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## G = C23⋊Dic5order 160 = 25·5

### The semidirect product of C23 and Dic5 acting via Dic5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C23⋊Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23.D5 — C23⋊Dic5
 Lower central C5 — C10 — C2×C10 — C23⋊Dic5
 Upper central C1 — C2 — C23 — C2×D4

Generators and relations for C23⋊Dic5
G = < a,b,c,d,e | a2=b2=c2=d10=1, e2=d5, ab=ba, dad-1=ac=ca, eae-1=abc, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Smallest permutation representation of C23⋊Dic5
On 40 points
Generators in S40
(1 14)(2 35)(3 16)(4 37)(5 18)(6 39)(7 20)(8 31)(9 12)(10 33)(11 30)(13 22)(15 24)(17 26)(19 28)(21 32)(23 34)(25 36)(27 38)(29 40)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 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 14 6 19)(2 13 7 18)(3 12 8 17)(4 11 9 16)(5 20 10 15)(21 36 26 31)(22 35 27 40)(23 34 28 39)(24 33 29 38)(25 32 30 37)

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

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

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

31 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 5A 5B 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D order 1 2 2 2 2 2 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 20 20 20 size 1 1 2 2 2 4 4 20 20 20 20 2 2 2 ··· 2 4 ··· 4 4 4 4 4

31 irreducible representations

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

Matrix representation of C23⋊Dic5 in GL4(𝔽41) generated by

 24 1 38 23 40 17 36 23 0 0 18 40 0 0 36 23
,
 24 1 0 0 40 17 0 0 0 0 23 1 0 0 5 18
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 40 17 0 0 24 3 0 0 40 33 18 24 7 6 38 21
,
 40 0 36 40 7 1 35 6 11 29 35 40 4 18 35 6
G:=sub<GL(4,GF(41))| [24,40,0,0,1,17,0,0,38,36,18,36,23,23,40,23],[24,40,0,0,1,17,0,0,0,0,23,5,0,0,1,18],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,24,40,7,17,3,33,6,0,0,18,38,0,0,24,21],[40,7,11,4,0,1,29,18,36,35,35,35,40,6,40,6] >;

C23⋊Dic5 in GAP, Magma, Sage, TeX

C_2^3\rtimes {\rm Dic}_5
% in TeX

G:=Group("C2^3:Dic5");
// GroupNames label

G:=SmallGroup(160,41);
// by ID

G=gap.SmallGroup(160,41);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,188,579,4613]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^10=1,e^2=d^5,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e^-1=a*b*c,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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