p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: D4.10D4, Q8.10D4, 2- 1+4.C2, C42.15C22, M4(2).2C22, C4wrC2:4C2, C4:Q8:1C2, (C2xC4).6D4, C4.28(C2xD4), C8.C22:2C2, (C2xC4).7C23, C2.17C22wrC2, C4.10D4:2C2, C4oD4.4C22, C22.15(C2xD4), (C2xQ8).6C22, SmallGroup(64,137)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4.10D4
G = < a,b,c,d | a4=b2=1, c4=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a-1b, dcd-1=c3 >
Subgroups: 121 in 71 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C42, C4:C4, M4(2), SD16, Q16, C2xQ8, C2xQ8, C4oD4, C4oD4, C4.10D4, C4wrC2, C4:Q8, C8.C22, 2- 1+4, D4.10D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C22wrC2, D4.10D4
Character table of D4.10D4
class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | |
size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | |
ρ1 | 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 | linear of order 2 |
ρ3 | 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 | linear of order 2 |
ρ5 | 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 | linear of order 2 |
ρ7 | 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 | linear of order 2 |
ρ9 | 2 | 2 | -2 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ11 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ12 | 2 | 2 | -2 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ13 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ14 | 2 | 2 | -2 | 2 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 15 5 11)(2 10 6 14)(3 13 7 9)(4 16 8 12)
G:=sub<Sym(16)| (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,15,5,11)(2,10,6,14)(3,13,7,9)(4,16,8,12)>;
G:=Group( (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,15,5,11)(2,10,6,14)(3,13,7,9)(4,16,8,12) );
G=PermutationGroup([[(1,7,5,3),(2,4,6,8),(9,15,13,11),(10,12,14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,15,5,11),(2,10,6,14),(3,13,7,9),(4,16,8,12)]])
G:=TransitiveGroup(16,137);
D4.10D4 is a maximal subgroup of
C42.4D4 Q8wrC2 C42.313C23 D8:6D4 D4.S4 2- 1+4.D5
D4p.D4: D8.13D4 D8oQ16 D12.4D4 Q8.14D12 D12.15D4 D12.40D4 D20.4D4 D4.9D20 ...
(CpxD4).D4: C42.16D4 C42.17D4 M4(2).C23 C42.13C23 2- 1+4.2S3 2- 1+4.2D5 2- 1+4.D7 ...
D4.10D4 is a maximal quotient of
(C2xC4):Q16 (C2xC4).SD16 Q8:4Q16 Q8.SD16 C42.9C23 C4.10D4:2C4 C4:Q8:15C4 (C2xQ8):2Q8 M4(2):Q8
D4.D4p: D4.7D8 Q8.14D12 D4.9D20 D4.9D28 ...
C42.D2p: C42.130D4 D12.15D4 D20.15D4 D28.15D4 ...
M4(2).D2p: M4(2).7D4 D12.4D4 D12.40D4 D20.4D4 D20.40D4 D28.4D4 D28.40D4 ...
(CpxD4).D4: C4:C4.6D4 Q8:D4:C2 C4:C4.12D4 (C2xC4).5D8 D4:4Q16 C42.211C23 Q8:4SD16 C42.213C23 ...
Matrix representation of D4.10D4 ►in GL4(F3) generated by
1 | 1 | 0 | 0 |
1 | 2 | 0 | 0 |
0 | 0 | 0 | 1 |
0 | 0 | 2 | 0 |
0 | 1 | 2 | 0 |
2 | 0 | 1 | 1 |
1 | 0 | 1 | 1 |
2 | 2 | 1 | 2 |
2 | 0 | 2 | 2 |
2 | 1 | 0 | 2 |
2 | 1 | 0 | 0 |
0 | 2 | 0 | 0 |
2 | 0 | 2 | 2 |
2 | 1 | 0 | 2 |
2 | 1 | 0 | 1 |
0 | 2 | 1 | 0 |
G:=sub<GL(4,GF(3))| [1,1,0,0,1,2,0,0,0,0,0,2,0,0,1,0],[0,2,1,2,1,0,0,2,2,1,1,1,0,1,1,2],[2,2,2,0,0,1,1,2,2,0,0,0,2,2,0,0],[2,2,2,0,0,1,1,2,2,0,0,1,2,2,1,0] >;
D4.10D4 in GAP, Magma, Sage, TeX
D_4._{10}D_4
% in TeX
G:=Group("D4.10D4");
// GroupNames label
G:=SmallGroup(64,137);
// by ID
G=gap.SmallGroup(64,137);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,362,158,963,489,255,117,730]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^4=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^3>;
// generators/relations
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