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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

 Derived series C1 — C2 — C2×C2.C25
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C2×C2.C25
 Lower central C1 — C2 — C2×C2.C25
 Upper central C1 — C2×C4 — C2×C2.C25
 Jennings C1 — C2 — 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, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C23×C4, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×C4○D4, C2×2+ 1+4, C2×2- 1+4, C2.C25, C2×C2.C25
Quotients: C1, C2, C22, C23, C24, C25, C2.C25, 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 2A 2B 2C 2D ··· 2AG 4A 4B 4C 4D 4E ··· 4AH order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 4 type + + + + + image C1 C2 C2 C2 C2 C2.C25 kernel C2×C2.C25 C22×C4○D4 C2×2+ 1+4 C2×2- 1+4 C2.C25 C2 # reps 1 15 10 6 32 4

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

 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 0 0 0 0 0 0 4 0 2 4 4 1 0 0 4 0 0
,
 1 0 0 0 0 0 2 3 3 0 0 4 3 3 2 0 0 0 0 3 0 0 0 2 0
,
 4 0 0 0 0 0 1 0 0 0 0 2 4 4 1 0 0 0 0 4 0 0 0 4 0
,
 4 0 0 0 0 0 1 0 4 1 0 0 1 0 0 0 0 0 4 0 0 0 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

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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