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G = C23.636C24order 128 = 27

353rd central stem extension by C23 of C24

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

Aliases: C25.62C22, C24.425C23, C23.636C24, C22.4092+ 1+4, (C2×C42)⋊9C22, C23⋊Q852C2, (C22×Q8)⋊9C22, C243C4.14C2, C23.180(C4○D4), (C22×C4).563C23, C23.11D4103C2, C2.C4241C22, C2.8(C24⋊C22), C24.C22151C2, C2.78(C22.32C24), C2.88(C22.45C24), (C2×C4⋊C4)⋊37C22, C22.497(C2×C4○D4), (C2×C22⋊C4).298C22, SmallGroup(128,1468)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.636C24
C1C2C22C23C24C25C243C4 — C23.636C24
C1C23 — C23.636C24
C1C23 — C23.636C24
C1C23 — C23.636C24

Generators and relations for C23.636C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=ca=ac, e2=ba=ab, 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: 676 in 280 conjugacy classes, 88 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×12], C22, C22 [×6], C22 [×46], C2×C4 [×36], Q8 [×4], C23, C23 [×6], C23 [×46], C42 [×2], C22⋊C4 [×24], C4⋊C4 [×3], C22×C4 [×12], C2×Q8 [×3], C24 [×3], C24 [×10], C2.C42, C2.C42 [×9], C2×C42 [×2], C2×C22⋊C4 [×18], C2×C4⋊C4 [×3], C22×Q8, C25, C243C4 [×3], C24.C22 [×6], C23⋊Q8 [×3], C23.11D4 [×3], C23.636C24
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], C24⋊C22, C23.636C24

Smallest permutation representation of C23.636C24
On 32 points
Generators in S32
(1 10)(2 11)(3 12)(4 9)(5 21)(6 22)(7 23)(8 24)(13 27)(14 28)(15 25)(16 26)(17 32)(18 29)(19 30)(20 31)
(1 20)(2 17)(3 18)(4 19)(5 15)(6 16)(7 13)(8 14)(9 30)(10 31)(11 32)(12 29)(21 25)(22 26)(23 27)(24 28)
(1 12)(2 9)(3 10)(4 11)(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 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 5 31 25)(2 22 32 16)(3 7 29 27)(4 24 30 14)(6 17 26 11)(8 19 28 9)(10 21 20 15)(12 23 18 13)
(1 29)(2 9)(3 31)(4 11)(5 15)(7 13)(10 18)(12 20)(17 30)(19 32)(21 25)(23 27)
(1 12)(2 19)(3 10)(4 17)(5 7)(6 28)(8 26)(9 32)(11 30)(13 15)(14 22)(16 24)(18 31)(20 29)(21 23)(25 27)

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

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

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

dim1111124
type++++++
imageC1C2C2C2C2C4○D42+ 1+4
kernelC23.636C24C243C4C24.C22C23⋊Q8C23.11D4C23C22
# reps13633124

Matrix representation of C23.636C24 in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
200000
020000
000300
003000
000001
000010
,
010000
100000
000100
004000
000020
000002
,
400000
010000
001000
000100
000040
000001
,
400000
040000
001000
000400
000010
000004

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

C23.636C24 in GAP, Magma, Sage, TeX

C_2^3._{636}C_2^4
% in TeX

G:=Group("C2^3.636C2^4");
// GroupNames label

G:=SmallGroup(128,1468);
// by ID

G=gap.SmallGroup(128,1468);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,344,758,723,1571,346]);
// Polycyclic

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