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G = C23.C24  order 128 = 27

3rd non-split extension by C23 of C24 acting via C24/C22=C22

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

Aliases: C23.3C24, C24.129C23, C24:4(C2xC4), D4o(C23:C4), (C2xD4).290D4, (C22xD4):13C4, C23:C4:12C22, C23.228(C2xD4), C22.11C24:6C2, D4.16(C22:C4), C42:C2:3C22, (C2xD4).351C23, C22:C4.67C23, C22.12(C23xC4), C23.58(C22xC4), C22.25(C22xD4), (C22xC4).271C23, (C2x2+ 1+4).4C2, C23.C23:16C2, (C22xD4).318C22, (C2xC4oD4):9C4, (C2xD4):45(C2xC4), (C22xC4):4(C2xC4), (C2xQ8):36(C2xC4), (C2xC23:C4):15C2, (C2xC4).441(C2xD4), C4.29(C2xC22:C4), (C2xC22:C4):7C22, C22.3(C2xC22:C4), (C2xC4).106(C22xC4), (C2xC4oD4).80C22, C2.26(C22xC22:C4), SmallGroup(128,1615)

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

Derived series
C1 — C22 — C23.C24
Chief series
C1 — C2 — C22 — C23 — C24 — C22xD4 — C2x2+ 1+4 — C23.C24
Lower central
C1 — C2 — C22 — C23.C24
Upper central
C1 — C2 — C22xD4 — C23.C24
Jennings
C1 — C2 — C23 — C23.C24

Generators and relations for C23.C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=g2=1, d2=b, ab=ba, dad-1=ac=ca, ae=ea, af=fa, ag=ga, ebe=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, ce=ec, gfg=cf=fc, cg=gc, ede=acd, df=fd, dg=gd, ef=fe, eg=ge >

Subgroups: 820 in 400 conjugacy classes, 170 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2xC4, C2xC4, D4, D4, Q8, C23, C23, C23, C42, C22:C4, C22:C4, C4:C4, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C23:C4, C2xC22:C4, C42:C2, C4xD4, C22xD4, C22xD4, C2xC4oD4, 2+ 1+4, C2xC23:C4, C23.C23, C22.11C24, C2x2+ 1+4, C23.C24
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C24, C2xC22:C4, C23xC4, C22xD4, C22xC22:C4, C23.C24

Permutation representations of C23.C24
►On 16 points - transitive group 16T199
Generators in S16
(1 3)(2 14)(4 16)(5 7)(6 11)(8 9)(10 12)(13 15)

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

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

(5 12)(6 9)(7 10)(8 11)
 
G:=sub<Sym(16)| (1,3)(2,14)(4,16)(5,7)(6,11)(8,9)(10,12)(13,15), (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), (1,2)(3,14)(4,13)(5,9)(6,12)(7,8)(10,11)(15,16), (1,10)(2,11)(3,12)(4,9)(5,13)(6,14)(7,15)(8,16), (5,12)(6,9)(7,10)(8,11)>;
 
G:=Group( (1,3)(2,14)(4,16)(5,7)(6,11)(8,9)(10,12)(13,15), (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), (1,2)(3,14)(4,13)(5,9)(6,12)(7,8)(10,11)(15,16), (1,10)(2,11)(3,12)(4,9)(5,13)(6,14)(7,15)(8,16), (5,12)(6,9)(7,10)(8,11) );
 
G=PermutationGroup([[(1,3),(2,14),(4,16),(5,7),(6,11),(8,9),(10,12),(13,15)], [(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)], [(1,2),(3,14),(4,13),(5,9),(6,12),(7,8),(10,11),(15,16)], [(1,10),(2,11),(3,12),(4,9),(5,13),(6,14),(7,15),(8,16)], [(5,12),(6,9),(7,10),(8,11)]])
 
G:=TransitiveGroup(16,199);
 

►On 16 points - transitive group 16T202
Generators in S16
(1 6)(4 8)(9 11)(13 15)

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

(1 6)(2 5)(3 7)(4 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)

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

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

(3 7)(4 8)(9 11)(10 12)
 
G:=sub<Sym(16)| (1,6)(4,8)(9,11)(13,15), (9,11)(10,12)(13,15)(14,16), (1,6)(2,5)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,15)(2,16)(3,12)(4,11)(5,14)(6,13)(7,10)(8,9), (1,8)(2,7)(3,5)(4,6)(9,15)(10,16)(11,13)(12,14), (3,7)(4,8)(9,11)(10,12)>;
 
G:=Group( (1,6)(4,8)(9,11)(13,15), (9,11)(10,12)(13,15)(14,16), (1,6)(2,5)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,15)(2,16)(3,12)(4,11)(5,14)(6,13)(7,10)(8,9), (1,8)(2,7)(3,5)(4,6)(9,15)(10,16)(11,13)(12,14), (3,7)(4,8)(9,11)(10,12) );
 
G=PermutationGroup([[(1,6),(4,8),(9,11),(13,15)], [(9,11),(10,12),(13,15),(14,16)], [(1,6),(2,5),(3,7),(4,8),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,15),(2,16),(3,12),(4,11),(5,14),(6,13),(7,10),(8,9)], [(1,8),(2,7),(3,5),(4,6),(9,15),(10,16),(11,13),(12,14)], [(3,7),(4,8),(9,11),(10,12)]])
 
G:=TransitiveGroup(16,202);
 

►On 16 points - transitive group 16T203
Generators in S16
(2 16)(4 14)(6 9)(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 8)(3 5)(4 6)(9 14)(10 15)(11 16)(12 13)

(5 12)(6 9)(7 10)(8 11)
 
G:=sub<Sym(16)| (2,16)(4,14)(6,9)(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)(2,8)(3,5)(4,6)(9,14)(10,15)(11,16)(12,13), (5,12)(6,9)(7,10)(8,11)>;
 
G:=Group( (2,16)(4,14)(6,9)(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)(2,8)(3,5)(4,6)(9,14)(10,15)(11,16)(12,13), (5,12)(6,9)(7,10)(8,11) );
 
G=PermutationGroup([[(2,16),(4,14),(6,9),(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),(2,8),(3,5),(4,6),(9,14),(10,15),(11,16),(12,13)], [(5,12),(6,9),(7,10),(8,11)]])
 
G:=TransitiveGroup(16,203);
 

41 conjugacy classes

class 1 2A2B···2L2M2N2O2P4A4B4C4D4E···4X
order122···2222244444···4
size112···2444422224···4

41 irreducible representations

dim111111128
type+++++++
imageC1C2C2C2C2C4C4D4C23.C24
kernelC23.C24C2xC23:C4C23.C23C22.11C24C2x2+ 1+4C22xD4C2xC4oD4C2xD4C1
# reps184218881

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

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

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

C23.C24 in GAP, Magma, Sage, TeX 

C_2^3.C_2^4
 
% in TeX
 
G:=Group("C2^3.C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1615);
 
// by ID
 
G=gap.SmallGroup(128,1615);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,521,2804,2028]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=g^2=1,d^2=b,a*b=b*a,d*a*d^-1=a*c=c*a,a*e=e*a,a*f=f*a,a*g=g*a,e*b*e=b*c=c*b,b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,g*f*g=c*f=f*c,c*g=g*c,e*d*e=a*c*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e>;
 
// generators/relations
 

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