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G = C24.C23  order 128 = 27

1st non-split extension by C24 of C23 acting faithfully

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

Aliases: C24.1C23, C24.5(C2×C4), C42⋊C2⋊5C4, (C2×D4).269D4, C23.9D4⋊3C2, C23.126(C2×D4), C23.64(C22×C4), C23.115(C4○D4), C23.34(C22⋊C4), C22.11C24.1C2, (C22×D4).11C22, C22.6(C42⋊C2), C2.16(C23.34D4), C22.2(C22.D4), (C2×C22⋊C4)⋊3C4, (C2×C23⋊C4).4C2, C22⋊C4.49(C2×C4), (C22×C4).17(C2×C4), (C2×C4).47(C22⋊C4), (C2×C22⋊C4).8C22, C22.36(C2×C22⋊C4), SmallGroup(128,560)

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

Derived series
C1 — C23 — C24.C23
Chief series
C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4 — C22.11C24 — C24.C23
Lower central
C1 — C2 — C23 — C24.C23
Upper central
C1 — C2 — C24 — C24.C23
Jennings
C1 — C2 — C24 — C24.C23

Generators and relations for C24.C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=c, g2=a, ab=ba, ac=ca, eae-1=ad=da, af=fa, ag=ga, bc=cb, gbg-1=bd=db, be=eb, bf=fb, fcf=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef=ade, geg-1=bde, fg=gf >

Subgroups: 380 in 148 conjugacy classes, 50 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C23⋊C4, C2×C22⋊C4, C42⋊C2, C4×D4, C22×D4, C23.9D4, C2×C23⋊C4, C22.11C24, C24.C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C42⋊C2, C22.D4, C23.34D4, C24.C23

Character table of C24.C23

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R
 size 11222222244444444444488888888
ρ111111111111111111111111111111    trivial
ρ2111111111-1-1111-1-1-1-1-1-11-111-1-111-1    linear of order 2
ρ311111111111-1-1-11-11-1-1-1-1-11-11-11-11    linear of order 2
ρ4111111111-1-1-1-1-1-11-1111-111-1-111-1-1    linear of order 2
ρ511111111111-1-1-11-11-1-1-1-11-11-11-11-1    linear of order 2
ρ6111111111-1-1-1-1-1-11-1111-1-1-111-1-111    linear of order 2
ρ7111111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ8111111111-1-1111-1-1-1-1-1-111-1-111-1-11    linear of order 2
ρ911-1-11-11-11-1-111-11-111-11-1-iiiii-i-i-i    linear of order 4
ρ1011-1-11-11-11-1-1-1-11111-11-11ii-ii-i-ii-i    linear of order 4
ρ1111-1-11-11-1111-1-11-1-1-11-111-ii-i-ii-iii    linear of order 4
ρ1211-1-11-11-111111-1-11-1-11-1-1iii-i-i-i-ii    linear of order 4
ρ1311-1-11-11-11-1-1-1-11111-11-11-i-ii-iii-ii    linear of order 4
ρ1411-1-11-11-11-1-111-11-111-11-1i-i-i-i-iiii    linear of order 4
ρ1511-1-11-11-111111-1-11-1-11-1-1-i-i-iiiii-i    linear of order 4
ρ1611-1-11-11-1111-1-11-1-1-11-111i-iii-ii-i-i    linear of order 4
ρ1722-22-2-222-2-22000-202000000000000    orthogonal lifted from D4
ρ18222-2-222-2-22-2000-202000000000000    orthogonal lifted from D4
ρ19222-2-222-2-2-2200020-2000000000000    orthogonal lifted from D4
ρ2022-22-2-222-22-200020-2000000000000    orthogonal lifted from D4
ρ2122-2222-2-2-200-2i2i2i000000-2i00000000    complex lifted from C4○D4
ρ22222-22-2-22-200-2i2i-2i0000002i00000000    complex lifted from C4○D4
ρ23222-22-2-22-2002i-2i2i000000-2i00000000    complex lifted from C4○D4
ρ242222-2-2-2-22000000-2i0-2i2i2i000000000    complex lifted from C4○D4
ρ2522-2-2-22-222000000-2i02i2i-2i000000000    complex lifted from C4○D4
ρ2622-2222-2-2-2002i-2i-2i0000002i00000000    complex lifted from C4○D4
ρ272222-2-2-2-220000002i02i-2i-2i000000000    complex lifted from C4○D4
ρ2822-2-2-22-2220000002i0-2i-2i2i000000000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    orthogonal faithful

Permutation representations of C24.C23
►On 16 points - transitive group 16T212
Generators in S16
(2 4)(5 8)(10 12)(14 16)

(5 8)(6 7)(13 15)(14 16)

(9 11)(10 12)(13 15)(14 16)

(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)

(1 11)(2 10)(3 9)(4 12)(5 16)(6 13)(7 15)(8 14)

(1 7)(2 5 4 8)(3 6)(9 13)(10 16 12 14)(11 15)
 
G:=sub<Sym(16)| (2,4)(5,8)(10,12)(14,16), (5,8)(6,7)(13,15)(14,16), (9,11)(10,12)(13,15)(14,16), (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), (1,11)(2,10)(3,9)(4,12)(5,16)(6,13)(7,15)(8,14), (1,7)(2,5,4,8)(3,6)(9,13)(10,16,12,14)(11,15)>;
 
G:=Group( (2,4)(5,8)(10,12)(14,16), (5,8)(6,7)(13,15)(14,16), (9,11)(10,12)(13,15)(14,16), (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), (1,11)(2,10)(3,9)(4,12)(5,16)(6,13)(7,15)(8,14), (1,7)(2,5,4,8)(3,6)(9,13)(10,16,12,14)(11,15) );
 
G=PermutationGroup([[(2,4),(5,8),(10,12),(14,16)], [(5,8),(6,7),(13,15),(14,16)], [(9,11),(10,12),(13,15),(14,16)], [(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)], [(1,11),(2,10),(3,9),(4,12),(5,16),(6,13),(7,15),(8,14)], [(1,7),(2,5,4,8),(3,6),(9,13),(10,16,12,14),(11,15)]])
 
G:=TransitiveGroup(16,212);
 

►On 16 points - transitive group 16T229
Generators in S16
(1 8)(2 7)(3 6)(4 5)(9 10)(11 12)(13 14)(15 16)

(1 2)(7 8)(13 15)(14 16)

(9 11)(10 12)(13 15)(14 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)

(1 15)(2 13)(3 11)(4 9)(5 10)(6 12)(7 14)(8 16)

(1 3 8 6)(2 4 7 5)(9 14 10 13)(11 16 12 15)
 
G:=sub<Sym(16)| (1,8)(2,7)(3,6)(4,5)(9,10)(11,12)(13,14)(15,16), (1,2)(7,8)(13,15)(14,16), (9,11)(10,12)(13,15)(14,16), (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), (1,15)(2,13)(3,11)(4,9)(5,10)(6,12)(7,14)(8,16), (1,3,8,6)(2,4,7,5)(9,14,10,13)(11,16,12,15)>;
 
G:=Group( (1,8)(2,7)(3,6)(4,5)(9,10)(11,12)(13,14)(15,16), (1,2)(7,8)(13,15)(14,16), (9,11)(10,12)(13,15)(14,16), (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), (1,15)(2,13)(3,11)(4,9)(5,10)(6,12)(7,14)(8,16), (1,3,8,6)(2,4,7,5)(9,14,10,13)(11,16,12,15) );
 
G=PermutationGroup([[(1,8),(2,7),(3,6),(4,5),(9,10),(11,12),(13,14),(15,16)], [(1,2),(7,8),(13,15),(14,16)], [(9,11),(10,12),(13,15),(14,16)], [(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)], [(1,15),(2,13),(3,11),(4,9),(5,10),(6,12),(7,14),(8,16)], [(1,3,8,6),(2,4,7,5),(9,14,10,13),(11,16,12,15)]])
 
G:=TransitiveGroup(16,229);
 

►On 16 points - transitive group 16T231
Generators in S16
(2 16)(4 14)(6 9)(8 11)

(5 12)(6 9)(7 10)(8 11)

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)

(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(2 16)(3 13)(5 12)(8 11)

(1 7)(2 11 16 8)(3 5)(4 9 14 6)(10 15)(12 13)
 
G:=sub<Sym(16)| (2,16)(4,14)(6,9)(8,11), (5,12)(6,9)(7,10)(8,11), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (2,16)(3,13)(5,12)(8,11), (1,7)(2,11,16,8)(3,5)(4,9,14,6)(10,15)(12,13)>;
 
G:=Group( (2,16)(4,14)(6,9)(8,11), (5,12)(6,9)(7,10)(8,11), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (2,16)(3,13)(5,12)(8,11), (1,7)(2,11,16,8)(3,5)(4,9,14,6)(10,15)(12,13) );
 
G=PermutationGroup([[(2,16),(4,14),(6,9),(8,11)], [(5,12),(6,9),(7,10),(8,11)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(2,16),(3,13),(5,12),(8,11)], [(1,7),(2,11,16,8),(3,5),(4,9,14,6),(10,15),(12,13)]])
 
G:=TransitiveGroup(16,231);
 

►On 16 points - transitive group 16T286
Generators in S16
(2 16)(4 14)(6 9)(8 11)

(1 7)(2 8)(3 5)(4 6)(9 14)(10 15)(11 16)(12 13)

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)

(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 7)(2 11)(3 12)(4 6)(5 13)(8 16)(9 14)(10 15)

(1 3)(2 6 16 9)(4 8 14 11)(5 10)(7 12)(13 15)
 
G:=sub<Sym(16)| (2,16)(4,14)(6,9)(8,11), (1,7)(2,8)(3,5)(4,6)(9,14)(10,15)(11,16)(12,13), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,7)(2,11)(3,12)(4,6)(5,13)(8,16)(9,14)(10,15), (1,3)(2,6,16,9)(4,8,14,11)(5,10)(7,12)(13,15)>;
 
G:=Group( (2,16)(4,14)(6,9)(8,11), (1,7)(2,8)(3,5)(4,6)(9,14)(10,15)(11,16)(12,13), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,7)(2,11)(3,12)(4,6)(5,13)(8,16)(9,14)(10,15), (1,3)(2,6,16,9)(4,8,14,11)(5,10)(7,12)(13,15) );
 
G=PermutationGroup([[(2,16),(4,14),(6,9),(8,11)], [(1,7),(2,8),(3,5),(4,6),(9,14),(10,15),(11,16),(12,13)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,7),(2,11),(3,12),(4,6),(5,13),(8,16),(9,14),(10,15)], [(1,3),(2,6,16,9),(4,8,14,11),(5,10),(7,12),(13,15)]])
 
G:=TransitiveGroup(16,286);
 

►On 16 points - transitive group 16T316
Generators in S16
(2 3)(5 8)(9 11)(13 15)

(1 7)(2 8)(3 5)(4 6)(9 13)(10 14)(11 15)(12 16)

(9 11)(10 12)(13 15)(14 16)

(1 4)(2 3)(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)

(1 16)(2 15)(3 13)(4 14)(5 9)(6 10)(7 12)(8 11)

(2 8 3 5)(6 7)(9 15 11 13)(10 12)
 
G:=sub<Sym(16)| (2,3)(5,8)(9,11)(13,15), (1,7)(2,8)(3,5)(4,6)(9,13)(10,14)(11,15)(12,16), (9,11)(10,12)(13,15)(14,16), (1,4)(2,3)(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), (1,16)(2,15)(3,13)(4,14)(5,9)(6,10)(7,12)(8,11), (2,8,3,5)(6,7)(9,15,11,13)(10,12)>;
 
G:=Group( (2,3)(5,8)(9,11)(13,15), (1,7)(2,8)(3,5)(4,6)(9,13)(10,14)(11,15)(12,16), (9,11)(10,12)(13,15)(14,16), (1,4)(2,3)(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), (1,16)(2,15)(3,13)(4,14)(5,9)(6,10)(7,12)(8,11), (2,8,3,5)(6,7)(9,15,11,13)(10,12) );
 
G=PermutationGroup([[(2,3),(5,8),(9,11),(13,15)], [(1,7),(2,8),(3,5),(4,6),(9,13),(10,14),(11,15),(12,16)], [(9,11),(10,12),(13,15),(14,16)], [(1,4),(2,3),(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)], [(1,16),(2,15),(3,13),(4,14),(5,9),(6,10),(7,12),(8,11)], [(2,8,3,5),(6,7),(9,15,11,13),(10,12)]])
 
G:=TransitiveGroup(16,316);
 

Matrix representation of C24.C23 ►in GL8(ℤ)

10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100
,
01000000
10000000
00010000
00100000
00000100
00001000
00000001
00000010
,
-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
00001000
00000100
00000010
00000001
01000000
10000000
00010000
00100000
,
10000000
0-1000000
00100000
000-10000
00001000
00000-100
00000010
0000000-1
,
10000000
01000000
00-100000
000-10000
000000-10
0000000-1
00001000
00000100

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0] >;
 

C24.C23 in GAP, Magma, Sage, TeX 

C_2^4.C_2^3
 
% in TeX
 
G:=Group("C2^4.C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,560);
 
// by ID
 
G=gap.SmallGroup(128,560);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,58,2019,718,2028]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=c,g^2=a,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,a*f=f*a,a*g=g*a,b*c=c*b,g*b*g^-1=b*d=d*b,b*e=e*b,b*f=f*b,f*c*f=c*d=d*c,c*e=e*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,f*e*f=a*d*e,g*e*g^-1=b*d*e,f*g=g*f>;
 
// generators/relations
 

Export

Character table of C24.C23 in TeX

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