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G = C22.73C25order 128 = 27

54th central stem extension by C22 of C25

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

Aliases: C243C23, C422C23, C256C22, C22.73C25, C23.33C24, C2212+ 1+4, D4211C2, D4C22≀C2, D413(C2×D4), (C2×D4)⋊53D4, C4⋊C48C23, C237(C2×D4), D45D412C2, (D4×C23)⋊16C2, (C2×D4)⋊19C23, (C4×D4)⋊32C22, C233D44C2, C22⋊C49C23, (C2×C4).67C24, (C22×C4)⋊3C23, (C2×Q8)⋊18C23, C2.25(D4×C23), C22≀C23C22, C4⋊D478C22, C41D415C22, (C23×C4)⋊38C22, C4.114(C22×D4), C22⋊Q893C22, C22.9(C22×D4), C4.4D419C22, (C22×D4)⋊32C22, (C2×2+ 1+4)⋊7C2, C22.11C2413C2, C22.29C2418C2, C42⋊C230C22, C22.19C2420C2, C2.26(C2×2+ 1+4), C22.D448C22, (C2×C4)⋊12(C2×D4), (C2×D4)C22≀C2, (C2×C22≀C2)⋊25C2, (C2×C4○D4)⋊22C22, (C2×C22⋊C4)⋊43C22, SmallGroup(128,2216)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.73C25
C1C2C22C23C24C25D4×C23 — C22.73C25
C1C22 — C22.73C25
C1C22 — C22.73C25
C1C22 — C22.73C25

Generators and relations for C22.73C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g2=1, ab=ba, dcd=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: 1836 in 950 conjugacy classes, 432 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×22], C4 [×4], C4 [×14], C22, C22 [×14], C22 [×102], C2×C4 [×20], C2×C4 [×34], D4 [×16], D4 [×84], Q8 [×4], C23, C23 [×22], C23 [×106], C42 [×4], C22⋊C4 [×44], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×20], C22×C4 [×4], C2×D4 [×62], C2×D4 [×92], C2×Q8 [×2], C4○D4 [×24], C24, C24 [×16], C24 [×16], C2×C22⋊C4 [×12], C42⋊C2 [×2], C4×D4 [×16], C22≀C2 [×36], C4⋊D4 [×28], C22⋊Q8 [×4], C22.D4 [×12], C4.4D4 [×4], C41D4 [×4], C23×C4, C22×D4 [×2], C22×D4 [×26], C22×D4 [×8], C2×C4○D4 [×6], 2+ 1+4 [×8], C25 [×2], C22.11C24, C2×C22≀C2 [×4], C22.19C24 [×2], C233D4 [×4], C22.29C24 [×2], D42 [×8], D45D4 [×8], D4×C23, C2×2+ 1+4, C22.73C25
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], 2+ 1+4 [×4], C25, D4×C23, C2×2+ 1+4 [×2], C22.73C25

Permutation representations of C22.73C25
On 16 points - transitive group 16T198
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 5)(2 6)(3 15)(4 16)(7 10)(8 9)(11 14)(12 13)
(1 12)(2 11)(3 10)(4 9)(5 13)(6 14)(7 15)(8 16)
(1 8)(2 7)(3 13)(4 14)(5 9)(6 10)(11 16)(12 15)
(1 6)(2 5)(3 4)(7 9)(8 10)(11 12)(13 14)(15 16)
(1 5)(2 6)(3 16)(4 15)(7 9)(8 10)(11 14)(12 13)
(1 5)(2 6)(3 16)(4 15)(7 10)(8 9)(11 13)(12 14)

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

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

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

G:=TransitiveGroup(16,198);

44 conjugacy classes

class 1 2A2B2C2D···2Q2R···2Y4A4B4C4D4E···4R
order12222···22···244444···4
size11112···24···422224···4

44 irreducible representations

dim111111111124
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D42+ 1+4
kernelC22.73C25C22.11C24C2×C22≀C2C22.19C24C233D4C22.29C24D42D45D4D4×C23C2×2+ 1+4C2×D4C22
# reps114242881184

Matrix representation of C22.73C25 in GL6(ℤ)

100000
010000
00-1000
000-100
0000-10
00000-1
,
-100000
0-10000
001000
000100
000010
000001
,
010000
100000
000010
000001
001000
000100
,
100000
010000
000100
001000
00000-1
0000-10
,
-100000
010000
001000
000100
000010
000001
,
-100000
0-10000
001000
000-100
000010
00000-1
,
100000
010000
001000
000100
0000-10
00000-1

G:=sub<GL(6,Integers())| [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,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,-1] >;

C22.73C25 in GAP, Magma, Sage, TeX

C_2^2._{73}C_2^5
% in TeX

G:=Group("C2^2.73C2^5");
// GroupNames label

G:=SmallGroup(128,2216);
// by ID

G=gap.SmallGroup(128,2216);
# by ID

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

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