Copied to
clipboard

G = C2×C423C4order 128 = 27

Direct product of C2 and C423C4

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

Aliases: C2×C423C4, C24.38D4, C425(C2×C4), (C2×C42)⋊10C4, (C22×Q8)⋊9C4, C23.8(C2×D4), (C2×D4).130D4, C4.4D420C4, (C2×D4).19C23, C23⋊C4.11C22, C22.51(C23⋊C4), C23.23(C22⋊C4), C4.4D4.120C22, (C22×D4).102C22, (C2×Q8)⋊3(C2×C4), C2.37(C2×C23⋊C4), (C2×D4).127(C2×C4), (C2×C23⋊C4).10C2, (C2×C4).94(C22×C4), (C22×C4).80(C2×C4), (C2×C4).50(C22⋊C4), (C2×C4.4D4).13C2, C22.61(C2×C22⋊C4), SmallGroup(128,857)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×C423C4
C1C2C22C23C2×D4C22×D4C2×C4.4D4 — C2×C423C4
C1C2C22C2×C4 — C2×C423C4
C1C22C23C22×D4 — C2×C423C4
C1C2C22C2×D4 — C2×C423C4

Generators and relations for C2×C423C4
 G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=b2c-1 >

Subgroups: 388 in 142 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×10], C22 [×3], C22 [×16], C2×C4 [×2], C2×C4 [×18], D4 [×4], Q8 [×4], C23, C23 [×4], C23 [×8], C42 [×2], C42, C22⋊C4 [×14], C22×C4, C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8 [×3], C24 [×2], C23⋊C4 [×4], C23⋊C4 [×2], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×4], C4.4D4 [×2], C22×D4, C22×Q8, C423C4 [×4], C2×C23⋊C4 [×2], C2×C4.4D4, C2×C423C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C423C4 [×2], C2×C23⋊C4, C2×C423C4

Character table of C2×C423C4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 11112244444444448888888888
ρ111111111111111111111111111    trivial
ρ21-1-11-11-1-111-1-11-111-11-11-111-11-1    linear of order 2
ρ31-1-11-11-1-1111-1-111-1-1-111-11-11-11    linear of order 2
ρ41111111111-11-1-11-11-1-1111-1-1-1-1    linear of order 2
ρ51111111111-11-1-11-1-11-1-1-1-1-1111    linear of order 2
ρ61-1-11-11-1-1111-1-111-1111-11-1-1-11-1    linear of order 2
ρ71-1-11-11-1-111-1-11-1111-1-1-11-111-11    linear of order 2
ρ81111111111111111-1-11-1-1-11-1-1-1    linear of order 2
ρ91-1-11-1111-1-1-1-11-111ii1-i-ii-1i-i-i    linear of order 4
ρ10111111-1-1-1-1111111-ii-1-iii-1-i-ii    linear of order 4
ρ11111111-1-1-1-1111111i-i-1i-i-i-1ii-i    linear of order 4
ρ121-1-11-1111-1-1-1-11-111-i-i1ii-i-1-iii    linear of order 4
ρ131-1-11-1111-1-11-1-111-1-ii-1ii-i1i-i-i    linear of order 4
ρ14111111-1-1-1-1-11-1-11-1ii1i-i-i1-i-ii    linear of order 4
ρ15111111-1-1-1-1-11-1-11-1-i-i1-iii1ii-i    linear of order 4
ρ161-1-11-1111-1-11-1-111-1i-i-1-i-ii1-iii    linear of order 4
ρ17222222-222-20-200-200000000000    orthogonal lifted from D4
ρ182-2-22-222-22-20200-200000000000    orthogonal lifted from D4
ρ192-2-22-22-22-220200-200000000000    orthogonal lifted from D4
ρ202222222-2-220-200-200000000000    orthogonal lifted from D4
ρ214-4-444-400000000000000000000    orthogonal lifted from C23⋊C4
ρ224444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ234-44-40000002i0-2i-2i02i0000000000    complex lifted from C423C4
ρ2444-4-4000000-2i0-2i2i02i0000000000    complex lifted from C423C4
ρ2544-4-40000002i02i-2i0-2i0000000000    complex lifted from C423C4
ρ264-44-4000000-2i02i2i0-2i0000000000    complex lifted from C423C4

Smallest permutation representation of C2×C423C4
On 32 points
Generators in S32
(1 7)(2 8)(3 5)(4 6)(9 26)(10 27)(11 28)(12 25)(13 22)(14 23)(15 24)(16 21)(17 31)(18 32)(19 29)(20 30)
(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 27 30 14)(2 28 31 15)(3 25 32 16)(4 26 29 13)(5 12 18 21)(6 9 19 22)(7 10 20 23)(8 11 17 24)
(1 9 14 8)(2 7 26 23)(3 24 16 19)(4 18 28 12)(5 15 21 29)(6 32 11 25)(10 31 20 13)(17 30 22 27)

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

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

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

Matrix representation of C2×C423C4 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
040000
400000
002322
000300
003230
003203
,
400000
040000
004030
000041
001010
001410
,
020000
300000
002222
002022
000303
003003

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,2,0,3,3,0,0,3,3,2,2,0,0,2,0,3,0,0,0,2,0,0,3],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,1,1,0,0,0,0,0,4,0,0,3,4,1,1,0,0,0,1,0,0],[0,3,0,0,0,0,2,0,0,0,0,0,0,0,2,2,0,3,0,0,2,0,3,0,0,0,2,2,0,0,0,0,2,2,3,3] >;

C2×C423C4 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes_3C_4
% in TeX

G:=Group("C2xC4^2:3C4");
// GroupNames label

G:=SmallGroup(128,857);
// by ID

G=gap.SmallGroup(128,857);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,1123,1018,248,1971,375,4037]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations

Export

Character table of C2×C423C4 in TeX

׿
×
𝔽