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G = C23.568C24  order 128 = 27

285th central stem extension by C23 of C24

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

Aliases: C25.54C22, C24.381C23, C23.568C24, C22.3422+ (1+4), (C2×D4)⋊11D4, C2.32(D42), C23.56(C2×D4), C24⋊3C4⋊21C2, C23⋊2D4⋊31C2, (C23×C4)⋊23C22, C2.81(D4⋊5D4), (C22×D4)⋊10C22, C23.4Q8⋊38C2, C23.165(C4○D4), C23.11D4⋊73C2, C23.23D4⋊75C2, C23.10D4⋊68C2, C2.35(C23⋊3D4), (C22×C4).173C23, C22.377(C22×D4), C2.C42⋊32C22, C2.6(C22.54C24), C2.56(C22.32C24), (C2×C4⋊D4)⋊27C2, (C2×C4⋊C4)⋊28C22, (C2×C4).409(C2×D4), (C2×C22≀C2)⋊11C2, (C2×C22⋊C4)⋊25C22, C22.435(C2×C4○D4), SmallGroup(128,1400)

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

Derived series
C1 — C23 — C23.568C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C2×C22≀C2 — C23.568C24
Lower central
C1 — C23 — C23.568C24
Upper central
C1 — C23 — C23.568C24
Jennings
C1 — C23 — C23.568C24

Subgroups: 996 in 408 conjugacy classes, 104 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×10], C4 [×12], C22 [×3], C22 [×4], C22 [×66], C2×C4 [×4], C2×C4 [×36], D4 [×28], C23, C23 [×10], C23 [×58], C22⋊C4 [×26], C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×8], C22×C4 [×8], C2×D4 [×8], C2×D4 [×29], C24, C24 [×4], C24 [×10], C2.C42 [×2], C2.C42 [×4], C2×C22⋊C4 [×2], C2×C22⋊C4 [×14], C2×C4⋊C4, C2×C4⋊C4 [×2], C22≀C2 [×8], C4⋊D4 [×8], C23×C4 [×2], C22×D4, C22×D4 [×6], C25, C24⋊3C4, C23.23D4 [×4], C23⋊2D4, C23.10D4 [×2], C23.11D4 [×2], C23.4Q8, C2×C22≀C2 [×2], C2×C4⋊D4 [×2], C23.568C24

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], C23⋊3D4 [×2], C22.32C24, D42, D4⋊5D4 [×2], C22.54C24, C23.568C24

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
020000
300000
001000
000100
000001
000010
,
010000
100000
001300
000400
000010
000001
,
100000
010000
004000
004100
000040
000001
,
100000
040000
004000
000400
000010
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[0,3,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4] >;
 

32 conjugacy classes

class 1 2A···2G2H···2Q4A···4H4I···4N
order12···22···24···44···4
size11···14···44···48···8

32 irreducible representations

dim111111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ (1+4)
kernelC23.568C24C24⋊3C4C23.23D4C23⋊2D4C23.10D4C23.11D4C23.4Q8C2×C22≀C2C2×C4⋊D4C2×D4C23C22
# reps114122122844

In GAP, Magma, Sage, TeX 

C_2^3._{568}C_2^4
 
% in TeX
 
G:=Group("C2^3.568C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1400);
 
// by ID
 
G=gap.SmallGroup(128,1400);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,1571,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^2=1,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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