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G = C2×C23.D4order 128 = 27

Direct product of C2 and C23.D4

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

Aliases: C2×C23.D4, C24.35D4, (C23×C4)⋊5C4, C23.2(C2×D4), (C2×D4).124D4, C24.31(C2×C4), (C22×C4).89D4, (C2×D4).13C23, C22.D45C4, C23⋊C4.6C22, C23.52(C22×C4), C4.D4.7C22, C22.49(C23⋊C4), (C22×D4).99C22, C23.202(C22⋊C4), C22.D4.23C22, (C2×C4).2(C2×D4), (C2×C22⋊C4)⋊8C4, C22⋊C42(C2×C4), (C22×C4)⋊6(C2×C4), (C2×C23⋊C4).9C2, C2.31(C2×C23⋊C4), (C2×D4).122(C2×C4), (C2×C4).24(C22⋊C4), (C2×C4.D4).13C2, C22.55(C2×C22⋊C4), (C2×C22.D4).4C2, SmallGroup(128,851)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C2×C23.D4
C1C2C22C23C2×D4C22×D4C2×C22.D4 — C2×C23.D4
C1C2C22C23 — C2×C23.D4
C1C22C23C22×D4 — C2×C23.D4
C1C2C22C2×D4 — C2×C23.D4

Generators and relations for C2×C23.D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=bcd, bf=fb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=be3 >

Subgroups: 356 in 139 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C22 [×3], C22 [×17], C8 [×2], C2×C4 [×2], C2×C4 [×18], D4 [×4], C23 [×5], C23 [×8], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8, M4(2) [×3], C22×C4, C22×C4 [×2], C22×C4 [×6], C2×D4 [×4], C2×D4 [×2], C24 [×2], C23⋊C4 [×2], C23⋊C4, C4.D4 [×2], C4.D4, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, C22.D4 [×4], C22.D4 [×2], C2×M4(2), C23×C4, C22×D4, C23.D4 [×4], C2×C23⋊C4, C2×C4.D4, C2×C22.D4, C2×C23.D4
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, C23.D4 [×2], C2×C23⋊C4, C2×C23.D4

Character table of C2×C23.D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D
 size 11112244444444448888888888
ρ111111111111111111111111111    trivial
ρ21111111111-1-1-11-111111-1-1-1-1-1-1    linear of order 2
ρ31111111111-1-1-11-11-1-1-1-1-1-11111    linear of order 2
ρ41111111111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ51-11-1-11-1-111-11-1-111-11-111-1-11-11    linear of order 2
ρ61-11-1-11-1-1111-11-1-11-11-11-111-11-1    linear of order 2
ρ71-11-1-11-1-1111-11-1-111-11-1-11-11-11    linear of order 2
ρ81-11-1-11-1-111-11-1-1111-11-11-11-11-1    linear of order 2
ρ91-11-1-111-1-11-11-111-1i-i-ii-11i-i-ii    linear of order 4
ρ101-11-1-111-1-111-111-1-1i-i-ii1-1-iii-i    linear of order 4
ρ11111111-11-11-1-1-1-1-1-1ii-i-i11-i-iii    linear of order 4
ρ12111111-11-11111-11-1ii-i-i-1-1ii-i-i    linear of order 4
ρ13111111-11-11111-11-1-i-iii-1-1-i-iii    linear of order 4
ρ14111111-11-11-1-1-1-1-1-1-i-iii11ii-i-i    linear of order 4
ρ151-11-1-111-1-111-111-1-1-iii-i1-1i-i-ii    linear of order 4
ρ161-11-1-111-1-11-11-111-1-iii-i-11-iii-i    linear of order 4
ρ172222222-22-2000-20-20000000000    orthogonal lifted from D4
ρ18222222-2-2-2-20002020000000000    orthogonal lifted from D4
ρ192-22-2-22-222-200020-20000000000    orthogonal lifted from D4
ρ202-22-2-2222-2-2000-2020000000000    orthogonal lifted from D4
ρ214-44-44-400000000000000000000    orthogonal lifted from C23⋊C4
ρ224444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ2344-4-4000000-2i-2i2i02i00000000000    complex lifted from C23.D4
ρ244-4-44000000-2i2i2i0-2i00000000000    complex lifted from C23.D4
ρ2544-4-40000002i2i-2i0-2i00000000000    complex lifted from C23.D4
ρ264-4-440000002i-2i-2i02i00000000000    complex lifted from C23.D4

Smallest permutation representation of C2×C23.D4
On 32 points
Generators in S32
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(2 26)(3 7)(4 32)(6 30)(8 28)(9 23)(10 14)(11 21)(13 19)(15 17)(20 24)(27 31)
(1 29)(2 26)(3 31)(4 28)(5 25)(6 30)(7 27)(8 32)(9 23)(10 20)(11 17)(12 22)(13 19)(14 24)(15 21)(16 18)
(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 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 16)(2 21 26 11)(3 20 7 24)(4 9 32 23)(5 12)(6 17 30 15)(8 13 28 19)(10 31 14 27)(18 25)(22 29)

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

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

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

Matrix representation of C2×C23.D4 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
100000
010000
001000
0001600
000010
0000016
,
100000
010000
0016000
0001600
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
010000
1600000
0000013
0000130
0013000
000400
,
010000
100000
0000160
0000016
0016000
000100

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[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],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,13,0,0,0,0,13,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,16,0,0] >;

C2×C23.D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3.D_4
% in TeX

G:=Group("C2xC2^3.D4");
// GroupNames label

G:=SmallGroup(128,851);
// by ID

G=gap.SmallGroup(128,851);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,1123,851,375,4037]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,b*f=f*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*e^3>;
// generators/relations

Export

Character table of C2×C23.D4 in TeX

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