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G = (C2×C4).5D8  order 128 = 27

5th non-split extension by C2×C4 of D8 acting via D8/C2=D4

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

Aliases: (C2×C4).5D8, C4⋊C4.13D4, (C2×D4).17D4, C22.16(C2×D8), (C22×C4).50D4, C2.9(C22⋊D8), C23.527(C2×D4), C22.D8⋊2C2, C22.SD16⋊8C2, C22⋊C8.6C22, C4⋊D4.11C22, (C22×C4).16C23, C22.137C22≀C2, C22.43(C8⋊C22), C2.7(C23.7D4), C2.10(D4.10D4), C23.81C23⋊2C2, C22.M4(2)⋊4C2, C22.31C24.2C2, C2.C42.23C22, (C2×C4).205(C2×D4), (C2×C4⋊C4).21C22, SmallGroup(128,342)

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

Derived series
C1 — C22×C4 — (C2×C4).5D8
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C22.31C24 — (C2×C4).5D8
Lower central
C1 — C22 — C22×C4 — (C2×C4).5D8
Upper central
C1 — C22 — C22×C4 — (C2×C4).5D8
Jennings
C1 — C2 — C22 — C22×C4 — (C2×C4).5D8

Generators and relations for (C2×C4).5D8
 G = < a,b,c,d | a2=b4=c8=1, d2=b2, cbc-1=dbd-1=ab=ba, cac-1=ab2, ad=da, dcd-1=b2c-1 >

Subgroups: 324 in 127 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C2.C42, C2.C42, C22⋊C8, D4⋊C4, C2.D8, C2×C4⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C2×C4○D4, C22.M4(2), C22.SD16, C23.81C23, C22.D8, C22.31C24, (C2×C4).5D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D4.10D4, C23.7D4, (C2×C4).5D8

Character table of (C2×C4).5D8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11112288444488888888888
ρ111111111111111111111111    trivial
ρ21111111-11-1-11-11-11-11-1-11-11    linear of order 2
ρ3111111111111-111-11-1-1-1-1-1-1    linear of order 2
ρ41111111-11-1-1111-1-1-1-111-11-1    linear of order 2
ρ5111111-1-111111-1-11111-1-1-1-1    linear of order 2
ρ6111111-111-1-11-1-111-11-11-11-1    linear of order 2
ρ7111111-1-11111-1-1-1-11-1-11111    linear of order 2
ρ8111111-111-1-111-11-1-1-11-11-11    linear of order 2
ρ92222-2-2-20-200202000000000    orthogonal lifted from D4
ρ102222-2-202200-200-200000000    orthogonal lifted from D4
ρ1122222200-2-2-2-200002000000    orthogonal lifted from D4
ρ122222-2-20-2200-200200000000    orthogonal lifted from D4
ρ1322222200-222-20000-2000000    orthogonal lifted from D4
ρ142222-2-220-20020-2000000000    orthogonal lifted from D4
ρ1522-2-2-22000-2200000000√2-√2-√2√2    orthogonal lifted from D8
ρ1622-2-2-22000-2200000000-√2√2√2-√2    orthogonal lifted from D8
ρ1722-2-2-220002-200000000-√2-√2√2√2    orthogonal lifted from D8
ρ1822-2-2-220002-200000000√2√2-√2-√2    orthogonal lifted from D8
ρ1944-4-44-400000000000000000    orthogonal lifted from C8⋊C22
ρ204-4-4400000000-20000020000    symplectic lifted from D4.10D4, Schur index 2
ρ214-4-4400000000200000-20000    symplectic lifted from D4.10D4, Schur index 2
ρ224-44-4000000000002i0-2i00000    complex lifted from C23.7D4
ρ234-44-400000000000-2i02i00000    complex lifted from C23.7D4

Smallest permutation representation of (C2×C4).5D8
►On 32 points
Generators in S32
(2 29)(4 31)(6 25)(8 27)(9 24)(11 18)(13 20)(15 22)

(1 21 28 14)(2 22 29 15)(3 16 30 23)(4 9 31 24)(5 17 32 10)(6 18 25 11)(7 12 26 19)(8 13 27 20)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)

(1 14 28 21)(2 20 29 13)(3 12 30 19)(4 18 31 11)(5 10 32 17)(6 24 25 9)(7 16 26 23)(8 22 27 15)
 
G:=sub<Sym(32)| (2,29)(4,31)(6,25)(8,27)(9,24)(11,18)(13,20)(15,22), (1,21,28,14)(2,22,29,15)(3,16,30,23)(4,9,31,24)(5,17,32,10)(6,18,25,11)(7,12,26,19)(8,13,27,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,14,28,21)(2,20,29,13)(3,12,30,19)(4,18,31,11)(5,10,32,17)(6,24,25,9)(7,16,26,23)(8,22,27,15)>;
 
G:=Group( (2,29)(4,31)(6,25)(8,27)(9,24)(11,18)(13,20)(15,22), (1,21,28,14)(2,22,29,15)(3,16,30,23)(4,9,31,24)(5,17,32,10)(6,18,25,11)(7,12,26,19)(8,13,27,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,14,28,21)(2,20,29,13)(3,12,30,19)(4,18,31,11)(5,10,32,17)(6,24,25,9)(7,16,26,23)(8,22,27,15) );
 
G=PermutationGroup([[(2,29),(4,31),(6,25),(8,27),(9,24),(11,18),(13,20),(15,22)], [(1,21,28,14),(2,22,29,15),(3,16,30,23),(4,9,31,24),(5,17,32,10),(6,18,25,11),(7,12,26,19),(8,13,27,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,14,28,21),(2,20,29,13),(3,12,30,19),(4,18,31,11),(5,10,32,17),(6,24,25,9),(7,16,26,23),(8,22,27,15)]])
 

Matrix representation of (C2×C4).5D8 ►in GL6(𝔽17)

100000
010000
001000
000100
0000160
0000016
,
100000
010000
004000
0001300
000040
0000013
,
3140000
330000
0000013
0000130
0013000
0001300
,
100000
0160000
0013000
0001300
0000013
0000130

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,13,0,0,0,0,13,0,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,13,0,0,0,0,13,0] >;
 

(C2×C4).5D8 in GAP, Magma, Sage, TeX 

(C_2\times C_4)._5D_8
 
% in TeX
 
G:=Group("(C2xC4).5D8");
 
// GroupNames label
 
G:=SmallGroup(128,342);
 
// by ID
 
G=gap.SmallGroup(128,342);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,422,352,1123,570,521,136,1411]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=b^2,c*b*c^-1=d*b*d^-1=a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,d*c*d^-1=b^2*c^-1>;
 
// generators/relations
 

Export

Character table of (C2×C4).5D8 in TeX

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