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G = C23.585C24order 128 = 27

302nd central stem extension by C23 of C24

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

Aliases: C25.59C22, C24.392C23, C23.585C24, C22.3592+ 1+4, C22⋊C4.11D4, C243C425C2, (C2×C42)⋊32C22, C23.209(C2×D4), C2.90(D45D4), C23.4Q842C2, C23.Q855C2, C23.169(C4○D4), C23.10D480C2, (C22×C4).180C23, C22.394(C22×D4), C2.C4238C22, C24.3C2277C2, (C22×D4).224C22, C24.C22123C2, C2.62(C22.32C24), C2.56(C22.29C24), C2.76(C22.45C24), C2.11(C22.54C24), (C2×C4).92(C2×D4), (C2×C4⋊C4)⋊34C22, (C2×C22⋊C4)⋊6C22, (C2×C422C2)⋊17C2, (C2×C22≀C2).14C2, C22.447(C2×C4○D4), SmallGroup(128,1417)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.585C24
C1C2C22C23C24C25C243C4 — C23.585C24
C1C23 — C23.585C24
C1C23 — C23.585C24
C1C23 — C23.585C24

Generators and relations for C23.585C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=a, e2=ba=ab, 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: 804 in 333 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×7], C4 [×13], C22 [×7], C22 [×53], C2×C4 [×4], C2×C4 [×31], D4 [×12], C23, C23 [×6], C23 [×53], C42 [×3], C22⋊C4 [×4], C22⋊C4 [×24], C4⋊C4 [×9], C22×C4 [×11], C2×D4 [×14], C24 [×4], C24 [×10], C2.C42 [×4], C2×C42 [×2], C2×C22⋊C4 [×19], C2×C4⋊C4 [×6], C22≀C2 [×4], C422C2 [×4], C22×D4 [×3], C25, C243C4 [×2], C24.C22 [×3], C24.3C22 [×2], C23.10D4 [×4], C23.Q8, C23.4Q8, C2×C22≀C2, C2×C422C2, C23.585C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4 [×4], C22.29C24, C22.32C24 [×2], D45D4 [×2], C22.45C24, C22.54C24, C23.585C24

Smallest permutation representation of C23.585C24
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 25)(2 26)(3 27)(4 28)(5 12)(6 9)(7 10)(8 11)(13 18)(14 19)(15 20)(16 17)(21 32)(22 29)(23 30)(24 31)
(1 11)(2 12)(3 9)(4 10)(5 26)(6 27)(7 28)(8 25)(13 24)(14 21)(15 22)(16 23)(17 30)(18 31)(19 32)(20 29)
(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 15 27 18)(2 14 28 17)(3 13 25 20)(4 16 26 19)(5 32 10 23)(6 31 11 22)(7 30 12 21)(8 29 9 24)
(1 8)(2 12)(3 6)(4 10)(5 26)(7 28)(9 27)(11 25)(13 18)(15 20)(22 29)(24 31)
(1 11)(2 7)(3 9)(4 5)(6 27)(8 25)(10 26)(12 28)(13 22)(14 32)(15 24)(16 30)(17 23)(18 29)(19 21)(20 31)

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

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

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

32 conjugacy classes

class 1 2A···2G2H···2M2N4A···4L4M···4Q
order12···22···224···44···4
size11···14···484···48···8

32 irreducible representations

dim111111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ 1+4
kernelC23.585C24C243C4C24.C22C24.3C22C23.10D4C23.Q8C23.4Q8C2×C22≀C2C2×C422C2C22⋊C4C23C22
# reps123241111484

Matrix representation of C23.585C24 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
020000
200000
004400
000100
000040
000004
,
010000
400000
002000
000200
000013
000004
,
100000
010000
004000
002100
000040
000041
,
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],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,0,4,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,3,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,2,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,0,1],[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] >;

C23.585C24 in GAP, Magma, Sage, TeX

C_2^3._{585}C_2^4
% in TeX

G:=Group("C2^3.585C2^4");
// GroupNames label

G:=SmallGroup(128,1417);
// by ID

G=gap.SmallGroup(128,1417);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,1571,346]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=a,e^2=b*a=a*b,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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