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G = C23×D5order 80 = 24·5

Direct product of C23 and D5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C5 — C23×D5
C1C5D5D10C22×D5 — C23×D5
C5 — C23×D5
C1C23

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···2G2H···2O5A5B10A···10N
order12···22···25510···10
size11···15···5222···2

32 irreducible representations

dim11122
type+++++
imageC1C2C2D5D10
kernelC23×D5C22×D5C22×C10C23C22
# reps1141214

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

1000
0100
00100
00010
,
10000
0100
00100
00010
,
1000
01000
00100
00010
,
1000
0100
0001
00103
,
10000
01000
00010
00100
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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