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G = D4.10D4order 64 = 26

5th non-split extension by D4 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial, rational

Aliases: D4.10D4, Q8.10D4, 2- 1+4.C2, C42.15C22, M4(2).2C22, C4≀C24C2, C4⋊Q81C2, (C2×C4).6D4, C4.28(C2×D4), C8.C222C2, (C2×C4).7C23, C2.17C22≀C2, C4.10D42C2, C4○D4.4C22, C22.15(C2×D4), (C2×Q8).6C22, SmallGroup(64,137)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D4.10D4
C1C2C22C2×C4C2×Q82- 1+4 — D4.10D4
C1C2C2×C4 — D4.10D4
C1C2C2×C4 — D4.10D4
C1C2C2C2×C4 — D4.10D4

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 [×3], C4 [×2], C4 [×7], C22, C22 [×2], C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×8], D4 [×2], D4 [×4], Q8 [×2], Q8 [×6], C42, C4⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×2], C4○D4 [×4], C4.10D4, C4≀C2 [×2], C4⋊Q8, C8.C22 [×2], 2- 1+4, D4.10D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, D4.10D4

Character table of D4.10D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I8A8B
 size 1124422444444888
ρ11111111111111111    trivial
ρ2111-1-111-11-1-11-1-111    linear of order 2
ρ3111-1-111-11-11111-1-1    linear of order 2
ρ41111111111-11-1-1-1-1    linear of order 2
ρ51111-111-1-11-1-1-11-11    linear of order 2
ρ6111-11111-1-11-11-1-11    linear of order 2
ρ7111-11111-1-1-1-1-111-1    linear of order 2
ρ81111-111-1-111-11-11-1    linear of order 2
ρ922-2-20-22002000000    orthogonal lifted from D4
ρ1022200-2-20200-20000    orthogonal lifted from D4
ρ1122-2022-2-200000000    orthogonal lifted from D4
ρ1222-20-22-2200000000    orthogonal lifted from D4
ρ1322200-2-20-20020000    orthogonal lifted from D4
ρ1422-220-2200-2000000    orthogonal lifted from D4
ρ154-40000000020-2000    symplectic faithful, Schur index 2
ρ164-400000000-202000    symplectic faithful, Schur index 2

Permutation representations of D4.10D4
On 16 points - transitive group 16T137
Generators in S16
(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 11 5 15)(2 14 6 10)(3 9 7 13)(4 12 8 16)

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), (1,11,5,15)(2,14,6,10)(3,9,7,13)(4,12,8,16)>;

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), (1,11,5,15)(2,14,6,10)(3,9,7,13)(4,12,8,16) );

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)], [(1,11,5,15),(2,14,6,10),(3,9,7,13),(4,12,8,16)])

G:=TransitiveGroup(16,137);

D4.10D4 is a maximal subgroup of
C42.4D4  Q8≀C2  C42.313C23  D86D4  D4.S4  2- 1+4.D5
 D4p.D4: D8.13D4  D8○Q16  D12.4D4  Q8.14D12  D12.15D4  D12.40D4  D20.4D4  D4.9D20 ...
 (Cp×D4).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
(C2×C4)⋊Q16  (C2×C4).SD16  Q84Q16  Q8.SD16  C42.9C23  C4.10D42C4  C4⋊Q815C4  (C2×Q8)⋊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 ...
 (Cp×D4).D4: C4⋊C4.6D4  Q8⋊D4⋊C2  C4⋊C4.12D4  (C2×C4).5D8  D44Q16  C42.211C23  Q84SD16  C42.213C23 ...

Matrix representation of D4.10D4 in GL4(𝔽3) generated by

1100
1200
0001
0020
,
0120
2011
1011
2212
,
2022
2102
2100
0200
,
2022
2102
2101
0210
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

Export

Character table of D4.10D4 in TeX

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