p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).41D4, C4wrC2:3C4, (C2xC8).25D4, (C2xD4).7Q8, (C2xQ8).4Q8, D4.6(C4:C4), C4oD4.43D4, C4.142(C4xD4), Q8.6(C4:C4), C22.32(C4xD4), C42:6C4:18C2, C4.118C22wrC2, C42.146(C2xC4), Q8oM4(2).2C2, M4(2).3(C2xC4), C4.97(C22:Q8), M4(2).C4:2C2, M4(2):4C4:1C2, C4:M4(2):25C2, C22.2(C22:Q8), C23.117(C4oD4), (C2xC42).267C22, (C22xC4).677C23, C42:C22.3C2, C42:C2.12C22, C2.17(C23.8Q8), (C2xM4(2)).175C22, C22.3(C22.D4), C4.11(C2xC4:C4), (C2xC4wrC2).3C2, C4oD4.6(C2xC4), (C2xC4).11(C2xQ8), (C2xC4).987(C2xD4), (C2xC4).52(C4oD4), (C2xC4).183(C22xC4), (C2xC4oD4).13C22, SmallGroup(128,593)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).41D4
G = < a,b,c,d | a8=b2=c4=1, d2=a6, bab=a5, cac-1=dad-1=a-1b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >
Subgroups: 236 in 131 conjugacy classes, 54 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C4:C4, C2xC8, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C4wrC2, C4wrC2, C4:C8, C8.C4, C2xC42, C42:C2, C2xM4(2), C2xM4(2), C8oD4, C2xC4oD4, C42:6C4, M4(2):4C4, C2xC4wrC2, C42:C22, C4:M4(2), M4(2).C4, Q8oM4(2), M4(2).41D4
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C23, C4:C4, C22xC4, C2xD4, C2xQ8, C4oD4, C2xC4:C4, C4xD4, C22wrC2, C22:Q8, C22.D4, C23.8Q8, M4(2).41D4
►On 16 points - transitive group 16T215
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(9 13)(11 15)
(1 4 5 8)(2 3 6 7)(9 12)(10 15)(11 14)(13 16)
(1 9 7 15 5 13 3 11)(2 16 8 14 6 12 4 10)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,13)(11,15), (1,4,5,8)(2,3,6,7)(9,12)(10,15)(11,14)(13,16), (1,9,7,15,5,13,3,11)(2,16,8,14,6,12,4,10)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(9,13)(11,15), (1,4,5,8)(2,3,6,7)(9,12)(10,15)(11,14)(13,16), (1,9,7,15,5,13,3,11)(2,16,8,14,6,12,4,10) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(9,13),(11,15)], [(1,4,5,8),(2,3,6,7),(9,12),(10,15),(11,14),(13,16)], [(1,9,7,15,5,13,3,11),(2,16,8,14,6,12,4,10)]])
G:=TransitiveGroup(16,215);
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4K | 4L | 4M | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | Q8 | D4 | C4oD4 | C4oD4 | M4(2).41D4 |
| kernel | M4(2).41D4 | C42:6C4 | M4(2):4C4 | C2xC4wrC2 | C42:C22 | C4:M4(2) | M4(2).C4 | Q8oM4(2) | C4wrC2 | C2xC8 | M4(2) | C2xD4 | C2xQ8 | C4oD4 | C2xC4 | C23 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of M4(2).41D4 ►in GL4(F5) generated by
| 0 | 0 | 0 | 4 |
| 0 | 0 | 3 | 0 |
| 0 | 4 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 2 | 0 | 0 | 3 |
| 0 | 2 | 4 | 0 |
| 0 | 3 | 3 | 0 |
| 4 | 0 | 0 | 3 |
| 3 | 0 | 0 | 2 |
| 0 | 4 | 3 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 4 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 3 | 0 | 0 | 1 |
| 0 | 3 | 3 | 0 |
G:=sub<GL(4,GF(5))| [0,0,0,3,0,0,4,0,0,3,0,0,4,0,0,0],[2,0,0,4,0,2,3,0,0,4,3,0,3,0,0,3],[3,0,0,0,0,4,0,0,0,3,1,0,2,0,0,2],[0,2,3,0,4,0,0,3,0,0,0,3,0,0,1,0] >;
M4(2).41D4 in GAP, Magma, Sage, TeX
M_4(2)._{41}D_4 % in TeX
G:=Group("M4(2).41D4"); // GroupNames label
G:=SmallGroup(128,593);
// by ID
G=gap.SmallGroup(128,593);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,248,2804,718,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^6,b*a*b=a^5,c*a*c^-1=d*a*d^-1=a^-1*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;
// generators/relations