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G = C23.568C24order 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, C2.32D42, (C2×D4)⋊11D4, C23.56(C2×D4), C243C421C2, C232D431C2, (C23×C4)⋊23C22, C2.81(D45D4), (C22×D4)⋊10C22, C23.4Q838C2, C23.165(C4○D4), C23.10D468C2, C23.23D475C2, C23.11D473C2, C2.35(C233D4), (C22×C4).173C23, C22.377(C22×D4), C2.C4232C22, 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

C1C23 — C23.568C24
C1C2C22C23C24C25C2×C22≀C2 — C23.568C24
C1C23 — C23.568C24
C1C23 — C23.568C24
C1C23 — C23.568C24

Generators and relations for C23.568C24
 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 >

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, C243C4, C23.23D4 [×4], C232D4, 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], C233D4 [×2], C22.32C24, D42, D45D4 [×2], C22.54C24, C23.568C24

Smallest permutation representation of C23.568C24
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 17)(2 18)(3 11)(4 12)(5 20)(6 19)(7 30)(8 29)(9 14)(10 13)(15 24)(16 23)(21 27)(22 28)(25 31)(26 32)
(1 25)(2 26)(3 22)(4 21)(5 15)(6 16)(7 10)(8 9)(11 28)(12 27)(13 30)(14 29)(17 31)(18 32)(19 23)(20 24)
(1 20)(2 19)(3 30)(4 29)(5 17)(6 18)(7 11)(8 12)(9 27)(10 28)(13 22)(14 21)(15 31)(16 32)(23 26)(24 25)
(1 29)(2 30)(3 20)(4 19)(5 11)(6 12)(7 18)(8 17)(9 31)(10 32)(13 26)(14 25)(15 28)(16 27)(21 23)(22 24)
(1 17)(2 18)(3 22)(4 21)(7 13)(8 14)(9 29)(10 30)(11 28)(12 27)(25 31)(26 32)
(1 26)(2 25)(3 27)(4 28)(5 24)(6 23)(7 10)(8 9)(11 21)(12 22)(13 30)(14 29)(15 20)(16 19)(17 32)(18 31)

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

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

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

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
kernelC23.568C24C243C4C23.23D4C232D4C23.10D4C23.11D4C23.4Q8C2×C22≀C2C2×C4⋊D4C2×D4C23C22
# reps114122122844

Matrix representation of C23.568C24 in 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] >;

C23.568C24 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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