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G = C2×C2.C25order 128 = 27

Direct product of C2 and C2.C25

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

Aliases: C2×C2.C25, C2.6C26, C4.24C25, D4.12C24, C22.2C25, Q8.12C24, C23.98C24, C24.519C23, 2+ 1+413C22, 2- 1+412C22, C4○D410C23, (C2×D4)⋊23C23, (C2×Q8)⋊24C23, (C2×C4)2- 1+4, C4(C2×2- 1+4), C4(C2×2+ 1+4), (C2×C4)2+ 1+4, C4(C2.C25), (C2×C4).619C24, (C23×C4)⋊54C22, (C22×C4)⋊20C23, (C22×D4)⋊67C22, (C22×Q8)⋊72C22, (C2×2+ 1+4)⋊17C2, (C2×2- 1+4)⋊15C2, D4(C2×C4○D4), Q8(C2×C4○D4), (C2×D4)(C4○D4), (C2×Q8)(C4○D4), C4○D42(C4○D4), (C2×C4○D4)⋊81C22, (C22×C4○D4)⋊30C2, (C2×C4)(C2×2- 1+4), C4○D4(C2×C4○D4), (C2×Q8)(C2×C4○D4), (C2×C4○D4)(C2×C4○D4), SmallGroup(128,2325)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×C2.C25
C1C2C22C2×C4C22×C4C23×C4C22×C4○D4 — C2×C2.C25
C1C2 — C2×C2.C25
C1C2×C4 — C2×C2.C25
C1C2 — C2×C2.C25

Generators and relations for C2×C2.C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, dcd=fcf=bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, ce=ec, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 3308 in 2978 conjugacy classes, 2828 normal (5 characteristic)
C1, C2, C2 [×2], C2 [×30], C4 [×32], C22, C22 [×30], C22 [×90], C2×C4, C2×C4 [×255], D4 [×240], Q8 [×80], C23 [×75], C23 [×30], C22×C4 [×180], C2×D4 [×420], C2×Q8 [×140], C4○D4 [×640], C24 [×15], C23×C4 [×15], C22×D4 [×45], C22×Q8 [×15], C2×C4○D4 [×320], 2+ 1+4 [×160], 2- 1+4 [×96], C22×C4○D4 [×15], C2×2+ 1+4 [×10], C2×2- 1+4 [×6], C2.C25 [×32], C2×C2.C25
Quotients: C1, C2 [×63], C22 [×651], C23 [×1395], C24 [×651], C25 [×63], C2.C25 [×2], C26, C2×C2.C25

Smallest permutation representation of C2×C2.C25
On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 30)(6 31)(7 32)(8 29)(13 18)(14 19)(15 20)(16 17)(21 26)(22 27)(23 28)(24 25)
(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 23)(3 24)(4 21)(5 16)(6 13)(7 14)(8 15)(9 27)(10 28)(11 25)(12 26)(17 30)(18 31)(19 32)(20 29)
(1 13)(2 14)(3 15)(4 16)(5 23)(6 24)(7 21)(8 22)(9 18)(10 19)(11 20)(12 17)(25 31)(26 32)(27 29)(28 30)
(1 14)(2 15)(3 16)(4 13)(5 24)(6 21)(7 22)(8 23)(9 19)(10 20)(11 17)(12 18)(25 30)(26 31)(27 32)(28 29)
(5 7)(6 8)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(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)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D···2AG4A4B4C4D4E···4AH
order12222···244444···4
size11112···211112···2

68 irreducible representations

dim111114
type+++++
imageC1C2C2C2C2C2.C25
kernelC2×C2.C25C22×C4○D4C2×2+ 1+4C2×2- 1+4C2.C25C2
# reps115106324

Matrix representation of C2×C2.C25 in GL5(𝔽5)

40000
01000
00100
00010
00001
,
10000
04000
00400
00040
00004
,
40000
01000
00004
02441
00400
,
10000
02330
04332
00003
00020
,
40000
01000
02441
00004
00040
,
40000
01041
00100
00040
00004
,
40000
03000
00300
00030
00003

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,2,0,0,0,0,4,4,0,0,0,4,0,0,0,4,1,0],[1,0,0,0,0,0,2,4,0,0,0,3,3,0,0,0,3,3,0,2,0,0,2,3,0],[4,0,0,0,0,0,1,2,0,0,0,0,4,0,0,0,0,4,0,4,0,0,1,4,0],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,4,0,4,0,0,1,0,0,4],[4,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3] >;

C2×C2.C25 in GAP, Magma, Sage, TeX

C_2\times C_2.C_2^5
% in TeX

G:=Group("C2xC2.C2^5");
// GroupNames label

G:=SmallGroup(128,2325);
// by ID

G=gap.SmallGroup(128,2325);
# by ID

G:=PCGroup([7,-2,2,2,2,2,2,-2,925,723,2019,172]);
// Polycyclic

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

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