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G = C23.240C24order 128 = 27

93rd central extension by C23 of C24

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

Aliases: C25.24C22, C24.213C23, C23.240C24, C22.742+ 1+4, C2.3D42, C222(C4×D4), C22≀C28C4, C2412(C2×C4), C22⋊C443D4, (C23×C4)⋊7C22, C2.6(D45D4), (C2×C42)⋊16C22, C23.417(C2×D4), C23.8Q815C2, C23.292(C4○D4), C23.23D412C2, C22.131(C23×C4), (C22×C4).762C23, C23.131(C22×C4), C22.111(C22×D4), C2.C4212C22, C24.3C2219C2, C24.C2217C2, (C22×D4).485C22, C2.4(C22.45C24), C2.29(C22.11C24), (C2×C4×D4)⋊11C2, C2.34(C2×C4×D4), (C2×D4)⋊18(C2×C4), (C4×C22⋊C4)⋊39C2, C22⋊C429(C2×C4), (C2×C4).886(C2×D4), (C2×C4⋊C4)⋊104C22, (C2×C22≀C2).5C2, C22⋊C44(C22⋊C4), (C22×C22⋊C4)⋊9C2, (C2×C4).39(C22×C4), (C2×C22⋊C4)⋊74C22, C22.125(C2×C4○D4), C22⋊C43(C2×C22⋊C4), SmallGroup(128,1090)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.240C24
C1C2C22C23C24C25C22×C22⋊C4 — C23.240C24
C1C22 — C23.240C24
C1C23 — C23.240C24
C1C23 — C23.240C24

Generators and relations for C23.240C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=e2=c, 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, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 940 in 464 conjugacy classes, 164 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×12], C4 [×18], C22 [×3], C22 [×12], C22 [×60], C2×C4 [×14], C2×C4 [×46], D4 [×20], C23, C23 [×16], C23 [×56], C42 [×5], C22⋊C4 [×20], C22⋊C4 [×17], C4⋊C4 [×5], C22×C4, C22×C4 [×10], C22×C4 [×25], C2×D4 [×12], C2×D4 [×10], C24 [×2], C24 [×6], C24 [×10], C2.C42 [×6], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×12], C2×C22⋊C4 [×8], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×D4 [×8], C22≀C2 [×8], C23×C4, C23×C4 [×4], C22×D4, C22×D4 [×2], C25, C4×C22⋊C4 [×2], C23.8Q8 [×2], C23.23D4, C23.23D4 [×2], C24.C22 [×2], C24.3C22, C22×C22⋊C4 [×2], C2×C4×D4 [×2], C2×C22≀C2, C23.240C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22×C4 [×14], C2×D4 [×12], C4○D4 [×4], C24, C4×D4 [×8], C23×C4, C22×D4 [×2], C2×C4○D4 [×2], 2+ 1+4 [×2], C2×C4×D4 [×2], C22.11C24, D42, D45D4 [×2], C22.45C24, C23.240C24

Smallest permutation representation of C23.240C24
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 21 3 23)(2 5 4 7)(6 11 8 9)(10 24 12 22)(13 17 15 19)(14 32 16 30)(18 28 20 26)(25 31 27 29)
(1 3)(2 26)(4 28)(5 18)(6 8)(7 20)(9 11)(10 16)(12 14)(13 15)(17 19)(21 23)(22 32)(24 30)(25 27)(29 31)
(1 25)(2 26)(3 27)(4 28)(5 32)(6 29)(7 30)(8 31)(9 15)(10 16)(11 13)(12 14)(17 21)(18 22)(19 23)(20 24)

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

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

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

50 conjugacy classes

class 1 2A···2G2H···2O2P2Q2R2S4A···4P4Q···4AD
order12···22···222224···44···4
size11···12···244442···24···4

50 irreducible representations

dim1111111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C4D4C4○D42+ 1+4
kernelC23.240C24C4×C22⋊C4C23.8Q8C23.23D4C24.C22C24.3C22C22×C22⋊C4C2×C4×D4C2×C22≀C2C22≀C2C22⋊C4C23C22
# reps12232122116882

Matrix representation of C23.240C24 in GL5(𝔽5)

10000
04000
00400
00010
00001
,
10000
01000
00100
00040
00004
,
40000
04000
00400
00010
00001
,
20000
03000
00200
00032
00012
,
20000
00100
04000
00010
00001
,
10000
04000
00400
00010
00024
,
10000
04000
00100
00040
00004

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

C23.240C24 in GAP, Magma, Sage, TeX

C_2^3._{240}C_2^4
% in TeX

G:=Group("C2^3.240C2^4");
// GroupNames label

G:=SmallGroup(128,1090);
// by ID

G=gap.SmallGroup(128,1090);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,346]);
// 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=c,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,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
// generators/relations

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