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G = C24.4C23order 128 = 27

4th non-split extension by C24 of C23 acting faithfully

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

Aliases: C24.4C23, C23.9D4.2C2, C23.137(C4○D4), C22.13(C422C2), C2.2(C23.84C23), (C2×C22⋊C4).21C22, SmallGroup(128,836)

Series: Derived Chief Lower central Upper central Jennings

C1C24 — C24.4C23
C1C2C22C23C24C2×C22⋊C4C23.9D4 — C24.4C23
C1C2C24 — C24.4C23
C1C2C24 — C24.4C23
C1C2C24 — C24.4C23

Generators and relations for C24.4C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=abc, f2=ba=ab, g2=a, ac=ca, eae-1=ad=da, af=fa, ag=ga, bc=cb, ebe-1=gbg-1=bd=db, bf=fb, fcf-1=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef-1=ade, geg-1=be, gfg-1=cdf >

Subgroups: 264 in 89 conjugacy classes, 32 normal (4 characteristic)
C1, C2, C2 [×7], C4 [×7], C22 [×7], C22 [×7], C2×C4 [×14], C23 [×7], C23, C22⋊C4 [×14], C22×C4 [×7], C24, C2×C22⋊C4 [×7], C23.9D4 [×7], C24.4C23
Quotients: C1, C2 [×7], C22 [×7], C23, C4○D4 [×7], C422C2 [×7], C23.84C23, C24.4C23

Character table of C24.4C23

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N
 size 11222222288888888888888
ρ111111111111111111111111    trivial
ρ21111111111-1-1-11-1111-1-1-1-11    linear of order 2
ρ311111111111-11-1-1-11-1-11-11-1    linear of order 2
ρ41111111111-11-1-11-11-11-11-1-1    linear of order 2
ρ5111111111-1-111-1-11-1-1-1-1111    linear of order 2
ρ6111111111-11-1-1-111-1-111-1-11    linear of order 2
ρ7111111111-1-1-1111-1-111-1-11-1    linear of order 2
ρ8111111111-111-11-1-1-11-111-1-1    linear of order 2
ρ922-2-2-22-222000-2i000000002i0    complex lifted from C4○D4
ρ102222-2-2-2-220000-2i0002i00000    complex lifted from C4○D4
ρ11222-22-2-22-20-2i000000002i000    complex lifted from C4○D4
ρ12222-2-222-2-2002i00000000-2i00    complex lifted from C4○D4
ρ1322-22-2-222-22i000000-2i000000    complex lifted from C4○D4
ρ1422-2-2-22-2220002i00000000-2i0    complex lifted from C4○D4
ρ152222-2-2-2-2200002i000-2i00000    complex lifted from C4○D4
ρ1622-2222-2-2-20000002i000000-2i    complex lifted from C4○D4
ρ1722-22-2-222-2-2i0000002i000000    complex lifted from C4○D4
ρ18222-22-2-22-202i00000000-2i000    complex lifted from C4○D4
ρ1922-2-22-22-2200000-2i0002i0000    complex lifted from C4○D4
ρ2022-2222-2-2-2000000-2i0000002i    complex lifted from C4○D4
ρ2122-2-22-22-22000002i000-2i0000    complex lifted from C4○D4
ρ22222-2-222-2-200-2i000000002i00    complex lifted from C4○D4
ρ238-8000000000000000000000    orthogonal faithful

Permutation representations of C24.4C23
On 16 points - transitive group 16T372
Generators in S16
(2 8)(3 6)(9 11)(14 16)
(2 8)(4 5)(10 12)(14 16)
(3 6)(4 5)(13 15)(14 16)
(1 7)(2 8)(3 6)(4 5)(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 13)(2 16)(3 11 6 9)(4 12 5 10)(7 15)(8 14)
(1 12)(2 9 8 11)(3 16 6 14)(4 13)(5 15)(7 10)

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

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

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

G:=TransitiveGroup(16,372);

On 16 points - transitive group 16T383
Generators in S16
(2 16)(4 14)(6 9)(8 11)
(2 16)(4 14)(5 12)(7 10)
(1 3)(2 4)(5 10)(6 11)(7 12)(8 9)(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 7 12 10)(6 11 9 8)
(1 10)(2 11 16 8)(3 5)(4 6 14 9)(7 15)(12 13)

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

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

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

G:=TransitiveGroup(16,383);

Matrix representation of C24.4C23 in GL8(ℤ)

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

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

C24.4C23 in GAP, Magma, Sage, TeX

C_2^4._4C_2^3
% in TeX

G:=Group("C2^4.4C2^3");
// GroupNames label

G:=SmallGroup(128,836);
// by ID

G=gap.SmallGroup(128,836);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,392,141,176,422,387,58,2019,2804,718,2028]);
// Polycyclic

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

Export

Character table of C24.4C23 in TeX

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