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G = C2×C23⋊C4order 64 = 26

Direct product of C2 and C23⋊C4

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

Aliases: C2×C23⋊C4, C242C4, C23.12D4, C23.1C23, C24.10C22, (C2×D4)⋊5C4, C231(C2×C4), (C22×C4)⋊3C4, C22.8(C2×D4), (C22×D4).4C2, C22⋊C411C22, (C2×D4).42C22, C22.6(C22×C4), C22.29(C22⋊C4), (C2×C4)⋊1(C2×C4), (C2×C22⋊C4)⋊4C2, C2.12(C2×C22⋊C4), SmallGroup(64,90)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C23⋊C4
C1C2C22C23C24C22×D4 — C2×C23⋊C4
C1C2C22 — C2×C23⋊C4
C1C22C24 — C2×C23⋊C4
C1C2C23 — C2×C23⋊C4

Generators and relations for C2×C23⋊C4
 G = < a,b,c,d,e | a2=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 217 in 105 conjugacy classes, 41 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×6], C22 [×3], C22 [×4], C22 [×18], C2×C4 [×2], C2×C4 [×10], D4 [×8], C23 [×3], C23 [×6], C23 [×6], C22⋊C4 [×4], C22⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×4], C24 [×2], C23⋊C4 [×4], C2×C22⋊C4 [×2], C22×D4, C2×C23⋊C4
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, C2×C23⋊C4

Character table of C2×C23⋊C4

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J
 size 1111222222444444444444
ρ11111111111111111111111    trivial
ρ21111111111-1-1-1-1-1-111-1-111    linear of order 2
ρ31-11-1-1111-1-11-1-11-111-1-11-11    linear of order 2
ρ41-11-1-1111-1-1-111-11-11-11-1-11    linear of order 2
ρ5111111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ61111111111-1-111-1-1-1-111-1-1    linear of order 2
ρ71-11-1-1111-1-11-11-1-11-111-11-1    linear of order 2
ρ81-11-1-1111-1-1-11-111-1-11-111-1    linear of order 2
ρ91-11-11-11-11-11-1i-i1-1i-i-iii-i    linear of order 4
ρ101-11-11-11-11-11-1-ii1-1-iii-i-ii    linear of order 4
ρ111111-1-11-1-1111-i-i-1-1iiii-i-i    linear of order 4
ρ121111-1-11-1-1111ii-1-1-i-i-i-iii    linear of order 4
ρ131-11-11-11-11-1-11i-i-11-ii-ii-ii    linear of order 4
ρ141-11-11-11-11-1-11-ii-11i-ii-ii-i    linear of order 4
ρ151111-1-11-1-11-1-1-i-i11-i-iiiii    linear of order 4
ρ161111-1-11-1-11-1-1ii11ii-i-i-i-i    linear of order 4
ρ1722222-2-22-2-2000000000000    orthogonal lifted from D4
ρ182-22-222-2-2-22000000000000    orthogonal lifted from D4
ρ192-22-2-2-2-2222000000000000    orthogonal lifted from D4
ρ202222-22-2-22-2000000000000    orthogonal lifted from D4
ρ2144-4-4000000000000000000    orthogonal lifted from C23⋊C4
ρ224-4-44000000000000000000    orthogonal lifted from C23⋊C4

Permutation representations of C2×C23⋊C4
On 16 points - transitive group 16T76
Generators in S16
(1 3)(2 4)(5 7)(6 8)(9 16)(10 13)(11 14)(12 15)
(1 12)(2 10)(3 15)(4 13)(5 9)(6 11)(7 16)(8 14)
(1 5)(2 6)(3 7)(4 8)(9 12)(10 11)(13 14)(15 16)
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)
(5 6)(7 8)(9 10 11 12)(13 14 15 16)

G:=sub<Sym(16)| (1,3)(2,4)(5,7)(6,8)(9,16)(10,13)(11,14)(12,15), (1,12)(2,10)(3,15)(4,13)(5,9)(6,11)(7,16)(8,14), (1,5)(2,6)(3,7)(4,8)(9,12)(10,11)(13,14)(15,16), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16), (5,6)(7,8)(9,10,11,12)(13,14,15,16)>;

G:=Group( (1,3)(2,4)(5,7)(6,8)(9,16)(10,13)(11,14)(12,15), (1,12)(2,10)(3,15)(4,13)(5,9)(6,11)(7,16)(8,14), (1,5)(2,6)(3,7)(4,8)(9,12)(10,11)(13,14)(15,16), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16), (5,6)(7,8)(9,10,11,12)(13,14,15,16) );

G=PermutationGroup([(1,3),(2,4),(5,7),(6,8),(9,16),(10,13),(11,14),(12,15)], [(1,12),(2,10),(3,15),(4,13),(5,9),(6,11),(7,16),(8,14)], [(1,5),(2,6),(3,7),(4,8),(9,12),(10,11),(13,14),(15,16)], [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12),(13,15),(14,16)], [(5,6),(7,8),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,76);

On 16 points - transitive group 16T78
Generators in S16
(1 15)(2 16)(3 13)(4 14)(5 7)(6 8)(9 11)(10 12)
(1 12)(2 7)(3 8)(4 11)(5 16)(6 13)(9 14)(10 15)
(2 14)(4 16)(5 11)(7 9)
(1 13)(2 14)(3 15)(4 16)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

G:=sub<Sym(16)| (1,15)(2,16)(3,13)(4,14)(5,7)(6,8)(9,11)(10,12), (1,12)(2,7)(3,8)(4,11)(5,16)(6,13)(9,14)(10,15), (2,14)(4,16)(5,11)(7,9), (1,13)(2,14)(3,15)(4,16)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;

G:=Group( (1,15)(2,16)(3,13)(4,14)(5,7)(6,8)(9,11)(10,12), (1,12)(2,7)(3,8)(4,11)(5,16)(6,13)(9,14)(10,15), (2,14)(4,16)(5,11)(7,9), (1,13)(2,14)(3,15)(4,16)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );

G=PermutationGroup([(1,15),(2,16),(3,13),(4,14),(5,7),(6,8),(9,11),(10,12)], [(1,12),(2,7),(3,8),(4,11),(5,16),(6,13),(9,14),(10,15)], [(2,14),(4,16),(5,11),(7,9)], [(1,13),(2,14),(3,15),(4,16),(5,11),(6,12),(7,9),(8,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,78);

On 16 points - transitive group 16T92
Generators in S16
(1 8)(2 7)(3 5)(4 6)(9 14)(10 15)(11 16)(12 13)
(1 11)(2 10)(3 9)(4 12)(5 14)(6 13)(7 15)(8 16)
(2 4)(6 7)(10 12)(13 15)
(1 3)(2 4)(5 8)(6 7)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

G:=sub<Sym(16)| (1,8)(2,7)(3,5)(4,6)(9,14)(10,15)(11,16)(12,13), (1,11)(2,10)(3,9)(4,12)(5,14)(6,13)(7,15)(8,16), (2,4)(6,7)(10,12)(13,15), (1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)>;

G:=Group( (1,8)(2,7)(3,5)(4,6)(9,14)(10,15)(11,16)(12,13), (1,11)(2,10)(3,9)(4,12)(5,14)(6,13)(7,15)(8,16), (2,4)(6,7)(10,12)(13,15), (1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16) );

G=PermutationGroup([(1,8),(2,7),(3,5),(4,6),(9,14),(10,15),(11,16),(12,13)], [(1,11),(2,10),(3,9),(4,12),(5,14),(6,13),(7,15),(8,16)], [(2,4),(6,7),(10,12),(13,15)], [(1,3),(2,4),(5,8),(6,7),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,92);

On 16 points - transitive group 16T93
Generators in S16
(1 11)(2 12)(3 9)(4 10)(5 15)(6 16)(7 13)(8 14)
(1 5)(3 9)(4 14)(7 13)(8 10)(11 15)
(1 11)(2 6)(3 9)(4 8)(5 15)(7 13)(10 14)(12 16)
(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

G:=sub<Sym(16)| (1,11)(2,12)(3,9)(4,10)(5,15)(6,16)(7,13)(8,14), (1,5)(3,9)(4,14)(7,13)(8,10)(11,15), (1,11)(2,6)(3,9)(4,8)(5,15)(7,13)(10,14)(12,16), (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;

G:=Group( (1,11)(2,12)(3,9)(4,10)(5,15)(6,16)(7,13)(8,14), (1,5)(3,9)(4,14)(7,13)(8,10)(11,15), (1,11)(2,6)(3,9)(4,8)(5,15)(7,13)(10,14)(12,16), (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );

G=PermutationGroup([(1,11),(2,12),(3,9),(4,10),(5,15),(6,16),(7,13),(8,14)], [(1,5),(3,9),(4,14),(7,13),(8,10),(11,15)], [(1,11),(2,6),(3,9),(4,8),(5,15),(7,13),(10,14),(12,16)], [(1,15),(2,16),(3,13),(4,14),(5,11),(6,12),(7,9),(8,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,93);

On 16 points - transitive group 16T94
Generators in S16
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)
(1 6)(2 12)(3 9)(4 5)(7 15)(8 16)(10 13)(11 14)
(2 15)(4 13)(5 10)(7 12)
(1 14)(2 15)(3 16)(4 13)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

G:=sub<Sym(16)| (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,6)(2,12)(3,9)(4,5)(7,15)(8,16)(10,13)(11,14), (2,15)(4,13)(5,10)(7,12), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;

G:=Group( (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,6)(2,12)(3,9)(4,5)(7,15)(8,16)(10,13)(11,14), (2,15)(4,13)(5,10)(7,12), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );

G=PermutationGroup([(1,11),(2,12),(3,9),(4,10),(5,13),(6,14),(7,15),(8,16)], [(1,6),(2,12),(3,9),(4,5),(7,15),(8,16),(10,13),(11,14)], [(2,15),(4,13),(5,10),(7,12)], [(1,14),(2,15),(3,16),(4,13),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,94);

On 16 points - transitive group 16T102
Generators in S16
(1 7)(2 8)(3 5)(4 6)(9 15)(10 16)(11 13)(12 14)
(1 11)(2 14)(3 9)(4 16)(5 15)(6 10)(7 13)(8 12)
(1 5)(2 8)(3 7)(4 6)(9 13)(10 16)(11 15)(12 14)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

G:=sub<Sym(16)| (1,7)(2,8)(3,5)(4,6)(9,15)(10,16)(11,13)(12,14), (1,11)(2,14)(3,9)(4,16)(5,15)(6,10)(7,13)(8,12), (1,5)(2,8)(3,7)(4,6)(9,13)(10,16)(11,15)(12,14), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)>;

G:=Group( (1,7)(2,8)(3,5)(4,6)(9,15)(10,16)(11,13)(12,14), (1,11)(2,14)(3,9)(4,16)(5,15)(6,10)(7,13)(8,12), (1,5)(2,8)(3,7)(4,6)(9,13)(10,16)(11,15)(12,14), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16) );

G=PermutationGroup([(1,7),(2,8),(3,5),(4,6),(9,15),(10,16),(11,13),(12,14)], [(1,11),(2,14),(3,9),(4,16),(5,15),(6,10),(7,13),(8,12)], [(1,5),(2,8),(3,7),(4,6),(9,13),(10,16),(11,15),(12,14)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,102);

C2×C23⋊C4 is a maximal subgroup of
C24.5D4  C24.6D4  C25⋊C4  C24.165C23  2+ 1+42C4  C24.167C23  C24.C23  C24.22D4  C25.C22  C24.26D4  C24.78D4  C24.174C23  C24.28D4  C24.175C23  C24⋊D4  C24.31D4  C24⋊Q8  C24.33D4  C24.36D4  C24.39D4  C23.C24  C24⋊C23  (C2×D4)⋊7F5
C2×C23⋊C4 is a maximal quotient of
C23.8M4(2)  C25.3C4  (C2×C4)⋊M4(2)  C23⋊M4(2)  C23⋊C8⋊C2  C24.(C2×C4)  C24.45(C2×C4)  C24.53D4  C24.150D4  C24.54D4  C24.55D4  C24.56D4  C24.57D4  C24.58D4  C24.59D4  C24.60D4  C24.61D4  C25⋊C4  C24.167C23  C24.68D4  C24.78D4  C24.175C23  C24.176C23  C4○C2≀C4  C24.36D4  C2≀C4⋊C2  C23.(C2×D4)  C4⋊Q829C4  C24.39D4  C4.4D4⋊C4  C4⋊Q8⋊C4  (C2×D4).135D4  C4⋊Q8.C4  C41D4.C4  (C2×D4).137D4  (C2×D4)⋊7F5

Matrix representation of C2×C23⋊C4 in GL6(ℤ)

-100000
0-10000
001000
000100
000010
000001
,
-100000
010000
00-1000
000100
000010
00000-1
,
-100000
0-10000
00-1000
000-100
000010
000001
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
0-10000
100000
0000-10
00000-1
000-100
00-1000

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,-1,0,0,0,0,0,0,-1,0,0] >;

C2×C23⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes C_4
% in TeX

G:=Group("C2xC2^3:C4");
// GroupNames label

G:=SmallGroup(64,90);
// by ID

G=gap.SmallGroup(64,90);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,963,730]);
// Polycyclic

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

Export

Character table of C2×C23⋊C4 in TeX

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