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## G = C23×F5order 160 = 25·5

### Direct product of C23 and F5

Aliases: C23×F5, D5.C24, D10.15C23, C5⋊(C23×C4), C10⋊(C22×C4), D5⋊(C22×C4), D109(C2×C4), (C22×C10)⋊5C4, (C22×D5)⋊6C4, (C23×D5).4C2, (C22×D5).40C22, (C2×C10)⋊3(C2×C4), SmallGroup(160,236)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C23×F5
 Chief series C1 — C5 — D5 — F5 — C2×F5 — C22×F5 — C23×F5
 Lower central C5 — C23×F5
 Upper central C1 — C23

Generators and relations for C23×F5
G = < a,b,c,d,e | a2=b2=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 644 in 236 conjugacy classes, 134 normal (7 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C23, C23, D5, D5, C10, C22×C4, C24, F5, D10, C2×C10, C23×C4, C2×F5, C22×D5, C22×C10, C22×F5, C23×D5, C23×F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, F5, C23×C4, C2×F5, C22×F5, C23×F5

Smallest permutation representation of C23×F5
On 40 points
Generators in S40
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(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 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)
(1 16)(2 18 5 19)(3 20 4 17)(6 11)(7 13 10 14)(8 15 9 12)(21 36)(22 38 25 39)(23 40 24 37)(26 31)(27 33 30 34)(28 35 29 32)

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

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

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

C23×F5 is a maximal subgroup of   D10⋊(C4⋊C4)  (C2×F5)⋊D4
C23×F5 is a maximal quotient of   Dic5.C24  D10.C24  Dic5.20C24  D5.2- 1+4  Dic5.21C24  Dic5.22C24  D5.2+ 1+4

40 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4P 5 10A ··· 10G order 1 2 ··· 2 2 ··· 2 4 ··· 4 5 10 ··· 10 size 1 1 ··· 1 5 ··· 5 5 ··· 5 4 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 4 4 type + + + + + image C1 C2 C2 C4 C4 F5 C2×F5 kernel C23×F5 C22×F5 C23×D5 C22×D5 C22×C10 C23 C22 # reps 1 14 1 14 2 1 7

Matrix representation of C23×F5 in GL7(𝔽41)

 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 40 0 0 0 1 0 0 40 0 0 0 0 1 0 40 0 0 0 0 0 1 40
,
 32 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0

G:=sub<GL(7,GF(41))| [1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,40,40,40,40],[32,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;

C23×F5 in GAP, Magma, Sage, TeX

C_2^3\times F_5
% in TeX

G:=Group("C2^3xF5");
// GroupNames label

G:=SmallGroup(160,236);
// by ID

G=gap.SmallGroup(160,236);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,2309,317]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^5=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,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations

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