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G = C23.7C24order 128 = 27

7th non-split extension by C23 of C24 acting via C24/C22=C22

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

Aliases: C23.7C24, C24.471C23, 2+ 1+4.13C22, C4C2≀C22, (C2×D4)⋊25D4, (C2×Q8)⋊20D4, (C22×C4)⋊6D4, C2≀C226C2, C4.88C22≀C2, C23⋊C47C22, C23.26(C2×D4), C2.C253C2, (C23×C4)⋊26C22, (C2×D4).41C23, C4(C23.7D4), C23.7D46C2, C22⋊C4.1C23, C22≀C224C22, C22.19C245C2, C22.41(C22×D4), C42⋊C212C22, (C22×C4).616C23, C23.C2318C2, C22.D429C22, (C2×C4).27(C2×D4), C2.62(C2×C22≀C2), (C2×C4○D4).111C22, SmallGroup(128,1757)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.7C24
C1C2C22C23C22×C4C2×C4○D4C2.C25 — C23.7C24
C1C2C23 — C23.7C24
C1C4C22×C4 — C23.7C24
C1C2C23 — C23.7C24

Generators and relations for C23.7C24
 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, cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 772 in 383 conjugacy classes, 106 normal (10 characteristic)
C1, C2, C2 [×11], C4, C4 [×3], C4 [×13], C22 [×3], C22 [×27], C2×C4 [×12], C2×C4 [×37], D4 [×42], Q8 [×10], C23, C23 [×6], C23 [×9], C42 [×3], C22⋊C4 [×6], C22⋊C4 [×12], C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×6], C22×C4 [×10], C2×D4 [×9], C2×D4 [×24], C2×Q8 [×3], C2×Q8 [×6], C4○D4 [×40], C24, C23⋊C4 [×12], C42⋊C2 [×3], C4×D4 [×6], C22≀C2 [×6], C4⋊D4 [×3], C22⋊Q8 [×3], C22.D4 [×6], C23×C4, 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 [×4], C23.7D4 [×4], C22.19C24 [×3], C2.C25, C23.7C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, C23.7C24

Permutation representations of C23.7C24
On 16 points - transitive group 16T223
Generators in S16
(1 9)(2 10)(3 11)(4 12)(5 14)(6 15)(7 16)(8 13)
(1 11)(2 12)(3 9)(4 10)(5 14)(6 15)(7 16)(8 13)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)(9 11)(10 12)
(1 14)(2 15)(3 16)(4 13)(5 9)(6 10)(7 11)(8 12)
(1 14)(2 15)(3 16)(4 13)(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)| (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13), (1,11)(2,12)(3,9)(4,10)(5,14)(6,15)(7,16)(8,13), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,11)(10,12), (1,14)(2,15)(3,16)(4,13)(5,9)(6,10)(7,11)(8,12), (1,14)(2,15)(3,16)(4,13)(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( (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13), (1,11)(2,12)(3,9)(4,10)(5,14)(6,15)(7,16)(8,13), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,11)(10,12), (1,14)(2,15)(3,16)(4,13)(5,9)(6,10)(7,11)(8,12), (1,14)(2,15)(3,16)(4,13)(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([(1,9),(2,10),(3,11),(4,12),(5,14),(6,15),(7,16),(8,13)], [(1,11),(2,12),(3,9),(4,10),(5,14),(6,15),(7,16),(8,13)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,3),(2,4),(5,16),(6,13),(7,14),(8,15),(9,11),(10,12)], [(1,14),(2,15),(3,16),(4,13),(5,9),(6,10),(7,11),(8,12)], [(1,14),(2,15),(3,16),(4,13),(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(16,223);

On 16 points - transitive group 16T270
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)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)
(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), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12), (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), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12), (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)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12)], [(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(16,270);

32 conjugacy classes

class 1 2A2B2C2D2E···2L4A4B4C4D4E4F···4M4N···4S
order122222···2444444···44···4
size112224···4112224···48···8

32 irreducible representations

dim1111112224
type+++++++++
imageC1C2C2C2C2C2D4D4D4C23.7C24
kernelC23.7C24C23.C23C2≀C22C23.7D4C22.19C24C2.C25C22×C4C2×D4C2×Q8C1
# reps1344316334

Matrix representation of C23.7C24 in GL4(𝔽5) generated by

4200
0100
1401
1410
,
1300
0400
0101
0110
,
4000
0400
0040
0004
,
1000
0100
0001
0010
,
4030
4044
0010
1410
,
4030
0041
0010
0110
,
3000
0300
0030
0003
G:=sub<GL(4,GF(5))| [4,0,1,1,2,1,4,4,0,0,0,1,0,0,1,0],[1,0,0,0,3,4,1,1,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[4,4,0,1,0,0,0,4,3,4,1,1,0,4,0,0],[4,0,0,0,0,0,0,1,3,4,1,1,0,1,0,0],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3] >;

C23.7C24 in GAP, Magma, Sage, TeX

C_2^3._7C_2^4
% in TeX

G:=Group("C2^3.7C2^4");
// GroupNames label

G:=SmallGroup(128,1757);
// by ID

G=gap.SmallGroup(128,1757);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,248,718,2028]);
// 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,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations

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