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G = C22.132C25order 128 = 27

113rd central stem extension by C22 of C25

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

Aliases: C23.73C24, C42.115C23, C24.149C23, C22.132C25, C4.1602+ 1+4, C22.152+ 1+4, D4222C2, D45D434C2, Q86D427C2, (C4×D4)⋊66C22, (C4×Q8)⋊63C22, C41D426C22, C4⋊C4.320C23, C233D414C2, C4⋊D439C22, (C2×C4).122C24, (C2×C42)⋊71C22, C22≀C216C22, (C2×D4).324C23, C4.4D491C22, (C22×D4)⋊46C22, C22⋊C4.47C23, C22⋊Q8101C22, (C2×Q8).465C23, C42.C265C22, C422C243C22, C22.29C2431C2, C22.54C249C2, C42⋊C260C22, (C22×C4).392C23, C2.61(C2×2+ 1+4), C22.D462C22, C22.34C2421C2, C23.36C2346C2, C22.53C2421C2, (C2×C41D4)⋊29C2, (C2×C4○D4)⋊49C22, (C2×C22⋊C4)⋊62C22, SmallGroup(128,2275)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.132C25
C1C2C22C23C22×C4C2×C42C2×C41D4 — C22.132C25
C1C22 — C22.132C25
C1C22 — C22.132C25
C1C22 — C22.132C25

Generators and relations for C22.132C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=b, f2=a, ab=ba, dcd=gcg=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf-1=bc=cb, ede-1=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1212 in 630 conjugacy classes, 384 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×14], C4 [×4], C4 [×16], C22, C22 [×2], C22 [×54], C2×C4 [×2], C2×C4 [×16], C2×C4 [×26], D4 [×68], Q8 [×4], C23, C23 [×12], C23 [×28], C42 [×2], C42 [×6], C22⋊C4 [×44], C4⋊C4 [×20], C22×C4, C22×C4 [×18], C2×D4 [×54], C2×D4 [×28], C2×Q8 [×2], C4○D4 [×8], C24 [×8], C2×C42, C2×C22⋊C4 [×4], C42⋊C2 [×4], C4×D4 [×14], C4×Q8 [×2], C22≀C2 [×24], C4⋊D4 [×42], C22⋊Q8 [×2], C22.D4 [×20], C4.4D4 [×6], C42.C2 [×2], C422C2 [×4], C41D4 [×2], C41D4 [×10], C22×D4 [×14], C2×C4○D4 [×4], C23.36C23 [×2], C2×C41D4, C233D4 [×4], C22.29C24 [×4], C22.34C24 [×4], D42, D42 [×4], D45D4 [×4], Q86D4 [×2], C22.53C24, C22.54C24 [×4], C22.132C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ 1+4 [×6], C25, C2×2+ 1+4 [×3], C22.132C25

Smallest permutation representation of C22.132C25
On 32 points
Generators in S32
(1 15)(2 16)(3 13)(4 14)(5 17)(6 18)(7 19)(8 20)(9 26)(10 27)(11 28)(12 25)(21 32)(22 29)(23 30)(24 31)
(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 22)(2 21)(3 24)(4 23)(5 25)(6 28)(7 27)(8 26)(9 20)(10 19)(11 18)(12 17)(13 31)(14 30)(15 29)(16 32)
(1 5)(2 8)(3 7)(4 6)(9 21)(10 24)(11 23)(12 22)(13 19)(14 18)(15 17)(16 20)(25 29)(26 32)(27 31)(28 30)
(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 18 15 6)(2 19 16 7)(3 20 13 8)(4 17 14 5)(9 29 26 22)(10 30 27 23)(11 31 28 24)(12 32 25 21)
(9 26)(10 27)(11 28)(12 25)(21 32)(22 29)(23 30)(24 31)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F···2Q4A4B4C4D4E···4T
order1222222···244444···4
size1111224···422224···4

38 irreducible representations

dim1111111111144
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C22+ 1+42+ 1+4
kernelC22.132C25C23.36C23C2×C41D4C233D4C22.29C24C22.34C24D42D45D4Q86D4C22.53C24C22.54C24C4C22
# reps1214445421442

Matrix representation of C22.132C25 in GL8(ℤ)

-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
-10-200000
00110000
00100000
01-100000
00000010
00000001
00001000
00000100
,
-1-2000000
01000000
110-10000
-1-1-100000
0000-1000
00000100
00000010
0000000-1
,
10000000
01000000
00100000
00010000
00000100
0000-1000
0000000-1
00000010
,
-1-2000000
11000000
01010000
-1-1-100000
00000100
0000-1000
0000000-1
00000010
,
10000000
01000000
-10-100000
100-10000
00001000
00000100
000000-10
0000000-1

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-2,1,1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[-1,0,1,-1,0,0,0,0,-2,1,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0],[-1,1,0,-1,0,0,0,0,-2,1,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0],[1,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;

C22.132C25 in GAP, Magma, Sage, TeX

C_2^2._{132}C_2^5
% in TeX

G:=Group("C2^2.132C2^5");
// GroupNames label

G:=SmallGroup(128,2275);
// by ID

G=gap.SmallGroup(128,2275);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,352,2019,570,136,1684]);
// Polycyclic

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

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