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G = (C2xC8).2D4  order 128 = 27

2nd non-split extension by C2xC8 of D4 acting faithfully

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

Aliases: (C2xC8).2D4, (C2xD4).95D4, (C2xQ8).86D4, C4.66C22wrC2, C4.58(C4:1D4), D8:C22:3C2, C4.10C42:4C2, C23.10(C4oD4), (C22xC8).64C22, C2.22(C23:2D4), C22.25(C4:D4), (C22xC4).715C23, (C2xM4(2)).20C22, (C2xC4).38(C2xD4), (C22xC8):C2:1C2, (C2xC4oD4).52C22, SmallGroup(128,749)

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

Derived series
C1 — C22xC4 — (C2xC8).2D4
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C2xC4oD4 — D8:C22 — (C2xC8).2D4
Lower central
C1 — C2 — C22xC4 — (C2xC8).2D4
Upper central
C1 — C4 — C22xC4 — (C2xC8).2D4
Jennings
C1 — C2 — C2 — C22xC4 — (C2xC8).2D4

Generators and relations for (C2xC8).2D4
 G = < a,b,c,d | a2=b8=d2=1, c4=b4, ab=ba, cac-1=ab4, ad=da, cbc-1=ab-1, dbd=ab5, dcd=b4c3 >

Subgroups: 376 in 175 conjugacy classes, 44 normal (8 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2xC4, C2xC4, D4, Q8, C23, C23, C2xC8, C2xC8, M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C22:C8, C22xC8, C2xM4(2), C4oD8, C8:C22, C8.C22, C2xC4oD4, C4.10C42, (C22xC8):C2, D8:C22, (C2xC8).2D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C22wrC2, C4:D4, C4:1D4, C23:2D4, (C2xC8).2D4

Character table of (C2xC8).2D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J
 size 11222888112228884444888888
ρ111111111111111111111111111    trivial
ρ2111111-1-111111-11-11111-1-111-1-1    linear of order 2
ρ3111111-111111111-1-1-1-1-111-1-1-1-1    linear of order 2
ρ41111111-111111-111-1-1-1-1-1-1-1-111    linear of order 2
ρ511111-11-111111-1-11111111-1-1-1-1    linear of order 2
ρ611111-1-11111111-1-11111-1-1-1-111    linear of order 2
ρ711111-1-1-111111-1-1-1-1-1-1-1111111    linear of order 2
ρ811111-111111111-11-1-1-1-1-1-111-1-1    linear of order 2
ρ9222-2-2200-2-2-2220-200000000000    orthogonal lifted from D4
ρ1022-2-2200022-2-22000000000002-2    orthogonal lifted from D4
ρ1122-2-2200-2-2-222-22000000000000    orthogonal lifted from D4
ρ1222-2-22002-2-222-2-2000000000000    orthogonal lifted from D4
ρ13222-2-2-200-2-2-2220200000000000    orthogonal lifted from D4
ρ1422-2-2200022-2-2200000000000-22    orthogonal lifted from D4
ρ1522-22-2020-2-22-2200-20000000000    orthogonal lifted from D4
ρ16222-2-2000222-2-2000000000-2200    orthogonal lifted from D4
ρ1722-22-200022-22-200000002-20000    orthogonal lifted from D4
ρ1822-22-200022-22-20000000-220000    orthogonal lifted from D4
ρ19222-2-2000222-2-20000000002-200    orthogonal lifted from D4
ρ2022-22-20-20-2-22-220020000000000    orthogonal lifted from D4
ρ2122222000-2-2-2-2-20002i-2i-2i2i000000    complex lifted from C4oD4
ρ2222222000-2-2-2-2-2000-2i2i2i-2i000000    complex lifted from C4oD4
ρ234-40000004i-4i0000002ζ872ζ852ζ82ζ83000000    complex faithful
ρ244-4000000-4i4i0000002ζ82ζ832ζ872ζ85000000    complex faithful
ρ254-4000000-4i4i0000002ζ852ζ872ζ832ζ8000000    complex faithful
ρ264-40000004i-4i0000002ζ832ζ82ζ852ζ87000000    complex faithful

Smallest permutation representation of (C2xC8).2D4
►On 32 points
Generators in S32
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)

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

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

Matrix representation of (C2xC8).2D4 ►in GL4(F17) generated by

01300
4000
0004
00130
,
0001
0010
1000
01600
,
15000
0200
0080
0009
,
00150
0002
8000
0900
G:=sub<GL(4,GF(17))| [0,4,0,0,13,0,0,0,0,0,0,13,0,0,4,0],[0,0,1,0,0,0,0,16,0,1,0,0,1,0,0,0],[15,0,0,0,0,2,0,0,0,0,8,0,0,0,0,9],[0,0,8,0,0,0,0,9,15,0,0,0,0,2,0,0] >;
 

(C2xC8).2D4 in GAP, Magma, Sage, TeX 

(C_2\times C_8)._2D_4
 
% in TeX
 
G:=Group("(C2xC8).2D4");
 
// GroupNames label
 
G:=SmallGroup(128,749);
 
// by ID
 
G=gap.SmallGroup(128,749);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,2019,1018,248,2804,718,172,2028,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=d^2=1,c^4=b^4,a*b=b*a,c*a*c^-1=a*b^4,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d=a*b^5,d*c*d=b^4*c^3>;
 
// generators/relations
 

Export

Character table of (C2xC8).2D4 in TeX

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