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G = C23.461C24  order 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⋊Q8⋊19C2, C23.619(C2×D4), (C22×C4).392D4, (C22×Q8)⋊5C22, C24⋊3C4.11C2, (C22×C4).99C23, C23.156(C4○D4), C23.34D4⋊36C2, C23.11D4⋊46C2, C2.14(C23⋊3D4), (C23×C4).402C22, C22.312(C22×D4), C24.C22⋊87C2, C2.C42⋊27C22, C22.30(C4.4D4), C2.62(C22.19C24), C2.50(C22.45C24), (C4×C22⋊C4)⋊91C2, (C2×C4⋊C4)⋊23C22, (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

Derived series
C1 — C23 — C23.461C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C23.461C24
Lower central
C1 — C23 — C23.461C24
Upper central
C1 — C23 — C23.461C24
Jennings
C1 — C23 — C23.461C24

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, C24⋊3C4 [×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, C23⋊3D4, C22.45C24 [×4], C23.461C24

Generators and relations
 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 >

Smallest permutation representation
►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)])
 

Matrix representation ►G ⊆ 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] >;
 

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⋊C4C24⋊3C4C23.34D4C24.C22C23⋊Q8C23.11D4C22×C22⋊C4C2×C22⋊Q8C22×C4C23C22
# reps1122422114162

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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