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G = C24.167C23  order 128 = 27

7th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.167C23, (C23×C4)⋊8C4, (C2×C42)⋊7C4, C4⋊2(C23⋊C4), C23⋊2(C4⋊C4), (C2×D4).20Q8, (C2×D4).196D4, C23.1(C2×Q8), C24.29(C2×C4), C23.3(C4○D4), C23.547(C2×D4), (C22×C4).265D4, C23.9D4⋊1C2, C23.7Q8⋊5C2, C22.37(C4⋊D4), C23.185(C22×C4), (C23×C4).236C22, C22.47(C22⋊Q8), C2.8(C23.7Q8), (C22×D4).454C22, C22.25(C42⋊C2), C2.25(C23.C23), (C2×C4⋊C4)⋊26C4, (C2×C4)⋊2(C4⋊C4), (C2×C4×D4).13C2, (C2×C22⋊C4)⋊18C4, (C2×C23⋊C4).3C2, C22.17(C2×C4⋊C4), C2.25(C2×C23⋊C4), (C22×C4).49(C2×C4), (C2×C22⋊C4).5C22, (C2×C4).355(C22⋊C4), C22.244(C2×C22⋊C4), SmallGroup(128,531)

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

Derived series
C1 — C23 — C24.167C23
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — C2×C4×D4 — C24.167C23
Lower central
C1 — C2 — C23 — C24.167C23
Upper central
C1 — C22 — C23×C4 — C24.167C23
Jennings
C1 — C2 — C24 — C24.167C23

Generators and relations for C24.167C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=c, f2=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=cd=dc, ce=ec, cf=fc, de=ed, df=fd, dg=gd, geg=bde, fg=gf >

Subgroups: 436 in 190 conjugacy classes, 62 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C23⋊C4, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C23.9D4, C23.7Q8, C2×C23⋊C4, C2×C4×D4, C24.167C23
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C23⋊C4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C23.7Q8, C2×C23⋊C4, C23.C23, C24.167C23

Smallest permutation representation of C24.167C23
►On 32 points
Generators in S32
(1 22)(2 23)(3 24)(4 21)(5 14)(6 15)(7 16)(8 13)(9 32)(10 29)(11 30)(12 31)(17 28)(18 25)(19 26)(20 27)

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A4B4C4D4E···4L4M···4T
order12222···22244444···44···4
size11112···24422224···48···8

32 irreducible representations

dim111111111222244
type+++++++-+
imageC1C2C2C2C2C4C4C4C4D4D4Q8C4○D4C23⋊C4C23.C23
kernelC24.167C23C23.9D4C23.7Q8C2×C23⋊C4C2×C4×D4C2×C42C2×C22⋊C4C2×C4⋊C4C23×C4C22×C4C2×D4C2×D4C23C4C2
# reps122212222422422

Matrix representation of C24.167C23 ►in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
000100
001000
000001
000010
,
400000
040000
000100
001000
000004
000040
,
100000
010000
004000
000400
000040
000004
,
200000
430000
000004
000010
001000
000400
,
410000
310000
004000
000400
000040
000004
,
100000
010000
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,4,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,4,0,0,0],[4,3,0,0,0,0,1,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C24.167C23 in GAP, Magma, Sage, TeX 

C_2^4._{167}C_2^3
 
% in TeX
 
G:=Group("C2^4.167C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,531);
 
// by ID
 
G=gap.SmallGroup(128,531);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2804,1027]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=c,f^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=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=b*d*e,f*g=g*f>;
 
// generators/relations
 

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