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G = (C2×C8).D4  order 128 = 27

5th non-split extension by C2×C8 of D4 acting faithfully

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

Aliases: (C2×C8).5D4, (C2×D4).122D4, C4.9C42⋊15C2, M4(2).C4⋊7C2, C4.160(C4⋊D4), C4.7(C42⋊2C2), (C22×C4).45C23, M4(2)⋊4C4⋊21C2, C23.134(C4○D4), C23.37D4.1C2, (C22×D4).94C22, C42⋊C2.69C22, C4.31(C22.D4), C22.27(C4.4D4), (C2×M4(2)).31C22, C22.10(C42⋊2C2), C2.16(C23.11D4), C22.59(C22.D4), (C2×C4).267(C2×D4), (C2×C4).354(C4○D4), (C2×C4.D4).12C2, SmallGroup(128,813)

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

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

Generators and relations for (C2×C8).D4
 G = < a,b,c,d | a2=b8=d2=1, c4=b4, cbc-1=ab=ba, cac-1=ab4, ad=da, dbd=ab-1, dcd=ab6c3 >

Subgroups: 264 in 102 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, C4.D4, D4⋊C4, C8.C4, C42⋊C2, C2×M4(2), C22×D4, C4.9C42, M4(2)⋊4C4, C2×C4.D4, C23.37D4, M4(2).C4, (C2×C8).D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C23.11D4, (C2×C8).D4

Character table of (C2×C8).D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 11222882222888888888888
ρ111111111111111111111111    trivial
ρ211111111111-11-11-11-1-11-1-1-1    linear of order 2
ρ311111-1-1111111111-1-1-1-1-1-11    linear of order 2
ρ411111-1-11111-11-11-1-111-111-1    linear of order 2
ρ511111111111-1-1-1-11-11-1-11-11    linear of order 2
ρ6111111111111-11-1-1-1-11-1-11-1    linear of order 2
ρ711111-1-11111-1-1-1-111-111-111    linear of order 2
ρ811111-1-111111-11-1-111-111-1-1    linear of order 2
ρ9222-2-2-222-22-2000000000000    orthogonal lifted from D4
ρ1022-22-200-2-2220000000200-20    orthogonal lifted from D4
ρ1122-22-200-2-2220000000-20020    orthogonal lifted from D4
ρ12222-2-22-22-22-2000000000000    orthogonal lifted from D4
ρ1322-22-20022-2-20-2i02i00000000    complex lifted from C4○D4
ρ14222-2-200-22-220000-2i0000002i    complex lifted from C4○D4
ρ152222200-2-2-2-2000000-2i002i00    complex lifted from C4○D4
ρ162222200-2-2-2-20000002i00-2i00    complex lifted from C4○D4
ρ17222-2-200-22-2200002i000000-2i    complex lifted from C4○D4
ρ1822-22-20022-2-202i0-2i00000000    complex lifted from C4○D4
ρ1922-2-22002-2-22000002i00-2i000    complex lifted from C4○D4
ρ2022-2-2200-222-22i0-2i000000000    complex lifted from C4○D4
ρ2122-2-22002-2-2200000-2i002i000    complex lifted from C4○D4
ρ2222-2-2200-222-2-2i02i000000000    complex lifted from C4○D4
ρ238-8000000000000000000000    orthogonal faithful

Permutation representations of (C2×C8).D4
►On 16 points - transitive group 16T412
Generators in S16
(9 13)(10 14)(11 15)(12 16)

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

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

(2 8)(3 7)(4 6)(9 13)(10 16)(12 14)
 
G:=sub<Sym(16)| (9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13,7,11,5,9,3,15)(2,14,4,16,6,10,8,12), (2,8)(3,7)(4,6)(9,13)(10,16)(12,14)>;
 
G:=Group( (9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13,7,11,5,9,3,15)(2,14,4,16,6,10,8,12), (2,8)(3,7)(4,6)(9,13)(10,16)(12,14) );
 
G=PermutationGroup([[(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,13,7,11,5,9,3,15),(2,14,4,16,6,10,8,12)], [(2,8),(3,7),(4,6),(9,13),(10,16),(12,14)]])
 
G:=TransitiveGroup(16,412);
 

Matrix representation of (C2×C8).D4 ►in GL8(ℤ)

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

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

(C2×C8).D4 in GAP, Magma, Sage, TeX 

(C_2\times C_8).D_4
 
% in TeX
 
G:=Group("(C2xC8).D4");
 
// GroupNames label
 
G:=SmallGroup(128,813);
 
// by ID
 
G=gap.SmallGroup(128,813);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,422,387,58,1018,248,1411,718,172,4037,1027]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=d^2=1,c^4=b^4,c*b*c^-1=a*b=b*a,c*a*c^-1=a*b^4,a*d=d*a,d*b*d=a*b^-1,d*c*d=a*b^6*c^3>;
 
// generators/relations
 

Export

Character table of (C2×C8).D4 in TeX

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