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G = C23.144C24  order 128 = 27

44th non-split extension by C23 of C24 acting via C24/C23=C2

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

Aliases: C23.144C24, C42.100C23, C24.511C23, C22.109C25, C4.832+ 1+4, C22.72- 1+4, (D4×Q8)⋊24C2, D4○3(C22⋊Q8), C4⋊Q8⋊96C22, D4⋊16(C4○D4), D4⋊6D4⋊28C2, D4⋊5D4⋊25C2, Q8⋊5D4⋊23C2, D4⋊3Q8⋊29C2, (C4×D4)⋊53C22, (C2×C4).99C24, (C4×Q8)⋊52C22, C4⋊D4⋊86C22, C4⋊C4.303C23, C22⋊Q8⋊38C22, C42⋊2C2⋊7C22, (C2×D4).481C23, C4.4D4⋊31C22, C22⋊C4.32C23, (C2×Q8).458C23, C42.C2⋊60C22, (C22×Q8)⋊36C22, C42⋊C2⋊46C22, C22.19C24⋊34C2, C22≀C2.30C22, (C23×C4).613C22, (C22×C4).378C23, (C2×C42).957C22, C22.45C24⋊10C2, C2.32(C2×2- 1+4), C2.43(C2×2+ 1+4), C22.33C24⋊8C2, (C22×D4).601C22, C22.D4⋊11C22, C22.50C24⋊28C2, C23.37C23⋊46C2, C23.33C23⋊29C2, C22.46C24⋊23C2, C22.36C24⋊20C2, C23.36C23⋊40C2, (C2×C4×D4)⋊95C2, (C2×C4⋊C4)⋊79C22, C4.282(C2×C4○D4), (C2×C22⋊Q8)⋊79C2, (C2×C4○D4)⋊38C22, C2.65(C22×C4○D4), C22.45(C2×C4○D4), (C2×C22⋊C4).385C22, SmallGroup(128,2252)

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

Derived series
C1 — C22 — C23.144C24
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C4×D4 — C23.144C24
Lower central
C1 — C22 — C23.144C24
Upper central
C1 — C22 — C23.144C24
Jennings
C1 — C22 — C23.144C24

Generators and relations for C23.144C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=b, g2=cb=bc, eae=ab=ba, ac=ca, ad=da, af=fa, ag=ga, ede=gdg-1=bd=db, be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, ef=fe, eg=ge, fg=gf >

Subgroups: 828 in 562 conjugacy classes, 392 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4×D4, C23.33C23, C2×C22⋊Q8, C22.19C24, C23.36C23, C23.37C23, C22.33C24, C22.36C24, D4⋊5D4, D4⋊6D4, Q8⋊5D4, D4×Q8, C22.45C24, C22.46C24, D4⋊3Q8, C22.50C24, C23.144C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C25, C22×C4○D4, C2×2+ 1+4, C2×2- 1+4, C23.144C24

Smallest permutation representation of C23.144C24
►On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 18)(6 19)(7 20)(8 17)(9 31)(10 32)(11 29)(12 30)(13 21)(14 22)(15 23)(16 24)

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

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

(2 28)(4 26)(6 17)(8 19)(10 30)(12 32)(14 22)(16 24)

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A···4L4M···4AD
order12222···222224···44···4
size11112···244442···24···4

44 irreducible representations

dim11111111111111111244
type++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC23.144C24C2×C4×D4C23.33C23C2×C22⋊Q8C22.19C24C23.36C23C23.37C23C22.33C24C22.36C24D4⋊5D4D4⋊6D4Q8⋊5D4D4×Q8C22.45C24C22.46C24D4⋊3Q8C22.50C24D4C4C22
# reps11222114222214221822

Matrix representation of C23.144C24 ►in GL6(𝔽5)

100000
010000
004000
004100
000040
000041
,
100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
040000
400000
003030
003203
000020
000023
,
400000
040000
001300
000400
000042
000001
,
100000
040000
001000
000100
003040
003204
,
300000
030000
001000
000100
003040
003204

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

C23.144C24 in GAP, Magma, Sage, TeX 

C_2^3._{144}C_2^4
 
% in TeX
 
G:=Group("C2^3.144C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2252);
 
// by ID
 
G=gap.SmallGroup(128,2252);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,570,1684,242]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=b,g^2=c*b=b*c,e*a*e=a*b=b*a,a*c=c*a,a*d=d*a,a*f=f*a,a*g=g*a,e*d*e=g*d*g^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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