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G = C2×C22.45C24order 128 = 27

Direct product of C2 and C22.45C24

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

Aliases: C2×C22.45C24, C4214C23, C22.58C25, C25.77C22, C24.495C23, C23.275C24, C22.1112+ 1+4, C4⋊C420C23, (C2×Q8)⋊7C23, C22⋊C48C23, (C2×C4).58C24, (C4×D4)⋊106C22, (C23×C4)⋊17C22, (C2×C42)⋊53C22, (C22×C4)⋊10C23, C22⋊Q887C22, (C2×D4).454C23, C4.4D473C22, (C22×Q8)⋊28C22, C42⋊C294C22, C422C228C22, C22≀C2.23C22, C23.319(C4○D4), C2.18(C2×2+ 1+4), (C22×D4).590C22, C22.D443C22, (C2×C4×D4)⋊84C2, (C2×C22⋊Q8)⋊71C2, (C2×C4⋊C4)⋊135C22, (C2×C4.4D4)⋊51C2, C2.30(C22×C4○D4), C22.24(C2×C4○D4), (C2×C42⋊C2)⋊59C2, (C2×C422C2)⋊34C2, (C2×C22≀C2).17C2, (C22×C22⋊C4)⋊33C2, (C2×C22⋊C4)⋊88C22, (C2×C22.D4)⋊56C2, SmallGroup(128,2201)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C22.45C24
C1C2C22C23C24C25C22×C22⋊C4 — C2×C22.45C24
C1C22 — C2×C22.45C24
C1C23 — C2×C22.45C24
C1C22 — C2×C22.45C24

Generators and relations for C2×C22.45C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=b, e2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 1084 in 664 conjugacy classes, 404 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×22], C22, C22 [×14], C22 [×60], C2×C4 [×22], C2×C4 [×50], D4 [×20], Q8 [×4], C23, C23 [×16], C23 [×56], C42 [×12], C22⋊C4 [×56], C4⋊C4 [×32], C22×C4, C22×C4 [×30], C22×C4 [×14], C2×D4 [×12], C2×D4 [×10], C2×Q8 [×4], C2×Q8 [×2], C24 [×2], C24 [×6], C24 [×10], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×28], C2×C4⋊C4 [×8], C42⋊C2 [×16], C4×D4 [×16], C22≀C2 [×8], C22⋊Q8 [×16], C22.D4 [×24], C4.4D4 [×8], C422C2 [×16], C23×C4, C23×C4 [×4], C22×D4, C22×D4 [×2], C22×Q8, C25, C22×C22⋊C4 [×2], C2×C42⋊C2 [×2], C2×C4×D4 [×2], C2×C22≀C2, C2×C22⋊Q8 [×2], C2×C22.D4, C2×C22.D4 [×2], C2×C4.4D4, C2×C422C2 [×2], C22.45C24 [×16], C2×C22.45C24
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×8], C24 [×31], C2×C4○D4 [×12], 2+ 1+4 [×2], C25, C22.45C24 [×4], C22×C4○D4 [×2], C2×2+ 1+4, C2×C22.45C24

Smallest permutation representation of C2×C22.45C24
On 32 points
Generators in S32
(1 25)(2 26)(3 27)(4 28)(5 12)(6 9)(7 10)(8 11)(13 31)(14 32)(15 29)(16 30)(17 22)(18 23)(19 24)(20 21)
(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 11)(2 12)(3 9)(4 10)(5 26)(6 27)(7 28)(8 25)(13 24)(14 21)(15 22)(16 23)(17 29)(18 30)(19 31)(20 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 11 19)(2 30 12 18)(3 29 9 17)(4 32 10 20)(5 23 26 16)(6 22 27 15)(7 21 28 14)(8 24 25 13)
(2 12)(4 10)(5 26)(7 28)(14 21)(16 23)(18 30)(20 32)
(1 11)(2 12)(3 9)(4 10)(5 26)(6 27)(7 28)(8 25)(13 22)(14 23)(15 24)(16 21)(17 31)(18 32)(19 29)(20 30)

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

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

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

50 conjugacy classes

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

50 irreducible representations

dim111111111124
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2C4○D42+ 1+4
kernelC2×C22.45C24C22×C22⋊C4C2×C42⋊C2C2×C4×D4C2×C22≀C2C2×C22⋊Q8C2×C22.D4C2×C4.4D4C2×C422C2C22.45C24C23C22
# reps12221231216162

Matrix representation of C2×C22.45C24 in GL5(𝔽5)

40000
04000
00400
00040
00004
,
10000
01000
00100
00040
00004
,
10000
04000
00400
00010
00001
,
10000
00100
01000
00030
00042
,
40000
02000
00200
00022
00013
,
10000
01000
00400
00010
00001
,
40000
04000
00400
00010
00034

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

C2×C22.45C24 in GAP, Magma, Sage, TeX

C_2\times C_2^2._{45}C_2^4
% in TeX

G:=Group("C2xC2^2.45C2^4");
// GroupNames label

G:=SmallGroup(128,2201);
// by ID

G=gap.SmallGroup(128,2201);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,456,1430,570]);
// Polycyclic

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