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G = C23.635C24order 128 = 27

352nd central stem extension by C23 of C24

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

Aliases: C25.61C22, C23.635C24, C24.424C23, C22.4082+ 1+4, (C2×C42)⋊8C22, C243C4.13C2, C23.Q875C2, C23.4Q854C2, C23.179(C4○D4), (C22×C4).562C23, C23.11D4102C2, C2.C4240C22, C24.C22150C2, C2.77(C22.32C24), C2.21(C22.54C24), C2.87(C22.45C24), (C2×C4⋊C4)⋊36C22, C22.496(C2×C4○D4), (C2×C22⋊C4).297C22, SmallGroup(128,1467)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.635C24
C1C2C22C23C24C25C243C4 — C23.635C24
C1C23 — C23.635C24
C1C23 — C23.635C24
C1C23 — C23.635C24

Generators and relations for C23.635C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=c, e2=b, 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: 660 in 276 conjugacy classes, 88 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×12], C22 [×3], C22 [×4], C22 [×46], C2×C4 [×36], C23, C23 [×6], C23 [×46], C42 [×2], C22⋊C4 [×24], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×10], C24, C24 [×2], C24 [×10], C2.C42 [×2], C2.C42 [×6], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×16], C2×C4⋊C4 [×2], C2×C4⋊C4 [×4], C25, C243C4, C243C4 [×2], C24.C22 [×6], C23.Q8 [×2], C23.11D4, C23.11D4 [×2], C23.4Q8, C23.635C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ 1+4 [×4], C22.32C24 [×3], C22.45C24 [×3], C22.54C24, C23.635C24

Smallest permutation representation of C23.635C24
On 32 points
Generators in S32
(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 29)(22 30)(23 31)(24 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 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 31 27 23)(2 18 28 7)(3 29 25 21)(4 20 26 5)(6 11 17 15)(8 9 19 13)(10 30 14 22)(12 32 16 24)
(1 25)(2 4)(3 27)(6 17)(8 19)(9 15)(10 12)(11 13)(14 16)(21 29)(23 31)(26 28)
(1 3)(2 14)(4 16)(5 18)(6 21)(7 20)(8 23)(9 11)(10 28)(12 26)(13 15)(17 29)(19 31)(22 32)(24 30)(25 27)

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

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

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

32 conjugacy classes

class 1 2A···2G2H···2M4A···4L4M···4R
order12···22···24···44···4
size11···14···44···48···8

32 irreducible representations

dim11111124
type+++++++
imageC1C2C2C2C2C2C4○D42+ 1+4
kernelC23.635C24C243C4C24.C22C23.Q8C23.11D4C23.4Q8C23C22
# reps136231124

Matrix representation of C23.635C24 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
200000
020000
000100
001000
000043
000001
,
010000
100000
003000
000300
000031
000022
,
400000
010000
004000
000100
000010
000001
,
400000
040000
001000
000400
000010
000044

G:=sub<GL(6,GF(5))| [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],[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],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,3,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,2,0,0,0,0,1,2],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,4,0,0,0,0,0,4] >;

C23.635C24 in GAP, Magma, Sage, TeX

C_2^3._{635}C_2^4
% in TeX

G:=Group("C2^3.635C2^4");
// GroupNames label

G:=SmallGroup(128,1467);
// by ID

G=gap.SmallGroup(128,1467);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,1571,346]);
// Polycyclic

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