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## G = C5×C23.D5order 400 = 24·52

### Direct product of C5 and C23.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×C23.D5
 Chief series C1 — C5 — C10 — C2×C10 — C102 — C10×Dic5 — C5×C23.D5
 Lower central C5 — C10 — C5×C23.D5
 Upper central C1 — C2×C10 — C22×C10

Generators and relations for C5×C23.D5
G = < a,b,c,d,e,f | a5=b2=c2=d2=e5=1, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 212 in 100 conjugacy classes, 38 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C5, C2×C4, C23, C10, C10, C10, C22⋊C4, Dic5, C20, C2×C10, C2×C10, C2×C10, C52, C2×Dic5, C2×C20, C22×C10, C22×C10, C5×C10, C5×C10, C5×C10, C23.D5, C5×C22⋊C4, C5×Dic5, C102, C102, C102, C10×Dic5, C2×C102, C5×C23.D5
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, D5, C10, C22⋊C4, Dic5, C20, D10, C2×C10, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×D5, C23.D5, C5×C22⋊C4, C5×Dic5, D5×C10, C10×Dic5, C5×C5⋊D4, C5×C23.D5

Smallest permutation representation of C5×C23.D5
On 40 points
Generators in S40
(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 33 34 35)(36 37 38 39 40)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 25 24 23 22)(26 30 29 28 27)(31 35 34 33 32)(36 40 39 38 37)
(1 32 11 22)(2 33 12 23)(3 34 13 24)(4 35 14 25)(5 31 15 21)(6 37 16 27)(7 38 17 28)(8 39 18 29)(9 40 19 30)(10 36 20 26)

G:=sub<Sym(40)| (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)(31,32,33,34,35)(36,37,38,39,40), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)(36,40,39,38,37), (1,32,11,22)(2,33,12,23)(3,34,13,24)(4,35,14,25)(5,31,15,21)(6,37,16,27)(7,38,17,28)(8,39,18,29)(9,40,19,30)(10,36,20,26)>;

G:=Group( (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)(31,32,33,34,35)(36,37,38,39,40), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)(36,40,39,38,37), (1,32,11,22)(2,33,12,23)(3,34,13,24)(4,35,14,25)(5,31,15,21)(6,37,16,27)(7,38,17,28)(8,39,18,29)(9,40,19,30)(10,36,20,26) );

G=PermutationGroup([[(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),(31,32,33,34,35),(36,37,38,39,40)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,25,24,23,22),(26,30,29,28,27),(31,35,34,33,32),(36,40,39,38,37)], [(1,32,11,22),(2,33,12,23),(3,34,13,24),(4,35,14,25),(5,31,15,21),(6,37,16,27),(7,38,17,28),(8,39,18,29),(9,40,19,30),(10,36,20,26)]])

130 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 5C 5D 5E ··· 5N 10A ··· 10L 10M ··· 10CL 20A ··· 20P order 1 2 2 2 2 2 4 4 4 4 5 5 5 5 5 ··· 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 10 10 10 10 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 10 ··· 10

130 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C4 C5 C10 C10 C20 D4 D5 Dic5 D10 C5⋊D4 C5×D4 C5×D5 C5×Dic5 D5×C10 C5×C5⋊D4 kernel C5×C23.D5 C10×Dic5 C2×C102 C102 C23.D5 C2×Dic5 C22×C10 C2×C10 C5×C10 C22×C10 C2×C10 C2×C10 C10 C10 C23 C22 C22 C2 # reps 1 2 1 4 4 8 4 16 2 2 4 2 8 8 8 16 8 32

Matrix representation of C5×C23.D5 in GL3(𝔽41) generated by

 10 0 0 0 10 0 0 0 10
,
 1 0 0 0 40 0 0 0 1
,
 40 0 0 0 40 0 0 0 40
,
 1 0 0 0 40 0 0 0 40
,
 1 0 0 0 10 0 0 0 37
,
 9 0 0 0 0 37 0 31 0
G:=sub<GL(3,GF(41))| [10,0,0,0,10,0,0,0,10],[1,0,0,0,40,0,0,0,1],[40,0,0,0,40,0,0,0,40],[1,0,0,0,40,0,0,0,40],[1,0,0,0,10,0,0,0,37],[9,0,0,0,0,31,0,37,0] >;

C5×C23.D5 in GAP, Magma, Sage, TeX

C_5\times C_2^3.D_5
% in TeX

G:=Group("C5xC2^3.D5");
// GroupNames label

G:=SmallGroup(400,91);
// by ID

G=gap.SmallGroup(400,91);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,505,11525]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^5=1,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

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