 0 | 0 | 6 | 3 | 0 | 1 |
| 9 | 3 | 0 | 0 | 0 | 0 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 | 11 | 0 |
| 0 | 0 | 7 | 10 | 0 | 11 |
| 0 | 0 | 12 | 0 | 10 | 7 |
| 0 | 0 | 0 | 12 | 6 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 7 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 10 | 3 | 12 | 12 |
| 0 | 0 | 10 | 7 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 7 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 3 | 12 | 12 |
| 0 | 0 | 6 | 3 | 0 | 1 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,10,6,0,0,0,12,7,3,0,0,7,12,1,0,0,0,1,6,0,1],[9,3,0,0,0,0,3,4,0,0,0,0,0,0,3,7,12,0,0,0,6,10,0,12,0,0,11,0,10,6,0,0,0,11,7,3],[1,7,0,0,0,0,0,12,0,0,0,0,0,0,1,12,10,10,0,0,1,0,3,7,0,0,0,0,12,1,0,0,0,0,12,0],[1,7,0,0,0,0,0,12,0,0,0,0,0,0,12,0,10,6,0,0,12,1,3,3,0,0,0,0,12,0,0,0,0,0,12,1] >;
C42:25D6 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{25}D_6 % in TeX
G:=Group("C4^2:25D6"); // GroupNames label
G:=SmallGroup(192,1263);
// by ID
G=gap.SmallGroup(192,1263);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,675,570,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations