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## G = C5×C23⋊C4order 160 = 25·5

### Direct product of C5 and C23⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C23⋊C4
 Chief series C1 — C2 — C22 — C23 — C22×C10 — C5×C22⋊C4 — C5×C23⋊C4
 Lower central C1 — C2 — C22 — C5×C23⋊C4
 Upper central C1 — C10 — C22×C10 — C5×C23⋊C4

Generators and relations for C5×C23⋊C4
G = < a,b,c,d,e | a5=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Smallest permutation representation of C5×C23⋊C4
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 36)(2 37)(3 38)(4 39)(5 40)(6 34)(7 35)(8 31)(9 32)(10 33)(11 22)(12 23)(13 24)(14 25)(15 21)(16 27)(17 28)(18 29)(19 30)(20 26)
(1 35)(2 31)(3 32)(4 33)(5 34)(6 40)(7 36)(8 37)(9 38)(10 39)(11 17)(12 18)(13 19)(14 20)(15 16)(21 27)(22 28)(23 29)(24 30)(25 26)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 37)(12 38)(13 39)(14 40)(15 36)(26 34)(27 35)(28 31)(29 32)(30 33)
(6 14 20 40)(7 15 16 36)(8 11 17 37)(9 12 18 38)(10 13 19 39)(26 34)(27 35)(28 31)(29 32)(30 33)

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

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

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

C5×C23⋊C4 is a maximal subgroup of   C53C2≀C4  (C2×C20).D4  C23.D20  C23.2D20  C23⋊C45D5  C23⋊D20  C23.5D20

55 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4E 5A 5B 5C 5D 10A 10B 10C 10D 10E ··· 10P 10Q 10R 10S 10T 20A ··· 20T order 1 2 2 2 2 2 4 ··· 4 5 5 5 5 10 10 10 10 10 ··· 10 10 10 10 10 20 ··· 20 size 1 1 2 2 2 4 4 ··· 4 1 1 1 1 1 1 1 1 2 ··· 2 4 4 4 4 4 ··· 4

55 irreducible representations

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

Matrix representation of C5×C23⋊C4 in GL4(𝔽41) generated by

 37 0 0 0 0 37 0 0 0 0 37 0 0 0 0 37
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 40 0 0 0 39 1 0 0 0 0 40 0 0 0 39 1
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 1 40 0 0 0 40 0 0 0 0 40 1 0 0 39 1
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[40,39,0,0,0,1,0,0,0,0,40,39,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,40,40,0,0,0,0,40,39,0,0,1,1] >;

C5×C23⋊C4 in GAP, Magma, Sage, TeX

C_5\times C_2^3\rtimes C_4
% in TeX

G:=Group("C5xC2^3:C4");
// GroupNames label

G:=SmallGroup(160,49);
// by ID

G=gap.SmallGroup(160,49);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,2403,1810]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations

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