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G = C23.570C24  order 128 = 27

287th central stem extension by C23 of C24

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

Aliases: C25.56C22, C23.570C24, C24.383C23, C22.3442+ (1+4), C2.34(D42), C22⋊C4⋊9D4, C24⋊3C4⋊22C2, C23⋊2D4⋊33C2, (C2×C42)⋊28C22, C23⋊Q8⋊37C2, C23.202(C2×D4), C2.82(D4⋊5D4), (C22×Q8)⋊7C22, (C22×D4)⋊12C22, C23.166(C4○D4), C23.10D4⋊70C2, (C22×C4).175C23, C22.379(C22×D4), C24.3C22⋊71C2, C2.C42⋊34C22, C2.3(C24⋊C22), C24.C22⋊115C2, C2.52(C22.29C24), C2.57(C22.32C24), (C2×C4⋊C4)⋊30C22, (C2×C4).410(C2×D4), (C2×C22≀C2)⋊13C2, (C2×C4.4D4)⋊24C2, (C2×C22⋊C4)⋊27C22, C22.436(C2×C4○D4), SmallGroup(128,1402)

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

Derived series
C1 — C23 — C23.570C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C2×C22≀C2 — C23.570C24
Lower central
C1 — C23 — C23.570C24
Upper central
C1 — C23 — C23.570C24
Jennings
C1 — C23 — C23.570C24

Subgroups: 980 in 398 conjugacy classes, 104 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×14], C22 [×3], C22 [×4], C22 [×60], C2×C4 [×8], C2×C4 [×26], D4 [×28], Q8 [×4], C23, C23 [×6], C23 [×60], C42 [×4], C22⋊C4 [×8], C22⋊C4 [×26], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×8], C2×D4 [×31], C2×Q8 [×5], C24, C24 [×4], C24 [×10], C2.C42 [×2], C2.C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×16], C2×C4⋊C4 [×2], C22≀C2 [×8], C4.4D4 [×8], C22×D4, C22×D4 [×6], C22×Q8, C25, C24⋊3C4, C24.C22 [×2], C24.3C22 [×2], C23⋊2D4, C23⋊2D4 [×2], C23⋊Q8, C23.10D4 [×2], C2×C22≀C2 [×2], C2×C4.4D4 [×2], C23.570C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C22×D4 [×2], C2×C4○D4, 2+ (1+4) [×4], C22.29C24 [×2], C22.32C24, D42, D4⋊5D4 [×2], C24⋊C22, C23.570C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=e2=a, 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, fef=ce=ec, cf=fc, cg=gc, gdg=abd, fg=gf >

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

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

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

(1 20)(2 30)(3 18)(4 32)(6 24)(8 22)(9 29)(10 17)(11 31)(12 19)(13 25)(15 27)

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
030000
300000
003300
004200
000010
000001
,
010000
400000
001000
000100
000001
000010
,
100000
010000
004000
002100
000040
000001
,
100000
040000
001000
003400
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],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,3,4,0,0,0,0,3,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,2,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,3,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
 

32 conjugacy classes

class 1 2A···2G2H···2M2N2O4A···4L4M4N4O4P
order12···22···2224···44444
size11···14···4884···48888

32 irreducible representations

dim111111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ (1+4)
kernelC23.570C24C24⋊3C4C24.C22C24.3C22C23⋊2D4C23⋊Q8C23.10D4C2×C22≀C2C2×C4.4D4C22⋊C4C23C22
# reps112231222844

In GAP, Magma, Sage, TeX 

C_2^3._{570}C_2^4
 
% in TeX
 
G:=Group("C2^3.570C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1402);
 
// by ID
 
G=gap.SmallGroup(128,1402);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,1571,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=e^2=a,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,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=a*b*d,f*g=g*f>;
 
// generators/relations
 

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