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G = C23.569C24  order 128 = 27

286th central stem extension by C23 of C24

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

Aliases: C25.55C22, C23.569C24, C24.382C23, C22.3432+ (1+4), (C2×D4)⋊12D4, C2.33(D42), C22⋊C4⋊8D4, C23.57(C2×D4), C23⋊2D4⋊32C2, (C2×C42)⋊27C22, (C23×C4)⋊24C22, (C22×D4)⋊11C22, C23.10D4⋊69C2, C23.23D4⋊76C2, C2.36(C23⋊3D4), (C22×C4).174C23, C22.378(C22×D4), C2.C42⋊33C22, C24.C22⋊114C2, C2.7(C22.54C24), C2.51(C22.29C24), (C2×C4⋊1D4)⋊9C2, (C2×C4).83(C2×D4), (C2×C4⋊D4)⋊28C2, (C2×C4⋊C4)⋊29C22, (C2×C22≀C2)⋊12C2, (C2×C22⋊C4)⋊26C22, SmallGroup(128,1401)

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

Derived series
C1 — C23 — C23.569C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C2×C22≀C2 — C23.569C24
Lower central
C1 — C23 — C23.569C24
Upper central
C1 — C23 — C23.569C24
Jennings
C1 — C23 — C23.569C24

Subgroups: 1156 in 466 conjugacy classes, 112 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×10], C4 [×14], C22 [×3], C22 [×4], C22 [×70], C2×C4 [×10], C2×C4 [×26], D4 [×48], C23, C23 [×8], C23 [×66], C42 [×2], C22⋊C4 [×8], C22⋊C4 [×21], C4⋊C4 [×4], C22×C4, C22×C4 [×8], C22×C4 [×5], C2×D4 [×4], C2×D4 [×54], C24 [×2], C24 [×4], C24 [×10], C2.C42 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×2], C22≀C2 [×12], C4⋊D4 [×8], C4⋊1D4 [×4], C23×C4, C22×D4 [×2], C22×D4 [×10], C25, C23.23D4, C24.C22 [×2], C23⋊2D4 [×4], C23.10D4 [×2], C2×C22≀C2, C2×C22≀C2 [×2], C2×C4⋊D4 [×2], C2×C4⋊1D4, C23.569C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22×D4 [×3], 2+ (1+4) [×4], C23⋊3D4, C22.29C24 [×2], D42 [×3], C22.54C24, C23.569C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=g2=1, d2=a, ab=ba, ac=ca, ede=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 25)(2 28)(3 27)(4 26)(5 30)(6 29)(7 32)(8 31)(9 13)(10 16)(11 15)(12 14)(17 23)(18 22)(19 21)(20 24)

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

100000
010000
001000
000100
0000-10
00000-1
,
100000
010000
00-1000
000-100
000010
000001
,
-100000
0-10000
001000
000100
000010
000001
,
-100000
0-10000
000100
001000
00000-1
000010
,
010000
100000
001000
000100
000001
000010
,
-100000
010000
00-1000
000100
000010
000001
,
-100000
0-10000
001000
000-100
000010
00000-1

G:=sub<GL(6,Integers())| [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,-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,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1] >;
 

32 conjugacy classes

class 1 2A···2G2H···2O2P2Q4A···4J4K4L4M4N
order12···22···2224···44444
size11···14···4884···48888

32 irreducible representations

dim11111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2D4D42+ (1+4)
kernelC23.569C24C23.23D4C24.C22C23⋊2D4C23.10D4C2×C22≀C2C2×C4⋊D4C2×C4⋊1D4C22⋊C4C2×D4C22
# reps11242321844

In GAP, Magma, Sage, TeX 

C_2^3._{569}C_2^4
 
% in TeX
 
G:=Group("C2^3.569C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1401);
 
// by ID
 
G=gap.SmallGroup(128,1401);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,1571,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=g^2=1,d^2=a,a*b=b*a,a*c=c*a,e*d*e=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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