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G = C2.7C2≀C4order 128 = 27

4th central stem extension by C2 of C2≀C4

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

Aliases: C2.7C2≀C4, (C2×C4).2Q16, (C2×C4).6SD16, C22.20C4≀C2, (C22×C4).34D4, (C22×Q8).3C4, C2.C42.6C4, C22.C42.9C2, C22.61(C23⋊C4), C2.5(C42.3C4), C23.161(C22⋊C4), C22.19(Q8⋊C4), C2.7(C23.31D4), C23.78C23.1C2, C22.M4(2).6C2, (C2×C4⋊C4).8C22, (C22×C4).8(C2×C4), SmallGroup(128,86)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C2.7C2≀C4
C1C2C22C23C22×C4C2×C4⋊C4C23.78C23 — C2.7C2≀C4
C1C2C23C22×C4 — C2.7C2≀C4
C1C22C23C2×C4⋊C4 — C2.7C2≀C4
C1C22C23C2×C4⋊C4 — C2.7C2≀C4

Generators and relations for C2.7C2≀C4
 G = < a,b,c,d,e,f | a2=d2=e2=1, b2=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: 180 in 70 conjugacy classes, 20 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C22 [×3], C22 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×14], Q8 [×4], C23, C4⋊C4 [×3], C2×C8 [×2], M4(2) [×3], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×3], C2.C42, C2.C42, C22⋊C8, C2×C4⋊C4, C2×C4⋊C4, C2×M4(2), C22×Q8, C22.M4(2), C22.C42, C23.78C23, C2.7C2≀C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, SD16, Q16, C23⋊C4, Q8⋊C4, C4≀C2, C23.31D4, C2≀C4, C42.3C4, C2.7C2≀C4

Character table of C2.7C2≀C4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H
 size 11112244448888888888888
ρ111111111111111111111111    trivial
ρ21111111111-1-11-1-1-111-111-1-1    linear of order 2
ρ3111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111111111-1-11-1-11-1-11-1-111    linear of order 2
ρ5111111-1-1-1-1-1111-1-iiii-i-i-ii    linear of order 4
ρ6111111-1-1-1-11-11-11iii-i-i-ii-i    linear of order 4
ρ7111111-1-1-1-1-1111-1i-i-i-iiii-i    linear of order 4
ρ8111111-1-1-1-11-11-11-i-i-iiii-ii    linear of order 4
ρ92222222-22-200-20000000000    orthogonal lifted from D4
ρ10222222-22-2200-20000000000    orthogonal lifted from D4
ρ112-22-22-2-202000000200200-2-2    symplectic lifted from Q16, Schur index 2
ρ122-22-22-2-202000000-200-20022    symplectic lifted from Q16, Schur index 2
ρ132-22-2-2202i0-2i0000001+i-1-i01-i-1+i00    complex lifted from C4≀C2
ρ142-22-2-220-2i02i0000001-i-1+i01+i-1-i00    complex lifted from C4≀C2
ρ152-22-2-220-2i02i000000-1+i1-i0-1-i1+i00    complex lifted from C4≀C2
ρ162-22-22-220-2000000--200-200-2--2    complex lifted from SD16
ρ172-22-22-220-2000000-200--200--2-2    complex lifted from SD16
ρ182-22-2-2202i0-2i000000-1-i1+i0-1+i1-i00    complex lifted from C4≀C2
ρ1944-4-4000000-2000200000000    orthogonal lifted from C2≀C4
ρ2044-4-40000002000-200000000    orthogonal lifted from C2≀C4
ρ214444-4-400000000000000000    orthogonal lifted from C23⋊C4
ρ224-4-44000000020-2000000000    symplectic lifted from C42.3C4, Schur index 2
ρ234-4-440000000-202000000000    symplectic lifted from C42.3C4, Schur index 2

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

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

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

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

Matrix representation of C2.7C2≀C4 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
030000
1100000
0001600
0016000
000010
000001
,
400000
0130000
0001600
0016000
000001
000010
,
100000
010000
0016000
0001600
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
800000
0150000
0000016
0000160
000100
0016000

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],[0,11,0,0,0,0,3,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[8,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0] >;

C2.7C2≀C4 in GAP, Magma, Sage, TeX

C_2._7C_2\wr C_4
% in TeX

G:=Group("C2.7C2wrC4");
// GroupNames label

G:=SmallGroup(128,86);
// by ID

G=gap.SmallGroup(128,86);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,568,422,387,520,1690,521,248,2804]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=d^2=e^2=1,b^2=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 C2.7C2≀C4 in TeX

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