Copied to
clipboard

G = C22.97C25order 128 = 27

78th central stem extension by C22 of C25

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

Aliases: C42.89C23, C22.97C25, C23.47C24, C24.138C23, C4.1562+ 1+4, D45D421C2, Q86D420C2, (C4×D4)⋊48C22, (C2×C4).87C24, C4⋊Q8109C22, (C4×Q8)⋊47C22, C4⋊D484C22, C41D418C22, C4⋊C4.492C23, (C2×C42)⋊62C22, C22≀C2.9C22, (C2×D4).305C23, C4.4D485C22, (C2×Q8).452C23, C42.C257C22, C22.29C2425C2, C22.11C2419C2, C22⋊C4.107C23, (C22×C4).368C23, C22.D49C22, C42⋊C2102C22, C22⋊Q8.228C22, C2.37(C2×2+ 1+4), C22.26C2440C2, (C22×D4).427C22, C22.34C2410C2, C23.37C2340C2, C22.53C2413C2, (C4×C4○D4)⋊29C2, (C2×C4)⋊7(C4○D4), (C2×C41D4)⋊27C2, C22⋊C4(C41D4), C4.180(C2×C4○D4), (C2×C4○D4)⋊34C22, C2.53(C22×C4○D4), C22.30(C2×C4○D4), (C2×C22⋊C4).383C22, SmallGroup(128,2240)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.97C25
C1C2C22C2×C4C22×C4C2×C42C4×C4○D4 — C22.97C25
C1C22 — C22.97C25
C1C22 — C22.97C25
C1C22 — C22.97C25

Generators and relations for C22.97C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=b, g2=a, ab=ba, dcd-1=gcg-1=ac=ca, fdf=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: 1052 in 616 conjugacy classes, 392 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×8], C4 [×16], C22, C22 [×2], C22 [×40], C2×C4 [×2], C2×C4 [×22], C2×C4 [×32], D4 [×62], Q8 [×10], C23, C23 [×10], C23 [×16], C42 [×2], C42 [×12], C22⋊C4 [×42], C4⋊C4 [×22], C22×C4, C22×C4 [×24], C2×D4, C2×D4 [×42], C2×D4 [×16], C2×Q8, C2×Q8 [×4], C4○D4 [×20], C24 [×4], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×8], C42⋊C2, C42⋊C2 [×4], C4×D4 [×30], C4×Q8 [×6], C22≀C2 [×8], C4⋊D4 [×32], C22⋊Q8 [×4], C22.D4 [×16], C4.4D4 [×12], C42.C2 [×2], C41D4 [×12], C4⋊Q8 [×2], C22×D4 [×6], C2×C4○D4, C2×C4○D4 [×8], C4×C4○D4, C22.11C24 [×2], C2×C41D4, C22.26C24 [×4], C23.37C23, C22.29C24 [×2], C22.34C24 [×4], D45D4 [×8], Q86D4 [×4], C22.53C24 [×4], C22.97C25
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], C22.97C25

Smallest permutation representation of C22.97C25
On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 29)(22 30)(23 31)(24 32)
(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 6)(2 18)(3 8)(4 20)(5 26)(7 28)(9 21)(10 30)(11 23)(12 32)(13 29)(14 22)(15 31)(16 24)(17 27)(19 25)
(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 21 3 23)(2 22 4 24)(5 10 7 12)(6 11 8 9)(13 17 15 19)(14 18 16 20)(25 31 27 29)(26 32 28 30)
(2 28)(4 26)(5 20)(7 18)(10 14)(12 16)(22 30)(24 32)
(1 15 27 11)(2 16 28 12)(3 13 25 9)(4 14 26 10)(5 22 20 30)(6 23 17 31)(7 24 18 32)(8 21 19 29)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F···2O4A···4P4Q···4AB
order1222222···24···44···4
size1111224···42···24···4

44 irreducible representations

dim1111111111124
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C4○D42+ 1+4
kernelC22.97C25C4×C4○D4C22.11C24C2×C41D4C22.26C24C23.37C23C22.29C24C22.34C24D45D4Q86D4C22.53C24C2×C4C4
# reps1121412484484

Matrix representation of C22.97C25 in GL6(𝔽5)

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

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],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;

C22.97C25 in GAP, Magma, Sage, TeX

C_2^2._{97}C_2^5
% in TeX

G:=Group("C2^2.97C2^5");
// GroupNames label

G:=SmallGroup(128,2240);
// by ID

G=gap.SmallGroup(128,2240);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,352,570,1684,102]);
// Polycyclic

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