Copied to
clipboard

G = (C2×C8).2D4  order 128 = 27

2nd non-split extension by C2×C8 of D4 acting faithfully

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

Aliases: (C2×C8).2D4, (C2×D4).95D4, (C2×Q8).86D4, C4.66C22≀C2, C4.58(C4⋊1D4), D8⋊C22⋊3C2, C4.10C42⋊4C2, C23.10(C4○D4), (C22×C8).64C22, C2.22(C23⋊2D4), C22.25(C4⋊D4), (C22×C4).715C23, (C2×M4(2)).20C22, (C2×C4).38(C2×D4), (C22×C8)⋊C2⋊1C2, (C2×C4○D4).52C22, SmallGroup(128,749)

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

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

Generators and relations for (C2×C8).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, C2×C4, C2×C4, D4, Q8, C23, C23, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C22⋊C8, C22×C8, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C4.10C42, (C22×C8)⋊C2, D8⋊C22, (C2×C8).2D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C4⋊1D4, C23⋊2D4, (C2×C8).2D4

Character table of (C2×C8).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 C4○D4
ρ2222222000-2-2-2-2-2000-2i2i2i-2i000000    complex lifted from C4○D4
ρ234-40000004i-4i0000002ζ872ζ852ζ82ζ83000000    complex faithful
ρ244-4000000-4i4i0000002ζ82ζ832ζ872ζ85000000    complex faithful
ρ254-4000000-4i4i0000002ζ852ζ872ζ832ζ8000000    complex faithful
ρ264-40000004i-4i0000002ζ832ζ82ζ852ζ87000000    complex faithful

Smallest permutation representation of (C2×C8).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 (C2×C8).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] >;
 

(C2×C8).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 (C2×C8).2D4 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁