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## G = C23×D5order 80 = 24·5

### Direct product of C23 and D5

Aliases: C23×D5, C5⋊C24, C10⋊C23, (C22×C10)⋊3C2, (C2×C10)⋊4C22, SmallGroup(80,51)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C23×D5
 Chief series C1 — C5 — D5 — D10 — C22×D5 — C23×D5
 Lower central C5 — C23×D5
 Upper central C1 — C23

Generators and relations for C23×D5
G = < a,b,c,d,e | a2=b2=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 338 in 134 conjugacy classes, 83 normal (5 characteristic)
C1, C2 [×7], C2 [×8], C22 [×7], C22 [×28], C5, C23, C23 [×14], D5 [×8], C10 [×7], C24, D10 [×28], C2×C10 [×7], C22×D5 [×14], C22×C10, C23×D5
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], C22×D5 [×7], C23×D5

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

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

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

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

C23×D5 is a maximal subgroup of   C22⋊D20  C23⋊D10
C23×D5 is a maximal quotient of   D46D10  Q8.10D10  D48D10  D4.10D10

32 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 5A 5B 10A ··· 10N order 1 2 ··· 2 2 ··· 2 5 5 10 ··· 10 size 1 1 ··· 1 5 ··· 5 2 2 2 ··· 2

32 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D5 D10 kernel C23×D5 C22×D5 C22×C10 C23 C22 # reps 1 14 1 2 14

Matrix representation of C23×D5 in GL4(𝔽11) generated by

 1 0 0 0 0 1 0 0 0 0 10 0 0 0 0 10
,
 10 0 0 0 0 1 0 0 0 0 10 0 0 0 0 10
,
 1 0 0 0 0 10 0 0 0 0 10 0 0 0 0 10
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 10 3
,
 10 0 0 0 0 10 0 0 0 0 0 10 0 0 10 0
G:=sub<GL(4,GF(11))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[10,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,0,10,0,0,1,3],[10,0,0,0,0,10,0,0,0,0,0,10,0,0,10,0] >;

C23×D5 in GAP, Magma, Sage, TeX

C_2^3\times D_5
% in TeX

G:=Group("C2^3xD5");
// GroupNames label

G:=SmallGroup(80,51);
// by ID

G=gap.SmallGroup(80,51);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,1604]);
// Polycyclic

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

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