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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C22.45C24
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C2×C22.45C24
 Lower central C1 — C22 — C2×C22.45C24
 Upper central C1 — C23 — C2×C22.45C24
 Jennings C1 — C22 — 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 ··· 2G 2H ··· 2O 2P 2Q 2R 2S 4A ··· 4P 4Q ··· 4AD order 1 2 ··· 2 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4○D4 2+ 1+4 kernel C2×C22.45C24 C22×C22⋊C4 C2×C42⋊C2 C2×C4×D4 C2×C22≀C2 C2×C22⋊Q8 C2×C22.D4 C2×C4.4D4 C2×C42⋊2C2 C22.45C24 C23 C22 # reps 1 2 2 2 1 2 3 1 2 16 16 2

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

 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 0 0 0 0 4 2
,
 4 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 2 2 0 0 0 1 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 0 0 0 0 3 4

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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