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G = C23.224C24  order 128 = 27

77th central extension by C23 of C24

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

Aliases: C25.23C22, C23.224C24, C24.551C23, C22.612+ 1+4, C24.70(C2×C4), (C2×C42)⋊4C22, C22.65(C4×D4), C2.3(D4⋊5D4), C24⋊3C4.5C2, C22⋊C4.182D4, C23.413(C2×D4), C23.8Q8⋊12C2, C23.7Q8⋊22C2, C22⋊2(C42⋊C2), C23.287(C4○D4), C23.34D4⋊15C2, (C22×C4).753C23, (C23×C4).298C22, C22.115(C23×C4), C23.215(C22×C4), C22.103(C22×D4), C2.C42⋊52C22, C24.C22⋊12C2, C2.1(C22.45C24), C2.24(C22.11C24), C2.29(C2×C4×D4), (C4×C22⋊C4)⋊8C2, (C2×C4⋊C4)⋊8C22, C22⋊C4⋊39(C2×C4), (C2×C22⋊C4)⋊23C4, (C22×C4)⋊30(C2×C4), (C2×C4).884(C2×D4), C22⋊C4○3(C22⋊C4), (C2×C42⋊C2)⋊11C2, (C2×C4).229(C22×C4), C2.25(C2×C42⋊C2), C22.109(C2×C4○D4), (C22×C22⋊C4).12C2, (C2×C22⋊C4).435C22, C22⋊C4○2(C2×C22⋊C4), SmallGroup(128,1074)

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

Derived series
C1 — C22 — C23.224C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C23.224C24
Lower central
C1 — C22 — C23.224C24
Upper central
C1 — C23 — C23.224C24
Jennings
C1 — C23 — C23.224C24

Generators and relations for C23.224C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=c, e2=b, ab=ba, ac=ca, ede-1=ad=da, geg=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 780 in 396 conjugacy classes, 156 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C23×C4, C25, C4×C22⋊C4, C4×C22⋊C4, C24⋊3C4, C23.7Q8, C23.34D4, C23.8Q8, C24.C22, C22×C22⋊C4, C2×C42⋊C2, C23.224C24
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C42⋊C2, C4×D4, C23×C4, C22×D4, C2×C4○D4, 2+ 1+4, C2×C42⋊C2, C2×C4×D4, C22.11C24, D4⋊5D4, C22.45C24, C23.224C24

Smallest permutation representation of C23.224C24
►On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)

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

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

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

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

50 conjugacy classes

class 1 2A···2G2H···2O2P2Q4A···4P4Q···4AF
order12···22···2224···44···4
size11···12···2442···24···4

50 irreducible representations

dim1111111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C4D4C4○D42+ 1+4
kernelC23.224C24C4×C22⋊C4C24⋊3C4C23.7Q8C23.34D4C23.8Q8C24.C22C22×C22⋊C4C2×C42⋊C2C2×C22⋊C4C22⋊C4C23C22
# reps131112421164122

Matrix representation of C23.224C24 ►in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
100000
340000
003000
000200
000001
000010
,
110000
040000
000100
001000
000020
000002
,
100000
010000
004000
000400
000010
000004
,
100000
340000
004000
000100
000040
000004

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

C23.224C24 in GAP, Magma, Sage, TeX 

C_2^3._{224}C_2^4
 
% in TeX
 
G:=Group("C2^3.224C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1074);
 
// by ID
 
G=gap.SmallGroup(128,1074);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=c,e^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

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