metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).25D6, C8.10(C4×S3), (C2×C8).68D6, C8.C4⋊3S3, C8⋊S3.1C4, C24.28(C2×C4), D6.7(C4⋊C4), (C4×S3).50D4, C4.211(S3×D4), C24.C4⋊7C2, C12.370(C2×D4), C22.4(S3×Q8), (C2×Dic3).4Q8, (C22×S3).3Q8, Dic3.9(C4⋊C4), C12.53(C22×C4), (C2×C24).40C22, (S3×M4(2)).3C2, C12.53D4⋊12C2, C3⋊1(M4(2).C4), (C2×C12).309C23, C4.Dic3.13C22, (C3×M4(2)).27C22, C3⋊C8.3(C2×C4), C4.83(S3×C2×C4), C6.17(C2×C4⋊C4), C2.18(S3×C4⋊C4), (C2×C6).2(C2×Q8), (C4×S3).8(C2×C4), (C3×C8.C4)⋊3C2, (C2×C8⋊S3).1C2, (C2×C3⋊C8).77C22, (S3×C2×C4).43C22, (C2×C4).412(C22×S3), SmallGroup(192,452)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).25D6
G = < a,b,c,d | a8=b2=1, c6=d2=a6b, bab=a5, cac-1=a-1b, dad-1=a3b, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 224 in 102 conjugacy classes, 51 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, C12, D6, D6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C3⋊C8, C3⋊C8, C24, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C8.C4, C8.C4, C2×M4(2), S3×C8, C8⋊S3, C8⋊S3, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), S3×C2×C4, M4(2).C4, C24.C4, C12.53D4, C3×C8.C4, C2×C8⋊S3, S3×M4(2), M4(2).25D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4×S3, C22×S3, C2×C4⋊C4, S3×C2×C4, S3×D4, S3×Q8, M4(2).C4, S3×C4⋊C4, M4(2).25D6
►On 48 points
(1 41 19 47 13 29 7 35)(2 48 20 30 14 36 8 42)(3 31 21 37 15 43 9 25)(4 38 22 44 16 26 10 32)(5 45 23 27 17 33 11 39)(6 28 24 34 18 40 12 46)
(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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 18 7 24 13 6 19 12)(2 23 8 5 14 11 20 17)(3 4 9 10 15 16 21 22)(25 38 31 44 37 26 43 32)(27 48 33 30 39 36 45 42)(28 29 34 35 40 41 46 47)
G:=sub<Sym(48)| (1,41,19,47,13,29,7,35)(2,48,20,30,14,36,8,42)(3,31,21,37,15,43,9,25)(4,38,22,44,16,26,10,32)(5,45,23,27,17,33,11,39)(6,28,24,34,18,40,12,46), (25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,18,7,24,13,6,19,12)(2,23,8,5,14,11,20,17)(3,4,9,10,15,16,21,22)(25,38,31,44,37,26,43,32)(27,48,33,30,39,36,45,42)(28,29,34,35,40,41,46,47)>;
G:=Group( (1,41,19,47,13,29,7,35)(2,48,20,30,14,36,8,42)(3,31,21,37,15,43,9,25)(4,38,22,44,16,26,10,32)(5,45,23,27,17,33,11,39)(6,28,24,34,18,40,12,46), (25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,18,7,24,13,6,19,12)(2,23,8,5,14,11,20,17)(3,4,9,10,15,16,21,22)(25,38,31,44,37,26,43,32)(27,48,33,30,39,36,45,42)(28,29,34,35,40,41,46,47) );
G=PermutationGroup([[(1,41,19,47,13,29,7,35),(2,48,20,30,14,36,8,42),(3,31,21,37,15,43,9,25),(4,38,22,44,16,26,10,32),(5,45,23,27,17,33,11,39),(6,28,24,34,18,40,12,46)], [(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(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)], [(1,18,7,24,13,6,19,12),(2,23,8,5,14,11,20,17),(3,4,9,10,15,16,21,22),(25,38,31,44,37,26,43,32),(27,48,33,30,39,36,45,42),(28,29,34,35,40,41,46,47)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 8A | ··· | 8F | 8G | ··· | 8L | 12A | 12B | 12C | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 6 | 6 | 2 | 1 | 1 | 2 | 6 | 6 | 2 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | Q8 | D6 | D6 | C4×S3 | S3×D4 | S3×Q8 | M4(2).C4 | M4(2).25D6 |
| kernel | M4(2).25D6 | C24.C4 | C12.53D4 | C3×C8.C4 | C2×C8⋊S3 | S3×M4(2) | C8⋊S3 | C8.C4 | C4×S3 | C2×Dic3 | C22×S3 | C2×C8 | M4(2) | C8 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 8 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 |
Matrix representation of M4(2).25D6 ►in GL4(𝔽5) generated by
| 0 | 0 | 3 | 3 |
| 0 | 0 | 1 | 2 |
| 2 | 2 | 0 | 0 |
| 4 | 3 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 4 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 3 | 2 |
| 0 | 0 | 2 | 4 |
| 1 | 4 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 1 | 4 |
G:=sub<GL(4,GF(5))| [0,0,2,4,0,0,2,3,3,1,0,0,3,2,0,0],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[2,4,0,0,4,4,0,0,0,0,3,2,0,0,2,4],[1,4,0,0,4,4,0,0,0,0,1,1,0,0,2,4] >;
M4(2).25D6 in GAP, Magma, Sage, TeX
M_4(2)._{25}D_6 % in TeX
G:=Group("M4(2).25D6"); // GroupNames label
G:=SmallGroup(192,452);
// by ID
G=gap.SmallGroup(192,452);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,219,58,136,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^6=d^2=a^6*b,b*a*b=a^5,c*a*c^-1=a^-1*b,d*a*d^-1=a^3*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations