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G = C23.D8  order 128 = 27

1st non-split extension by C23 of D8 acting via D8/C2=D4

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

Aliases: C23.1D8, C4.5C4≀C2, (C2×D8)⋊2C4, C4.Q8⋊1C4, (C2×C8).19D4, C8⋊2D4.1C2, C23.C8⋊6C2, (C2×C4).1SD16, C4.9C42⋊1C2, C4.1(C23⋊C4), (C22×C4).27D4, C2.4(C22.SD16), (C2×M4(2)).1C22, C22.13(D4⋊C4), (C2×C8).1(C2×C4), (C2×C4).52(C22⋊C4), SmallGroup(128,71)

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

Derived series
C1 — C2×C8 — C23.D8
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2) — C8⋊2D4 — C23.D8
Lower central
C1 — C2 — C2×C4 — C2×C8 — C23.D8
Upper central
C1 — C2 — C2×C4 — C2×M4(2) — C23.D8
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C2×M4(2) — C23.D8

Generators and relations for C23.D8
 G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=a, dad-1=ab=ba, ac=ca, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=acd7 >

C1
C2
2C2
4C2
16C2
C4
C22
C4
2C22
2C4
4C22
8C4
8C22
8C22
8C4
8C22
8C4
C2×C4
C23
C2×C4
2C2×C4
2C8
2C2×C4
2C8
4C23
4D4
4D4
4C2×C4
4C2×C4
4C2×C4
8D4
8D4
C2×C8
C22×C4
C2×C8
2C42
2C2×D4
2C4⋊C4
2C42
4C16
4C2×D4
4C4⋊C4
4D8
4M4(2)
4C22⋊C4
4C22⋊C4
C2×M4(2)
C2×D8
C4.Q8
2M5(2)
2D4⋊C4
2C4⋊D4
2C42⋊C2
C4.9C42
C23.C8
C8⋊2D4
C23.D8

Character table of C23.D8

 class 12A2B2C2D4A4B4C4D4E4F4G4H8A8B8C16A16B16C16D
 size 1124162248888164488888
ρ111111111111111111111    trivial
ρ21111-1111-1-1-1-1-11111111    linear of order 2
ρ311111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ41111-11111111-1111-1-1-1-1    linear of order 2
ρ5111-1111-1i-i-ii-1-1-11-iii-i    linear of order 4
ρ6111-1-111-1i-i-ii1-1-11i-i-ii    linear of order 4
ρ7111-1-111-1-iii-i1-1-11-iii-i    linear of order 4
ρ8111-1111-1-iii-i-1-1-11i-i-ii    linear of order 4
ρ92222022200000-2-2-20000    orthogonal lifted from D4
ρ10222-2022-20000022-20000    orthogonal lifted from D4
ρ1122220-2-2-200000000-√2√2-√2√2    orthogonal lifted from D8
ρ1222220-2-2-200000000√2-√2√2-√2    orthogonal lifted from D8
ρ1322-2002-20-1-i-1+i1-i1+i02i-2i00000    complex lifted from C4≀C2
ρ1422-2002-201+i1-i-1+i-1-i02i-2i00000    complex lifted from C4≀C2
ρ1522-2002-201-i1+i-1-i-1+i0-2i2i00000    complex lifted from C4≀C2
ρ16222-20-2-2200000000-√-2-√-2√-2√-2    complex lifted from SD16
ρ1722-2002-20-1+i-1-i1+i1-i0-2i2i00000    complex lifted from C4≀C2
ρ18222-20-2-2200000000√-2√-2-√-2-√-2    complex lifted from SD16
ρ1944-400-440000000000000    orthogonal lifted from C23⋊C4
ρ208-8000000000000000000    orthogonal faithful

Permutation representations of C23.D8
►On 16 points - transitive group 16T371
Generators in S16
(1 9)(4 12)(5 13)(8 16)

(1 9)(3 11)(5 13)(7 15)

(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)

(1 8 9 16)(2 7)(3 6)(4 13 12 5)(10 15)(11 14)
 
G:=sub<Sym(16)| (1,9)(4,12)(5,13)(8,16), (1,9)(3,11)(5,13)(7,15), (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), (1,8,9,16)(2,7)(3,6)(4,13,12,5)(10,15)(11,14)>;
 
G:=Group( (1,9)(4,12)(5,13)(8,16), (1,9)(3,11)(5,13)(7,15), (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), (1,8,9,16)(2,7)(3,6)(4,13,12,5)(10,15)(11,14) );
 
G=PermutationGroup([[(1,9),(4,12),(5,13),(8,16)], [(1,9),(3,11),(5,13),(7,15)], [(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)], [(1,8,9,16),(2,7),(3,6),(4,13,12,5),(10,15),(11,14)]])
 
G:=TransitiveGroup(16,371);
 

Matrix representation of C23.D8 ►in GL8(ℤ)

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

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],[0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-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] >;
 

C23.D8 in GAP, Magma, Sage, TeX 

C_2^3.D_8
 
% in TeX
 
G:=Group("C2^3.D8");
 
// GroupNames label
 
G:=SmallGroup(128,71);
 
// by ID
 
G=gap.SmallGroup(128,71);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,520,1690,521,1411,172,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^8=c,e^2=a,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*c*d^7>;
 
// generators/relations
 

Export

Subgroup lattice of C23.D8 in TeX
Character table of C23.D8 in TeX

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