metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2).31D6, C4.66(C2×D12), (C2×C4).50D12, C4.56(D6⋊C4), C12.418(C2×D4), (C2×C12).174D4, C23.26(C4×S3), (C6×M4(2))⋊20C2, (C2×M4(2))⋊12S3, C22.7(D6⋊C4), (C22×C4).156D6, C12.46D4⋊15C2, C12.47D4⋊15C2, C12.67(C22⋊C4), (C2×C12).417C23, (C2×D12).251C22, C4.Dic3.42C22, (C22×C12).189C22, (C2×Dic6).278C22, (C3×M4(2)).34C22, C3⋊3(M4(2).8C22), (S3×C2×C4).4C4, C2.31(C2×D6⋊C4), C22.21(S3×C2×C4), (C2×C4).160(C4×S3), (C2×C3⋊D4).13C4, C4.111(C2×C3⋊D4), C6.59(C2×C22⋊C4), (C2×C12).109(C2×C4), (C2×C4○D12).12C2, (C2×C4).77(C3⋊D4), (C22×S3).6(C2×C4), (C2×C4.Dic3)⋊16C2, (C2×C6).15(C22×C4), (C22×C6).71(C2×C4), (C2×Dic3).5(C2×C4), (C2×C6).66(C22⋊C4), (C2×C4).121(C22×S3), SmallGroup(192,691)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2).31D6
G = < a,b,c,d | a8=b2=1, c6=d2=a4, bab=a5, ac=ca, dad-1=ab, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 408 in 150 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4.D4, C4.10D4, C2×M4(2), C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3, C4.Dic3, C2×C24, C3×M4(2), C3×M4(2), C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, M4(2).8C22, C12.46D4, C12.47D4, C2×C4.Dic3, C6×M4(2), C2×C4○D12, M4(2).31D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, M4(2).8C22, C2×D6⋊C4, M4(2).31D6
►On 48 points
(1 39 29 16 7 45 35 22)(2 40 30 17 8 46 36 23)(3 41 31 18 9 47 25 24)(4 42 32 19 10 48 26 13)(5 43 33 20 11 37 27 14)(6 44 34 21 12 38 28 15)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 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 6 7 12)(2 11 8 5)(3 4 9 10)(13 18 19 24)(14 23 20 17)(15 16 21 22)(25 32 31 26)(27 30 33 36)(28 35 34 29)(37 40 43 46)(38 45 44 39)(41 48 47 42)
G:=sub<Sym(48)| (1,39,29,16,7,45,35,22)(2,40,30,17,8,46,36,23)(3,41,31,18,9,47,25,24)(4,42,32,19,10,48,26,13)(5,43,33,20,11,37,27,14)(6,44,34,21,12,38,28,15), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,18,19,24)(14,23,20,17)(15,16,21,22)(25,32,31,26)(27,30,33,36)(28,35,34,29)(37,40,43,46)(38,45,44,39)(41,48,47,42)>;
G:=Group( (1,39,29,16,7,45,35,22)(2,40,30,17,8,46,36,23)(3,41,31,18,9,47,25,24)(4,42,32,19,10,48,26,13)(5,43,33,20,11,37,27,14)(6,44,34,21,12,38,28,15), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,18,19,24)(14,23,20,17)(15,16,21,22)(25,32,31,26)(27,30,33,36)(28,35,34,29)(37,40,43,46)(38,45,44,39)(41,48,47,42) );
G=PermutationGroup([[(1,39,29,16,7,45,35,22),(2,40,30,17,8,46,36,23),(3,41,31,18,9,47,25,24),(4,42,32,19,10,48,26,13),(5,43,33,20,11,37,27,14),(6,44,34,21,12,38,28,15)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,18,19,24),(14,23,20,17),(15,16,21,22),(25,32,31,26),(27,30,33,36),(28,35,34,29),(37,40,43,46),(38,45,44,39),(41,48,47,42)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 2 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | C4×S3 | M4(2).8C22 | M4(2).31D6 |
| kernel | M4(2).31D6 | C12.46D4 | C12.47D4 | C2×C4.Dic3 | C6×M4(2) | C2×C4○D12 | S3×C2×C4 | C2×C3⋊D4 | C2×M4(2) | C2×C12 | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 1 | 4 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 4 |
Matrix representation of M4(2).31D6 ►in GL4(𝔽73) generated by
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 43 | 60 | 0 | 0 |
| 13 | 30 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 27 | 0 | 0 |
| 46 | 27 | 0 | 0 |
| 0 | 0 | 0 | 27 |
| 0 | 0 | 46 | 27 |
| 27 | 0 | 0 | 0 |
| 27 | 46 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 27 | 46 |
G:=sub<GL(4,GF(73))| [0,0,43,13,0,0,60,30,46,0,0,0,0,46,0,0],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[0,46,0,0,27,27,0,0,0,0,0,46,0,0,27,27],[27,27,0,0,0,46,0,0,0,0,27,27,0,0,0,46] >;
M4(2).31D6 in GAP, Magma, Sage, TeX
M_4(2)._{31}D_6 % in TeX
G:=Group("M4(2).31D6"); // GroupNames label
G:=SmallGroup(192,691);
// by ID
G=gap.SmallGroup(192,691);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,422,58,1123,136,438,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^6=d^2=a^4,b*a*b=a^5,a*c=c*a,d*a*d^-1=a*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations