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G = C22.79C25order 128 = 27

60th central stem extension by C22 of C25

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

Aliases: C423C23, C25.78C22, C22.79C25, C23.131C24, C24.132C23, C22.72+ 1+4, C4⋊C421C23, (C2×D4)⋊8C23, D45D416C2, (C2×Q8)⋊8C23, (C4×D4)⋊37C22, C236(C4○D4), C233D46C2, (C2×C4).72C24, (C22×C4)⋊4C23, C232Q84C2, C22≀C26C22, C4⋊D422C22, C22⋊C420C23, (C23×C4)⋊41C22, C22⋊Q825C22, C22.32C242C2, C422C21C22, C4.4D425C22, (C22×D4)⋊35C22, C22.11C2415C2, C42⋊C234C22, C22.45C242C2, C22.19C2426C2, C2.28(C2×2+ 1+4), C22.D450C22, C22⋊C4C22≀C2, (C2×C22≀C2)⋊26C2, (C2×C4○D4)⋊27C22, C2.44(C22×C4○D4), C22.26(C2×C4○D4), (C22×C22⋊C4)⋊34C2, (C2×C22⋊C4)⋊46C22, SmallGroup(128,2222)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.79C25
C1C2C22C23C24C25C22×C22⋊C4 — C22.79C25
C1C22 — C22.79C25
C1C22 — C22.79C25
C1C22 — C22.79C25

Generators and relations for C22.79C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=g2=1, d2=b, ab=ba, dcd-1=gcg=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1196 in 664 conjugacy classes, 392 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×16], C4 [×20], C22, C22 [×10], C22 [×64], C2×C4 [×20], C2×C4 [×32], D4 [×44], Q8 [×4], C23, C23 [×14], C23 [×56], C42 [×8], C22⋊C4 [×56], C4⋊C4 [×24], C22×C4 [×26], C22×C4 [×4], C2×D4 [×28], C2×D4 [×16], C2×Q8 [×4], C4○D4 [×8], C24 [×3], C24 [×6], C24 [×8], C2×C22⋊C4 [×2], C2×C22⋊C4 [×20], C42⋊C2 [×8], C4×D4 [×24], C22≀C2 [×20], C4⋊D4 [×16], C22⋊Q8 [×16], C22.D4 [×20], C4.4D4 [×8], C422C2 [×8], C23×C4 [×2], C22×D4 [×6], C2×C4○D4 [×4], C25, C22×C22⋊C4, C22.11C24 [×2], C2×C22≀C2 [×2], C22.19C24 [×4], C233D4, C22.32C24 [×4], C232Q8, D45D4 [×8], C22.45C24 [×8], C22.79C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2+ 1+4 [×4], C25, C22×C4○D4, C2×2+ 1+4 [×2], C22.79C25

Permutation representations of C22.79C25
On 16 points - transitive group 16T201
Generators in S16
(1 9)(2 10)(3 11)(4 12)(5 14)(6 15)(7 16)(8 13)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 5)(2 15)(3 7)(4 13)(6 10)(8 12)(9 14)(11 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 11)(2 12)(3 9)(4 10)(5 14)(6 15)(7 16)(8 13)
(1 3)(2 12)(4 10)(5 7)(6 13)(8 15)(9 11)(14 16)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)(9 11)(10 12)

G:=sub<Sym(16)| (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,5)(2,15)(3,7)(4,13)(6,10)(8,12)(9,14)(11,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,11)(2,12)(3,9)(4,10)(5,14)(6,15)(7,16)(8,13), (1,3)(2,12)(4,10)(5,7)(6,13)(8,15)(9,11)(14,16), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,11)(10,12)>;

G:=Group( (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,5)(2,15)(3,7)(4,13)(6,10)(8,12)(9,14)(11,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,11)(2,12)(3,9)(4,10)(5,14)(6,15)(7,16)(8,13), (1,3)(2,12)(4,10)(5,7)(6,13)(8,15)(9,11)(14,16), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,11)(10,12) );

G=PermutationGroup([(1,9),(2,10),(3,11),(4,12),(5,14),(6,15),(7,16),(8,13)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,5),(2,15),(3,7),(4,13),(6,10),(8,12),(9,14),(11,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,11),(2,12),(3,9),(4,10),(5,14),(6,15),(7,16),(8,13)], [(1,3),(2,12),(4,10),(5,7),(6,13),(8,15),(9,11),(14,16)], [(1,3),(2,4),(5,16),(6,13),(7,14),(8,15),(9,11),(10,12)])

G:=TransitiveGroup(16,201);

44 conjugacy classes

class 1 2A2B2C2D···2M2N···2S4A···4H4I···4X
order12222···22···24···44···4
size11112···24···42···24···4

44 irreducible representations

dim111111111124
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2C4○D42+ 1+4
kernelC22.79C25C22×C22⋊C4C22.11C24C2×C22≀C2C22.19C24C233D4C22.32C24C232Q8D45D4C22.45C24C23C22
# reps112241418884

Matrix representation of C22.79C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
130000
040000
000010
000001
001000
000100
,
300000
030000
000100
001000
000004
000040
,
400000
410000
001000
000100
000010
000001
,
400000
040000
001000
000400
000010
000004
,
100000
010000
001000
000100
000040
000004

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

C22.79C25 in GAP, Magma, Sage, TeX

C_2^2._{79}C_2^5
% in TeX

G:=Group("C2^2.79C2^5");
// GroupNames label

G:=SmallGroup(128,2222);
// by ID

G=gap.SmallGroup(128,2222);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,570,1684]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=g^2=1,d^2=b,a*b=b*a,d*c*d^-1=g*c*g=a*c=c*a,f*d*f=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*f=f*c,d*e=e*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations

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