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G = C23.9C24order 128 = 27

9th non-split extension by C23 of C24 acting via C24/C22=C22

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

Aliases: C23.9C24, C24.130C23, 2+ 1+4.15C22, (C2×D4)⋊27D4, (C2×Q8)⋊21D4, (C22×C4)⋊8D4, C2≀C225C2, C4.47C22≀C2, C23⋊C49C22, C23.28(C2×D4), C22≀C22C22, C2.C254C2, (C2×D4).43C23, C22⋊C4.2C23, C22.29C247C2, (C22×D4)⋊22C22, C22.43(C22×D4), C42⋊C213C22, (C22×C4).285C23, C23.C2319C2, (C2×C4).29(C2×D4), C2.64(C2×C22≀C2), (C2×C4○D4).112C22, SmallGroup(128,1759)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.9C24
C1C2C22C23C22×C4C2×C4○D4C2.C25 — C23.9C24
C1C2C23 — C23.9C24
C1C2C22×C4 — C23.9C24
C1C2C23 — C23.9C24

Generators and relations for C23.9C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=c, ab=ba, faf=ac=ca, ede=ad=da, ae=ea, ag=ga, ebe=bc=cb, fdf=bd=db, bf=fb, bg=gb, gdg-1=cd=dc, ce=ec, cf=fc, cg=gc, ef=fe, eg=ge, fg=gf >

Subgroups: 860 in 386 conjugacy classes, 106 normal (8 characteristic)
C1, C2, C2 [×11], C4, C4 [×3], C4 [×12], C22 [×3], C22 [×31], C2×C4 [×12], C2×C4 [×30], D4 [×46], Q8 [×10], C23, C23 [×6], C23 [×12], C42 [×3], C22⋊C4 [×6], C22⋊C4 [×12], C4⋊C4 [×3], C22×C4, C22×C4 [×6], C22×C4 [×4], C2×D4 [×9], C2×D4 [×36], C2×Q8 [×3], C2×Q8 [×6], C4○D4 [×40], C24 [×2], C23⋊C4 [×12], C42⋊C2 [×3], C22≀C2 [×12], C4⋊D4 [×6], C4.4D4 [×3], C41D4 [×3], C22×D4, C2×C4○D4 [×3], C2×C4○D4 [×6], 2+ 1+4 [×4], 2+ 1+4 [×3], 2- 1+4 [×3], C23.C23 [×3], C2≀C22 [×8], C22.29C24 [×3], C2.C25, C23.9C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, C23.9C24

Character table of C23.9C24

 class 12A2B2C2D2E2F2G2H2I2J2K2L4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 11222444444882222444444888888
ρ111111111111111111111111111111    trivial
ρ211111111111-1-11111111111-1-1-1-1-1-1    linear of order 2
ρ311111-1-111-11-11-1-1-1-1-11-11-111-11-1-11    linear of order 2
ρ411111-1-111-111-1-1-1-1-1-11-11-11-11-111-1    linear of order 2
ρ511111-11-1-1-11111111-1-1-1-111-1-11-11-1    linear of order 2
ρ611111-11-1-1-11-1-11111-1-1-1-11111-11-11    linear of order 2
ρ7111111-1-1-111-11-1-1-1-11-11-1-11-1111-1-1    linear of order 2
ρ8111111-1-1-1111-1-1-1-1-11-11-1-111-1-1-111    linear of order 2
ρ9111111-1-11-1-1-1-11111-111-1-1-11-1111-1    linear of order 2
ρ10111111-1-11-1-1111111-111-1-1-1-11-1-1-11    linear of order 2
ρ1111111-11-111-11-1-1-1-1-111-1-11-1111-1-1-1    linear of order 2
ρ1211111-11-111-1-11-1-1-1-111-1-11-1-1-1-1111    linear of order 2
ρ1311111-1-11-11-1-1-111111-1-11-1-1-111-111    linear of order 2
ρ1411111-1-11-11-11111111-1-11-1-11-1-11-1-1    linear of order 2
ρ1511111111-1-1-11-1-1-1-1-1-1-1111-1-1-111-11    linear of order 2
ρ1611111111-1-1-1-11-1-1-1-1-1-1111-111-1-11-1    linear of order 2
ρ1722-2-2200-20200022-2-2-200200000000    orthogonal lifted from D4
ρ18222-2-202000200-22-220000-2-2000000    orthogonal lifted from D4
ρ1922-22-220020000-222-20-2-2000000000    orthogonal lifted from D4
ρ2022-2-2200-20-2000-2-222200200000000    orthogonal lifted from D4
ρ21222-2-20-2000-200-22-22000022000000    orthogonal lifted from D4
ρ2222-22-2-200200002-2-220-22000000000    orthogonal lifted from D4
ρ23222-2-202000-2002-22-20000-22000000    orthogonal lifted from D4
ρ2422-2-2200202000-2-222-200-200000000    orthogonal lifted from D4
ρ2522-22-2200-200002-2-2202-2000000000    orthogonal lifted from D4
ρ26222-2-20-20002002-22-200002-2000000    orthogonal lifted from D4
ρ2722-22-2-200-20000-222-2022000000000    orthogonal lifted from D4
ρ2822-2-220020-200022-2-2200-200000000    orthogonal lifted from D4
ρ298-8000000000000000000000000000    orthogonal faithful

Permutation representations of C23.9C24
On 16 points - transitive group 16T239
Generators in S16
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)
(5 7)(6 8)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(2 4)(5 16)(6 15)(7 14)(8 13)(10 12)
(1 16)(2 13)(3 14)(4 15)(5 11)(6 12)(7 9)(8 10)
(5 7)(6 8)(9 11)(10 12)
(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)(5,16)(6,13)(7,14)(8,15), (5,7)(6,8)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (2,4)(5,16)(6,15)(7,14)(8,13)(10,12), (1,16)(2,13)(3,14)(4,15)(5,11)(6,12)(7,9)(8,10), (5,7)(6,8)(9,11)(10,12), (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)(5,16)(6,13)(7,14)(8,15), (5,7)(6,8)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (2,4)(5,16)(6,15)(7,14)(8,13)(10,12), (1,16)(2,13)(3,14)(4,15)(5,11)(6,12)(7,9)(8,10), (5,7)(6,8)(9,11)(10,12), (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),(5,16),(6,13),(7,14),(8,15)], [(5,7),(6,8),(13,15),(14,16)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(2,4),(5,16),(6,15),(7,14),(8,13),(10,12)], [(1,16),(2,13),(3,14),(4,15),(5,11),(6,12),(7,9),(8,10)], [(5,7),(6,8),(9,11),(10,12)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,239);

On 16 points - transitive group 16T313
Generators in S16
(5 7)(6 8)(13 15)(14 16)
(9 11)(10 12)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(2 4)(6 8)(10 12)(13 15)
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)
(1 7)(2 8)(3 5)(4 6)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

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

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

G:=TransitiveGroup(16,313);

Matrix representation of C23.9C24 in GL8(ℤ)

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

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],[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],[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,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[0,0,0,0,0,0,0,1,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,-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] >;

C23.9C24 in GAP, Magma, Sage, TeX

C_2^3._9C_2^4
% in TeX

G:=Group("C2^3.9C2^4");
// GroupNames label

G:=SmallGroup(128,1759);
// by ID

G=gap.SmallGroup(128,1759);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2019,248,718,2028]);
// Polycyclic

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

Export

Character table of C23.9C24 in TeX

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