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G = C23.SD16  order 128 = 27

1st non-split extension by C23 of SD16 acting via SD16/C2=D4

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C23.1SD16, C4.7C4≀C2, (C2×C4).1D8, (C2×D8).4C4, (C2×C8).21D4, C4.Q8.1C4, C8⋊2D4.2C2, C23.C8⋊7C2, C4.3(C23⋊C4), (C22×C4).29D4, C4.10C42⋊1C2, C2.6(C22.SD16), (C2×M4(2)).3C22, C22.15(D4⋊C4), (C2×C8).3(C2×C4), (C2×C4).54(C22⋊C4), SmallGroup(128,73)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C8 — C23.SD16
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2) — C8⋊2D4 — C23.SD16
Lower central
C1 — C2 — C2×C4 — C2×C8 — C23.SD16
Upper central
C1 — C2 — C2×C4 — C2×M4(2) — C23.SD16
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C2×M4(2) — C23.SD16

Generators and relations for C23.SD16
 G = < a,b,c,d,e | a2=b2=c2=e2=1, d8=c, dad-1=eae=ab=ba, ac=ca, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=abd3 >

C1
C2
2C2
4C2
16C2
C4
C4
C22
2C4
2C22
4C22
8C22
8C22
8C4
8C22
C23
C2×C4
C2×C4
2C2×C4
2C8
2C2×C4
2C8
4D4
4D4
4C23
4C8
4C8
4C2×C4
8D4
8D4
C22×C4
C2×C8
C2×C8
2C2×C8
2C4⋊C4
2C2×D4
2C2×C8
4C16
4C22⋊C4
4M4(2)
4M4(2)
4D8
4C2×D4
4M4(2)
C2×M4(2)
C2×D8
C4.Q8
2M5(2)
2D4⋊C4
2C4⋊D4
2C2×M4(2)
C8⋊2D4
C4.10C42
C23.C8
C23.SD16

Character table of C23.SD16

 class 12A2B2C2D4A4B4C4D8A8B8C8D8E8F8G16A16B16C16D
 size 1124162241644888888888
ρ111111111111111111111    trivial
ρ21111-1111-11111111-1-1-1-1    linear of order 2
ρ31111-1111-111-1-1-11-11111    linear of order 2
ρ411111111111-1-1-11-1-1-1-1-1    linear of order 2
ρ5111-1111-1-1-1-1-ii-i1ii-ii-i    linear of order 4
ρ6111-1-111-11-1-1i-ii1-ii-ii-i    linear of order 4
ρ7111-1-111-11-1-1-ii-i1i-ii-ii    linear of order 4
ρ8111-1111-1-1-1-1i-ii1-i-ii-ii    linear of order 4
ρ9222202220-2-2000-200000    orthogonal lifted from D4
ρ10222-2022-2022000-200000    orthogonal lifted from D4
ρ11222-20-2-2200000000-√2√2√2-√2    orthogonal lifted from D8
ρ12222-20-2-2200000000√2-√2-√2√2    orthogonal lifted from D8
ρ1322-2002-200-2i2i1+i1-i-1-i0-1+i0000    complex lifted from C4≀C2
ρ1422-2002-200-2i2i-1-i-1+i1+i01-i0000    complex lifted from C4≀C2
ρ1522220-2-2-200000000√-2√-2-√-2-√-2    complex lifted from SD16
ρ1622220-2-2-200000000-√-2-√-2√-2√-2    complex lifted from SD16
ρ1722-2002-2002i-2i-1+i-1-i1-i01+i0000    complex lifted from C4≀C2
ρ1822-2002-2002i-2i1-i1+i-1+i0-1-i0000    complex lifted from C4≀C2
ρ1944-400-440000000000000    orthogonal lifted from C23⋊C4
ρ208-8000000000000000000    orthogonal faithful

Permutation representations of C23.SD16
►On 16 points - transitive group 16T348
Generators in S16
(1 9)(2 10)(5 13)(6 14)

(2 10)(4 12)(6 14)(8 16)

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

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

(2 12)(3 7)(4 10)(5 13)(6 8)(11 15)(14 16)
 
G:=sub<Sym(16)| (1,9)(2,10)(5,13)(6,14), (2,10)(4,12)(6,14)(8,16), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,12)(3,7)(4,10)(5,13)(6,8)(11,15)(14,16)>;
 
G:=Group( (1,9)(2,10)(5,13)(6,14), (2,10)(4,12)(6,14)(8,16), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,12)(3,7)(4,10)(5,13)(6,8)(11,15)(14,16) );
 
G=PermutationGroup([[(1,9),(2,10),(5,13),(6,14)], [(2,10),(4,12),(6,14),(8,16)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(2,12),(3,7),(4,10),(5,13),(6,8),(11,15),(14,16)]])
 
G:=TransitiveGroup(16,348);
 

Matrix representation of C23.SD16 ►in GL8(ℤ)

10000000
01000000
00-100000
000-10000
0000-1000
00000-100
00000010
00000001
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
00001000
00000-100
00000001
00000010
00-100000
00010000
10000000
0-1000000
,
10000000
0-1000000
00010000
00100000
00000001
00000010
00000100
00001000

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;
 

C23.SD16 in GAP, Magma, Sage, TeX 

C_2^3.{\rm SD}_{16}
 
% in TeX
 
G:=Group("C2^3.SD16");
 
// GroupNames label
 
G:=SmallGroup(128,73);
 
// by ID
 
G=gap.SmallGroup(128,73);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,520,1690,521,248,1411,172,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^2=1,d^8=c,d*a*d^-1=e*a*e=a*b=b*a,a*c=c*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=a*b*d^3>;
 
// generators/relations
 

Export

Subgroup lattice of C23.SD16 in TeX
Character table of C23.SD16 in TeX

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