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G = C24.169C23  order 128 = 27

9th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.169C23, (C2×C42)⋊2C4, (C22×C4).274D4, C23.550(C2×D4), C23.9D4.1C2, C23.114(C4○D4), C23.189(C22×C4), (C23×C4).240C22, C23.34D4.5C2, C2.8(C23.34D4), C22.29(C42⋊C2), C2.26(C23.C23), C22.43(C22.D4), (C2×C4⋊C4)⋊27C4, (C22×C4).53(C2×C4), (C2×C22⋊C4).7C22, (C2×C4).124(C22⋊C4), (C2×C42⋊C2).15C2, C22.254(C2×C22⋊C4), SmallGroup(128,552)

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

Derived series
C1 — C23 — C24.169C23
Chief series
C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4 — C2×C42⋊C2 — C24.169C23
Lower central
C1 — C2 — C23 — C24.169C23
Upper central
C1 — C22 — C23×C4 — C24.169C23
Jennings
C1 — C2 — C24 — C24.169C23

Generators and relations for C24.169C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=c, f2=d, g2=a, ab=ba, ac=ca, ad=da, fef-1=ae=ea, af=fa, ag=ga, bc=cb, ebe-1=bd=db, bf=fb, bg=gb, gcg-1=cd=dc, ce=ec, cf=fc, de=ed, df=fd, dg=gd, geg-1=bde, fg=gf >

Subgroups: 324 in 150 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C23.9D4, C23.34D4, C2×C42⋊C2, C24.169C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C42⋊C2, C22.D4, C23.34D4, C23.C23, C24.169C23

Smallest permutation representation of C24.169C23
►On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 10)(6 11)(7 12)(8 9)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(29 31)(30 32)

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

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4N4O···4V
order12222···244444···44···4
size11112···222224···48···8

32 irreducible representations

dim111111224
type+++++
imageC1C2C2C2C4C4D4C4○D4C23.C23
kernelC24.169C23C23.9D4C23.34D4C2×C42⋊C2C2×C42C2×C4⋊C4C22×C4C23C2
# reps142144484

Matrix representation of C24.169C23 ►in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
400000
040000
000100
001000
000001
000010
,
100000
010000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
230000
430000
002000
000300
000003
000020
,
140000
040000
003000
000300
000030
000003
,
230000
030000
000010
000001
001000
000100

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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,4,0,0,0,0,3,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,0,0,3,0],[1,0,0,0,0,0,4,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[2,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C24.169C23 in GAP, Magma, Sage, TeX 

C_2^4._{169}C_2^3
 
% in TeX
 
G:=Group("C2^4.169C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,552);
 
// by ID
 
G=gap.SmallGroup(128,552);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,58,2804,1027]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=c,f^2=d,g^2=a,a*b=b*a,a*c=c*a,a*d=d*a,f*e*f^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*b*e^-1=b*d=d*b,b*f=f*b,b*g=g*b,g*c*g^-1=c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=b*d*e,f*g=g*f>;
 
// generators/relations
 

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