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G = C23.461C24order 128 = 27

178th central stem extension by C23 of C24

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C25.49C22, C24.582C23, C23.461C24, C22.2462+ 1+4, (C2×C42)⋊7C22, C23⋊Q819C2, (C22×C4).392D4, C23.619(C2×D4), (C22×Q8)⋊5C22, C243C4.11C2, (C22×C4).99C23, C23.156(C4○D4), C23.11D446C2, C23.34D436C2, C2.14(C233D4), (C23×C4).402C22, C22.312(C22×D4), C24.C2287C2, C2.C4227C22, C22.30(C4.4D4), C2.62(C22.19C24), C2.50(C22.45C24), (C2×C4⋊C4)⋊23C22, (C4×C22⋊C4)⋊91C2, (C2×C4).357(C2×D4), (C2×C22⋊Q8)⋊24C2, C2.23(C2×C4.4D4), C22.337(C2×C4○D4), (C22×C22⋊C4).25C2, (C2×C22⋊C4).184C22, SmallGroup(128,1293)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.461C24
C1C2C22C23C24C25C22×C22⋊C4 — C23.461C24
C1C23 — C23.461C24
C1C23 — C23.461C24
C1C23 — C23.461C24

Generators and relations for C23.461C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=cb=bc, e2=ca=ac, ab=ba, ede-1=gdg=ad=da, ae=ea, af=fa, ag=ga, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 740 in 336 conjugacy classes, 104 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×14], C22, C22 [×10], C22 [×48], C2×C4 [×4], C2×C4 [×42], Q8 [×4], C23, C23 [×10], C23 [×48], C42 [×2], C22⋊C4 [×28], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×14], C22×C4 [×8], C2×Q8 [×4], C24, C24 [×2], C24 [×12], C2.C42 [×10], C2×C42 [×2], C2×C22⋊C4 [×16], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×2], C22⋊Q8 [×4], C23×C4 [×2], C22×Q8, C25, C4×C22⋊C4, C243C4 [×2], C23.34D4 [×2], C24.C22 [×4], C23⋊Q8 [×2], C23.11D4 [×2], C22×C22⋊C4, C2×C22⋊Q8, C23.461C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×4], 2+ 1+4 [×2], C22.19C24, C2×C4.4D4, C233D4, C22.45C24 [×4], C23.461C24

Smallest permutation representation of C23.461C24
On 32 points
Generators in S32
(1 15)(2 16)(3 13)(4 14)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(17 29)(18 30)(19 31)(20 32)
(1 5)(2 6)(3 7)(4 8)(9 31)(10 32)(11 29)(12 30)(13 23)(14 24)(15 21)(16 22)(17 27)(18 28)(19 25)(20 26)
(1 7)(2 8)(3 5)(4 6)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 27 23 31)(2 12 24 20)(3 25 21 29)(4 10 22 18)(5 17 13 9)(6 30 14 26)(7 19 15 11)(8 32 16 28)
(1 5)(3 7)(9 11)(10 30)(12 32)(13 23)(15 21)(17 19)(18 26)(20 28)(25 27)(29 31)
(1 3)(2 14)(4 16)(5 7)(6 24)(8 22)(9 11)(10 28)(12 26)(13 15)(17 19)(18 32)(20 30)(21 23)(25 27)(29 31)

G:=sub<Sym(32)| (1,15)(2,16)(3,13)(4,14)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(17,29)(18,30)(19,31)(20,32), (1,5)(2,6)(3,7)(4,8)(9,31)(10,32)(11,29)(12,30)(13,23)(14,24)(15,21)(16,22)(17,27)(18,28)(19,25)(20,26), (1,7)(2,8)(3,5)(4,6)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,27,23,31)(2,12,24,20)(3,25,21,29)(4,10,22,18)(5,17,13,9)(6,30,14,26)(7,19,15,11)(8,32,16,28), (1,5)(3,7)(9,11)(10,30)(12,32)(13,23)(15,21)(17,19)(18,26)(20,28)(25,27)(29,31), (1,3)(2,14)(4,16)(5,7)(6,24)(8,22)(9,11)(10,28)(12,26)(13,15)(17,19)(18,32)(20,30)(21,23)(25,27)(29,31)>;

G:=Group( (1,15)(2,16)(3,13)(4,14)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(17,29)(18,30)(19,31)(20,32), (1,5)(2,6)(3,7)(4,8)(9,31)(10,32)(11,29)(12,30)(13,23)(14,24)(15,21)(16,22)(17,27)(18,28)(19,25)(20,26), (1,7)(2,8)(3,5)(4,6)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,27,23,31)(2,12,24,20)(3,25,21,29)(4,10,22,18)(5,17,13,9)(6,30,14,26)(7,19,15,11)(8,32,16,28), (1,5)(3,7)(9,11)(10,30)(12,32)(13,23)(15,21)(17,19)(18,26)(20,28)(25,27)(29,31), (1,3)(2,14)(4,16)(5,7)(6,24)(8,22)(9,11)(10,28)(12,26)(13,15)(17,19)(18,32)(20,30)(21,23)(25,27)(29,31) );

G=PermutationGroup([(1,15),(2,16),(3,13),(4,14),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(17,29),(18,30),(19,31),(20,32)], [(1,5),(2,6),(3,7),(4,8),(9,31),(10,32),(11,29),(12,30),(13,23),(14,24),(15,21),(16,22),(17,27),(18,28),(19,25),(20,26)], [(1,7),(2,8),(3,5),(4,6),(9,29),(10,30),(11,31),(12,32),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26),(19,27),(20,28)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,27,23,31),(2,12,24,20),(3,25,21,29),(4,10,22,18),(5,17,13,9),(6,30,14,26),(7,19,15,11),(8,32,16,28)], [(1,5),(3,7),(9,11),(10,30),(12,32),(13,23),(15,21),(17,19),(18,26),(20,28),(25,27),(29,31)], [(1,3),(2,14),(4,16),(5,7),(6,24),(8,22),(9,11),(10,28),(12,26),(13,15),(17,19),(18,32),(20,30),(21,23),(25,27),(29,31)])

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A4B4C4D4E···4R4S4T4U4V
order12···22222222244444···44444
size11···12222444422224···48888

38 irreducible representations

dim111111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ 1+4
kernelC23.461C24C4×C22⋊C4C243C4C23.34D4C24.C22C23⋊Q8C23.11D4C22×C22⋊C4C2×C22⋊Q8C22×C4C23C22
# reps1122422114162

Matrix representation of C23.461C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
010000
100000
000100
004000
000030
000003
,
200000
030000
001000
000100
000001
000040
,
100000
010000
004000
000100
000010
000004
,
100000
040000
004000
000400
000040
000004

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;

C23.461C24 in GAP, Magma, Sage, TeX

C_2^3._{461}C_2^4
% in TeX

G:=Group("C2^3.461C2^4");
// GroupNames label

G:=SmallGroup(128,1293);
// by ID

G=gap.SmallGroup(128,1293);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,568,758,723,675]);
// Polycyclic

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

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