p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).15D4, (C22xC4).Q8, C23.56(C2xQ8), C4.158(C4:D4), C4.6(C42:2C2), (C22xC4).44C23, C22.8(C22:Q8), C4.10C42.4C2, M4(2).C4.2C2, (C22xQ8).77C22, (C2xM4(2)).30C22, C2.11(C23.Q8), (C2xC4).266(C2xD4), (C2xC4).353(C4oD4), (C2xC4.10D4).4C2, SmallGroup(128,802)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).15D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a6b, bab=a5, cac-1=a3, dad-1=a3b, cbc-1=a4b, bd=db, dcd-1=a4c3 >
Subgroups: 176 in 98 conjugacy classes, 40 normal (8 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2xC4, C2xC4, Q8, C23, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C2xQ8, C4.10D4, C8.C4, C2xM4(2), C22xQ8, C4.10C42, C2xC4.10D4, M4(2).C4, M4(2).15D4
Quotients: C1, C2, C22, D4, Q8, C23, C2xD4, C2xQ8, C4oD4, C4:D4, C22:Q8, C42:2C2, C23.Q8, M4(2).15D4
Character table of M4(2).15D4
class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | |
size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
ρ9 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | orthogonal lifted from D4 |
ρ11 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | orthogonal lifted from D4 |
ρ12 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | orthogonal lifted from D4 |
ρ13 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ14 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | orthogonal lifted from D4 |
ρ15 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
ρ16 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
ρ17 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
ρ18 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | complex lifted from C4oD4 |
ρ19 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | complex lifted from C4oD4 |
ρ20 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | complex lifted from C4oD4 |
ρ21 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
ρ22 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | complex lifted from C4oD4 |
ρ23 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
(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)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)
(1 21 3 19 5 17 7 23)(2 24 4 22 6 20 8 18)(9 26 15 28 13 30 11 32)(10 29 16 31 14 25 12 27)
(1 16 7 14 5 12 3 10)(2 11 4 13 6 15 8 9)(17 25 19 27 21 29 23 31)(18 32 24 30 22 28 20 26)
G:=sub<Sym(32)| (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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31), (1,21,3,19,5,17,7,23)(2,24,4,22,6,20,8,18)(9,26,15,28,13,30,11,32)(10,29,16,31,14,25,12,27), (1,16,7,14,5,12,3,10)(2,11,4,13,6,15,8,9)(17,25,19,27,21,29,23,31)(18,32,24,30,22,28,20,26)>;
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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31), (1,21,3,19,5,17,7,23)(2,24,4,22,6,20,8,18)(9,26,15,28,13,30,11,32)(10,29,16,31,14,25,12,27), (1,16,7,14,5,12,3,10)(2,11,4,13,6,15,8,9)(17,25,19,27,21,29,23,31)(18,32,24,30,22,28,20,26) );
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)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(25,29),(27,31)], [(1,21,3,19,5,17,7,23),(2,24,4,22,6,20,8,18),(9,26,15,28,13,30,11,32),(10,29,16,31,14,25,12,27)], [(1,16,7,14,5,12,3,10),(2,11,4,13,6,15,8,9),(17,25,19,27,21,29,23,31),(18,32,24,30,22,28,20,26)]])
Matrix representation of M4(2).15D4 ►in GL8(F17)
12 | 12 | 12 | 12 | 5 | 5 | 0 | 10 |
5 | 5 | 5 | 5 | 12 | 12 | 7 | 7 |
0 | 0 | 0 | 0 | 5 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
0 | 0 | 12 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 12 | 5 | 0 | 0 | 0 | 0 |
5 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 5 | 5 | 0 | 0 | 12 | 12 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
1 | 1 | 0 | 0 | 16 | 16 | 0 | 16 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
16 | 16 | 16 | 16 | 1 | 1 | 1 | 2 |
0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 1 | 16 | 0 | 0 | 16 |
0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
16 | 16 | 16 | 16 | 1 | 1 | 1 | 2 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
G:=sub<GL(8,GF(17))| [12,5,0,0,0,0,5,0,12,5,0,0,0,0,12,5,12,5,0,0,12,12,0,5,12,5,0,0,12,5,0,0,5,12,5,12,0,0,0,0,5,12,12,12,0,0,0,12,0,7,0,0,0,0,0,12,10,7,0,0,0,0,0,0],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,1,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,0,16,0,0,16,0,0,0,0,0,16,0,16,0,0,0,16,0,0,0,16,0,0,1,0,0,0,0,16,0,1,0,0,0,0,0,1,0,16,0,0,0,0,0,1,16,0,0,0,0,0,16,1,0,0,0,0,0,0,0,2,0,16],[0,0,16,0,16,0,0,0,0,0,16,0,0,16,0,0,0,0,16,0,0,0,16,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,1,16,0,1,0,0,0,0,0,0,0,1,1,0,0,0,1,0,0,2,0,0,0,0,1] >;
M4(2).15D4 in GAP, Magma, Sage, TeX
M_4(2)._{15}D_4
% in TeX
G:=Group("M4(2).15D4");
// GroupNames label
G:=SmallGroup(128,802);
// by ID
G=gap.SmallGroup(128,802);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,64,422,387,352,2019,521,248,2804,4037,2028]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^6*b,b*a*b=a^5,c*a*c^-1=a^3,d*a*d^-1=a^3*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^4*c^3>;
// generators/relations
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