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## G = C2×C23.C23order 128 = 27

### Direct product of C2 and C23.C23

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C23.C23
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C4○D4 — C2×C23.C23
 Lower central C1 — C2 — C22 — C2×C23.C23
 Upper central C1 — C2×C4 — C23×C4 — C2×C23.C23
 Jennings C1 — C2 — C23 — C2×C23.C23

Generators and relations for C2×C23.C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=c, g2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ebe-1=bd=db, bf=fb, bg=gb, fcf=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef=bde, eg=ge, fg=gf >

Subgroups: 700 in 394 conjugacy classes, 172 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C23⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C23⋊C4, C23.C23, C2×C42⋊C2, C22×C4○D4, C2×C23.C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C23×C4, C22×D4, C23.C23, C22×C22⋊C4, C2×C23.C23

Smallest permutation representation of C2×C23.C23
On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 10)(6 11)(7 12)(8 9)(13 29)(14 30)(15 31)(16 32)(17 24)(18 21)(19 22)(20 23)
(1 6)(2 28)(3 8)(4 26)(5 10)(7 12)(9 25)(11 27)(13 17)(14 30)(15 19)(16 32)(18 21)(20 23)(22 31)(24 29)
(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 11)(2 12)(3 9)(4 10)(5 26)(6 27)(7 28)(8 25)(13 24)(14 21)(15 22)(16 23)(17 29)(18 30)(19 31)(20 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)
(1 17)(2 21)(3 31)(4 16)(5 20)(6 13)(7 30)(8 22)(9 19)(10 23)(11 29)(12 14)(15 25)(18 28)(24 27)(26 32)
(1 31 11 19)(2 32 12 20)(3 29 9 17)(4 30 10 18)(5 21 26 14)(6 22 27 15)(7 23 28 16)(8 24 25 13)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E ··· 4J 4K ··· 4AD order 1 2 2 2 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 size 1 1 1 1 2 ··· 2 4 4 4 4 1 1 1 1 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 4 type + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 D4 C23.C23 kernel C2×C23.C23 C2×C23⋊C4 C23.C23 C2×C42⋊C2 C22×C4○D4 C23×C4 C22×D4 C22×Q8 C2×C4○D4 C22×C4 C2 # reps 1 4 8 2 1 4 2 2 8 8 4

Matrix representation of C2×C23.C23 in GL6(𝔽5)

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

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,1,0,0,0,0,0,3,4],[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],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,1,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,2,2,0,0,0,0,1,3,0,0,0,0,0,0,2,2,0,0,0,0,1,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2] >;

C2×C23.C23 in GAP, Magma, Sage, TeX

C_2\times C_2^3.C_2^3
% in TeX

G:=Group("C2xC2^3.C2^3");
// GroupNames label

G:=SmallGroup(128,1614);
// by ID

G=gap.SmallGroup(128,1614);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,352,2804,2028]);
// Polycyclic

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

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