Copied to
clipboard

G = C22.95C25order 128 = 27

76th central stem extension by C22 of C25

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

Aliases: C42.87C23, C22.95C25, C23.46C24, C24.508C23, C4.812+ 1+4, C4⋊Q893C22, D46D423C2, Q86D419C2, (C4×D4)⋊47C22, (C2×C4).85C24, (C4×Q8)⋊46C22, C41D453C22, C4⋊D483C22, C4⋊C4.491C23, C22⋊Q896C22, C422C25C22, C22.32C245C2, (C2×D4).304C23, C4.4D427C22, (C2×Q8).290C23, C42.C256C22, C42⋊C241C22, C22.19C2431C2, C22.29C2424C2, C22≀C2.29C22, C22⋊C4.105C23, (C2×C42).948C22, (C22×C4).366C23, (C23×C4).612C22, C2.36(C2×2+ 1+4), C2.29(C2.C25), C22.26C2439C2, (C22×D4).600C22, C22.D452C22, C22.47C2416C2, C22.50C2420C2, C23.37C2338C2, C22.53C2411C2, C23.33C2321C2, C22.49C2413C2, (C2×C4×D4)⋊92C2, (C2×C4)⋊6(C4○D4), (C2×C4⋊C4)⋊76C22, C4.178(C2×C4○D4), (C2×C4○D4)⋊33C22, C22.16(C2×C4○D4), C2.51(C22×C4○D4), (C2×C22⋊C4).547C22, SmallGroup(128,2238)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.95C25
C1C2C22C23C22×C4C2×C42C2×C4×D4 — C22.95C25
C1C22 — C22.95C25
C1C22 — C22.95C25
C1C22 — C22.95C25

Generators and relations for C22.95C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=e2=b, f2=a, ab=ba, dcd-1=gcg=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 884 in 570 conjugacy classes, 390 normal (30 characteristic)
C1, C2 [×3], C2 [×10], C4 [×6], C4 [×19], C22, C22 [×2], C22 [×34], C2×C4 [×6], C2×C4 [×20], C2×C4 [×33], D4 [×42], Q8 [×10], C23, C23 [×8], C23 [×8], C42 [×4], C42 [×12], C22⋊C4 [×44], C4⋊C4 [×32], C22×C4 [×3], C22×C4 [×22], C22×C4 [×4], C2×D4 [×30], C2×D4 [×4], C2×Q8 [×6], C4○D4 [×16], C24 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×4], C42⋊C2 [×10], C4×D4 [×32], C4×Q8 [×8], C22≀C2 [×8], C4⋊D4 [×24], C22⋊Q8 [×8], C22.D4 [×16], C4.4D4 [×14], C42.C2 [×2], C422C2 [×8], C41D4, C41D4 [×4], C4⋊Q8 [×3], C23×C4 [×2], C22×D4, C2×C4○D4 [×6], C2×C4×D4, C23.33C23 [×2], C22.19C24 [×4], C22.26C24, C23.37C23, C22.29C24 [×2], C22.32C24 [×4], D46D4 [×2], Q86D4 [×2], C22.47C24 [×4], C22.49C24 [×2], C22.50C24 [×2], C22.53C24 [×4], C22.95C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2+ 1+4 [×2], C25, C22×C4○D4, C2×2+ 1+4, C2.C25, C22.95C25

Smallest permutation representation of C22.95C25
On 32 points
Generators in S32
(1 22)(2 23)(3 24)(4 21)(5 11)(6 12)(7 9)(8 10)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)
(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 14)(2 27)(3 16)(4 25)(5 31)(6 18)(7 29)(8 20)(9 19)(10 30)(11 17)(12 32)(13 21)(15 23)(22 26)(24 28)
(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 6 3 8)(2 7 4 5)(9 21 11 23)(10 22 12 24)(13 19 15 17)(14 20 16 18)(25 29 27 31)(26 30 28 32)
(1 9 22 7)(2 8 23 10)(3 11 24 5)(4 6 21 12)(13 32 25 18)(14 19 26 29)(15 30 27 20)(16 17 28 31)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 27)(14 28)(15 25)(16 26)(17 29)(18 30)(19 31)(20 32)(21 23)(22 24)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F···2M4A···4P4Q···4AD
order1222222···24···44···4
size1111224···42···24···4

44 irreducible representations

dim11111111111111244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D42+ 1+4C2.C25
kernelC22.95C25C2×C4×D4C23.33C23C22.19C24C22.26C24C23.37C23C22.29C24C22.32C24D46D4Q86D4C22.47C24C22.49C24C22.50C24C22.53C24C2×C4C4C2
# reps11241124224224822

Matrix representation of C22.95C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
040000
400000
000010
000001
001000
000100
,
200000
020000
000100
001000
000004
000040
,
300000
020000
001000
000100
000010
000001
,
100000
010000
000100
004000
000001
000040
,
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],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;

C22.95C25 in GAP, Magma, Sage, TeX

C_2^2._{95}C_2^5
% in TeX

G:=Group("C2^2.95C2^5");
// GroupNames label

G:=SmallGroup(128,2238);
// by ID

G=gap.SmallGroup(128,2238);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,352,570,136,1684]);
// Polycyclic

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

׿
×
𝔽