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G = C23.31D4order 64 = 26

2nd non-split extension by C23 of D4 acting via D4/C22=C2

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

Aliases: C23.31D4, C22.2Q16, C22.2SD16, C4⋊C42C4, C2.5C4≀C2, (C2×Q8)⋊1C4, (C2×C4).95D4, C22⋊C8.2C2, C22⋊Q8.1C2, C2.5(C23⋊C4), C2.3(Q8⋊C4), C2.C42.4C2, (C22×C4).17C22, C22.36(C22⋊C4), (C2×C4).9(C2×C4), SmallGroup(64,9)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C23.31D4
C1C2C22C23C22×C4C22⋊Q8 — C23.31D4
C1C22C2×C4 — C23.31D4
C1C22C22×C4 — C23.31D4
C1C2C22C22×C4 — C23.31D4

Generators and relations for C23.31D4
 G = < a,b,c,d,e | a2=b2=c2=1, d4=c, e2=a, ab=ba, ac=ca, dad-1=abc, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=abcd3 >

2C2
2C2
2C22
2C22
2C4
2C4
4C4
4C4
4C4
4C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
4C2×C4
4C2×C4
4Q8
4C2×C4
4C8
2C2×C8
2C22×C4
2C4⋊C4
2C22⋊C4

Character table of C23.31D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 1111222244444884444
ρ11111111111111111111    trivial
ρ211111111-1-11-1-1-1-11111    linear of order 2
ρ311111111-1-11-1-111-1-1-1-1    linear of order 2
ρ41111111111111-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-1-1-ii1i-i-11ii-i-i    linear of order 4
ρ61111-1-1-1-1i-i1-ii1-1ii-i-i    linear of order 4
ρ71111-1-1-1-1i-i1-ii-11-i-iii    linear of order 4
ρ81111-1-1-1-1-ii1i-i1-1-i-iii    linear of order 4
ρ92222-2-22200-200000000    orthogonal lifted from D4
ρ10222222-2-200-200000000    orthogonal lifted from D4
ρ112-2-22-22000000000-222-2    symplectic lifted from Q16, Schur index 2
ρ122-2-22-220000000002-2-22    symplectic lifted from Q16, Schur index 2
ρ1322-2-200-2i2i1-i1+i0-1-i-1+i000000    complex lifted from C4≀C2
ρ1422-2-200-2i2i-1+i-1-i01+i1-i000000    complex lifted from C4≀C2
ρ1522-2-2002i-2i1+i1-i0-1+i-1-i000000    complex lifted from C4≀C2
ρ1622-2-2002i-2i-1-i-1+i01-i1+i000000    complex lifted from C4≀C2
ρ172-2-222-2000000000-2--2-2--2    complex lifted from SD16
ρ182-2-222-2000000000--2-2--2-2    complex lifted from SD16
ρ194-44-4000000000000000    orthogonal lifted from C23⋊C4

Permutation representations of C23.31D4
On 16 points - transitive group 16T161
Generators in S16
(2 15)(4 9)(6 11)(8 13)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 4 15 9)(3 12)(6 8 11 13)(7 16)

G:=sub<Sym(16)| (2,15)(4,9)(6,11)(8,13), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,4,15,9)(3,12)(6,8,11,13)(7,16)>;

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

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

G:=TransitiveGroup(16,161);

C23.31D4 is a maximal subgroup of
C24.53D4  C24.150D4  C24.55D4  C24.57D4  C42.58D4  C24.58D4  C42.60D4  C24.59D4  C42.62D4  C24.61D4  C42.63D4  C23⋊SD16  C4⋊C4.D4  (C2×C4)⋊SD16  C24.9D4  C23⋊Q16  C4⋊C4.6D4  (C2×C4)⋊Q16  C24.12D4  C24.14D4  C4⋊C4.12D4  (C2×C4).SD16  C24.15D4  C24.17D4  C4⋊C4.18D4  C4⋊C4.20D4  C24.18D4  C23.7S4
 C2p.C4≀C2: C42.375D4  C42.404D4  C42.56D4  C42.57D4  C4⋊Dic3⋊C4  C23.35D12  (C6×Q8)⋊C4  C4⋊Dic5⋊C4 ...
C23.31D4 is a maximal quotient of
C23.Q16  C24.4D4  D10.1Q16  D10.Q16
 C23.D4p: C23.30D8  C23.35D12  C23.30D20  C23.30D28 ...
 (C2×C2p).Q16: C4⋊C4⋊C8  (C2×Q8)⋊C8  (C2×C4).Q16  C2.7C2≀C4  C4⋊Dic3⋊C4  (C6×Q8)⋊C4  C4⋊Dic5⋊C4  C10.29C4≀C2 ...

Matrix representation of C23.31D4 in GL4(𝔽17) generated by

16000
01600
0010
001516
,
16000
01600
0010
0001
,
16000
01600
00160
00016
,
14300
141400
0099
00168
,
13000
0400
00160
001413
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,15,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[14,14,0,0,3,14,0,0,0,0,9,16,0,0,9,8],[13,0,0,0,0,4,0,0,0,0,16,14,0,0,0,13] >;

C23.31D4 in GAP, Magma, Sage, TeX

C_2^3._{31}D_4
% in TeX

G:=Group("C2^3.31D4");
// GroupNames label

G:=SmallGroup(64,9);
// by ID

G=gap.SmallGroup(64,9);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,103,362,332,158,681]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=c,e^2=a,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*b*c*d^3>;
// generators/relations

Export

Subgroup lattice of C23.31D4 in TeX
Character table of C23.31D4 in TeX

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