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G = C23.570C24order 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, C2.34D42, C22⋊C49D4, C243C422C2, C232D433C2, (C2×C42)⋊28C22, C23⋊Q837C2, C23.202(C2×D4), C2.82(D45D4), (C22×Q8)⋊7C22, (C22×D4)⋊12C22, C23.166(C4○D4), C23.10D470C2, (C22×C4).175C23, C22.379(C22×D4), C2.C4234C22, C24.3C2271C2, C2.3(C24⋊C22), C24.C22115C2, C2.57(C22.32C24), C2.52(C22.29C24), (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

C1C23 — C23.570C24
C1C2C22C23C24C25C2×C22≀C2 — C23.570C24
C1C23 — C23.570C24
C1C23 — C23.570C24
C1C23 — C23.570C24

Generators and relations for C23.570C24
 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 >

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, C243C4, C24.C22 [×2], C24.3C22 [×2], C232D4, C232D4 [×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, D45D4 [×2], C24⋊C22, C23.570C24

Smallest permutation representation of C23.570C24
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)])

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
kernelC23.570C24C243C4C24.C22C24.3C22C232D4C23⋊Q8C23.10D4C2×C22≀C2C2×C4.4D4C22⋊C4C23C22
# reps112231222844

Matrix representation of C23.570C24 in 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] >;

C23.570C24 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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