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G = (C2×C8).6D4  order 128 = 27

6th non-split extension by C2×C8 of D4 acting faithfully

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

Aliases: (C2×C8).6D4, (C2×Q8).113D4, C4.9C42.5C2, C4.161(C4⋊D4), C4.8(C42⋊2C2), (C22×C4).46C23, M4(2).C4.3C2, C23.135(C4○D4), M4(2)⋊4C4.5C2, C23.38D4.1C2, (C22×Q8).80C22, C42⋊C2.70C22, C4.32(C22.D4), C22.28(C4.4D4), (C2×M4(2)).32C22, C22.11(C42⋊2C2), C2.17(C23.11D4), C22.60(C22.D4), (C2×C4).268(C2×D4), (C2×C4).355(C4○D4), (C2×C4.10D4).12C2, SmallGroup(128,814)

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

Derived series
C1 — C22×C4 — (C2×C8).6D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×M4(2) — C2×C4.10D4 — (C2×C8).6D4
Lower central
C1 — C2 — C22×C4 — (C2×C8).6D4
Upper central
C1 — C2 — C22×C4 — (C2×C8).6D4
Jennings
C1 — C2 — C2 — C22×C4 — (C2×C8).6D4

Generators and relations for (C2×C8).6D4
 G = < a,b,c,d | a2=b8=1, c4=d2=b4, cbc-1=ab=ba, cac-1=ab4, ad=da, dbd-1=ab-1, dcd-1=ab2c3 >

Subgroups: 184 in 92 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C4.10D4, Q8⋊C4, C8.C4, C42⋊C2, C2×M4(2), C22×Q8, C4.9C42, M4(2)⋊4C4, C2×C4.10D4, C23.38D4, M4(2).C4, (C2×C8).6D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C23.11D4, (C2×C8).6D4

Character table of (C2×C8).6D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 11222222288888888888888
ρ111111111111111111111111    trivial
ρ2111111111-11-1-1-11-1111-11-1-1    linear of order 2
ρ3111111111-1-1-1-1-1-11-1-111111    linear of order 2
ρ41111111111-1111-1-1-1-11-11-1-1    linear of order 2
ρ5111111111-111-1111-1-1-1-1-1-11    linear of order 2
ρ611111111111-11-11-1-1-1-11-11-1    linear of order 2
ρ71111111111-1-11-1-1111-1-1-1-11    linear of order 2
ρ8111111111-1-11-11-1-111-11-11-1    linear of order 2
ρ922-2-22-22-2200000000020-200    orthogonal lifted from D4
ρ1022-2-22-22-22000000000-20200    orthogonal lifted from D4
ρ1122-22-2-222-2200-20000000000    orthogonal lifted from D4
ρ1222-22-2-222-2-20020000000000    orthogonal lifted from D4
ρ1322-2-222-22-202i000-2i00000000    complex lifted from C4○D4
ρ14222-2-2-2-22200000000002i0-2i0    complex lifted from C4○D4
ρ15222-2-2-2-2220000000000-2i02i0    complex lifted from C4○D4
ρ1622-2-222-22-20-2i0002i00000000    complex lifted from C4○D4
ρ1722222-2-2-2-20000000-2i2i00000    complex lifted from C4○D4
ρ1822-22-22-2-220000002i000000-2i    complex lifted from C4○D4
ρ19222-2-222-2-2002i0-2i000000000    complex lifted from C4○D4
ρ2022222-2-2-2-200000002i-2i00000    complex lifted from C4○D4
ρ2122-22-22-2-22000000-2i0000002i    complex lifted from C4○D4
ρ22222-2-222-2-200-2i02i000000000    complex lifted from C4○D4
ρ238-8000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of (C2×C8).6D4
►On 32 points
Generators in S32
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)

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

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

Matrix representation of (C2×C8).6D4 ►in GL8(𝔽17)

160020000
160110000
161010000
00010000
160010001
000100160
000101600
160011000
,
100000707
0000512512
50000700
50001212512
50120012012
51205012012
0050012012
1012012012012
,
7000100100
70005555
1200000100
1200055512
1201205050
1212055050
001205050
75055050
,
100700000
101212120000
50700000
5121250000
100120120120
001200505
5012012050
5012005012

G:=sub<GL(8,GF(17))| [16,16,16,0,16,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0],[10,0,5,5,5,5,0,10,0,0,0,0,0,12,0,12,0,0,0,0,12,0,5,0,0,0,0,0,0,5,0,12,0,5,0,12,0,0,0,0,7,12,7,12,12,12,12,12,0,5,0,5,0,0,0,0,7,12,0,12,12,12,12,12],[7,7,12,12,12,12,0,7,0,0,0,0,0,12,0,5,0,0,0,0,12,0,12,0,0,0,0,0,0,5,0,5,10,5,0,5,5,5,5,5,0,5,0,5,0,0,0,0,10,5,10,5,5,5,5,5,0,5,0,12,0,0,0,0],[10,10,5,5,10,0,5,5,0,12,0,12,0,0,0,0,7,12,7,12,12,12,12,12,0,12,0,5,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,0,0,5,0,5,0,0,0,0,12,0,5,0,0,0,0,0,0,5,0,12] >;
 

(C2×C8).6D4 in GAP, Magma, Sage, TeX 

(C_2\times C_8)._6D_4
 
% in TeX
 
G:=Group("(C2xC8).6D4");
 
// GroupNames label
 
G:=SmallGroup(128,814);
 
// by ID
 
G=gap.SmallGroup(128,814);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,456,422,387,58,1018,248,1411,718,172,4037,1027]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=1,c^4=d^2=b^4,c*b*c^-1=a*b=b*a,c*a*c^-1=a*b^4,a*d=d*a,d*b*d^-1=a*b^-1,d*c*d^-1=a*b^2*c^3>;
 
// generators/relations
 

Export

Character table of (C2×C8).6D4 in TeX

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