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G = C2.C2≀C4  order 128 = 27

2nd central stem extension by C2 of C2≀C4

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C2.5C2≀C4, (C2×C4).3D8, (C2×C4).3SD16, C22.15C4≀C2, (C22×D4).3C4, (C22×C4).31D4, C2.C42.3C4, C22.C42⋊11C2, C22.56(C23⋊C4), C2.4(C42.C4), C23.10D4.1C2, C22.19(D4⋊C4), C23.156(C22⋊C4), C2.10(C22.SD16), C22.M4(2)⋊2C2, (C2×C4⋊C4).5C22, (C22×C4).3(C2×C4), SmallGroup(128,77)

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

Derived series
C1 — C22×C4 — C2.C2≀C4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.10D4 — C2.C2≀C4
Lower central
C1 — C2 — C23 — C22×C4 — C2.C2≀C4
Upper central
C1 — C22 — C23 — C2×C4⋊C4 — C2.C2≀C4
Jennings
C1 — C22 — C23 — C2×C4⋊C4 — C2.C2≀C4

Generators and relations for C2.C2≀C4
 G = < a,b,c,d,e,f | a2=b2=d2=e2=1, c2=f4=a, cbc-1=ab=ba, ac=ca, ad=da, ae=ea, af=fa, bd=db, be=eb, fbf-1=bcde, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >

Subgroups: 268 in 84 conjugacy classes, 20 normal (all characteristic)
C1, C2 [×3], C2 [×4], C4 [×5], C22 [×3], C22 [×12], C8 [×3], C2×C4 [×2], C2×C4 [×9], D4 [×4], C23, C23 [×8], C22⋊C4 [×4], C4⋊C4, C2×C8 [×2], M4(2) [×3], C22×C4 [×3], C22×C4, C2×D4 [×3], C24, C2.C42, C22⋊C8, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×M4(2), C22×D4, C22.M4(2), C22.C42, C23.10D4, C2.C2≀C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, C23⋊C4, D4⋊C4, C4≀C2, C22.SD16, C2≀C4, C42.C4, C2.C2≀C4

Character table of C2.C2≀C4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H
 size 11112288444488888888888
ρ111111111111111111111111    trivial
ρ2111111-1-111111-1-1-111-111-1-1    linear of order 2
ρ3111111-1-111111-1-11-1-11-1-111    linear of order 2
ρ4111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511111111-1-1-1-11-1-1-iiii-i-i-ii    linear of order 4
ρ6111111-1-1-1-1-1-1111iii-i-i-ii-i    linear of order 4
ρ711111111-1-1-1-11-1-1i-i-i-iiii-i    linear of order 4
ρ8111111-1-1-1-1-1-1111-i-i-iiii-ii    linear of order 4
ρ92222220022-2-2-20000000000    orthogonal lifted from D4
ρ1022222200-2-222-20000000000    orthogonal lifted from D4
ρ112-22-2-22002-200000-√200√200√2-√2    orthogonal lifted from D8
ρ122-22-2-22002-200000√200-√200-√2√2    orthogonal lifted from D8
ρ132-22-22-20000-2i2i00001+i-1-i01-i-1+i00    complex lifted from C4≀C2
ρ142-22-22-200002i-2i00001-i-1+i01+i-1-i00    complex lifted from C4≀C2
ρ152-22-22-200002i-2i0000-1+i1-i0-1-i1+i00    complex lifted from C4≀C2
ρ162-22-2-2200-2200000-√-200-√-200√-2√-2    complex lifted from SD16
ρ172-22-2-2200-2200000√-200√-200-√-2-√-2    complex lifted from SD16
ρ182-22-22-20000-2i2i0000-1-i1+i0-1+i1-i00    complex lifted from C4≀C2
ρ194-4-4400-22000000000000000    orthogonal lifted from C2≀C4
ρ204-4-44002-2000000000000000    orthogonal lifted from C2≀C4
ρ214444-4-400000000000000000    orthogonal lifted from C23⋊C4
ρ2244-4-4000000000-2i2i00000000    complex lifted from C42.C4
ρ2344-4-40000000002i-2i00000000    complex lifted from C42.C4

Smallest permutation representation of C2.C2≀C4
►On 32 points
Generators in S32
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)

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

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

(1 30)(3 32)(5 26)(7 28)(9 24)(11 18)(13 20)(15 22)

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

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

Matrix representation of C2.C2≀C4 ►in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
1600000
010000
001000
0001600
000010
0001601
,
020000
800000
001000
0001600
0000160
0001611
,
1600000
0160000
0016000
0001600
000010
00161601
,
100000
010000
0016000
0001600
0000160
0000016
,
1030000
570000
000010
00161612
000100
0001611

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,16,1,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,16,0,0,0,16,0,16,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[10,5,0,0,0,0,3,7,0,0,0,0,0,0,0,16,0,0,0,0,0,16,1,16,0,0,1,1,0,1,0,0,0,2,0,1] >;
 

C2.C2≀C4 in GAP, Magma, Sage, TeX 

C_2.C_2\wr C_4
 
% in TeX
 
G:=Group("C2.C2wrC4");
 
// GroupNames label
 
G:=SmallGroup(128,77);
 
// by ID
 
G=gap.SmallGroup(128,77);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,184,794,521,248,2804]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=d^2=e^2=1,c^2=f^4=a,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d*e,c*d=d*c,c*e=e*c,f*c*f^-1=c*d*e,f*d*f^-1=d*e=e*d,e*f=f*e>;
 
// generators/relations
 

Export

Character table of C2.C2≀C4 in TeX

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