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G = C23.D5order 80 = 24·5

The non-split extension by C23 of D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.D5, C22⋊Dic5, C10.11D4, C22.7D10, (C2×C10)⋊4C4, C53(C22⋊C4), C10.16(C2×C4), (C2×Dic5)⋊2C2, C2.3(C5⋊D4), C2.5(C2×Dic5), (C22×C10).2C2, (C2×C10).7C22, SmallGroup(80,19)

Series: Derived Chief Lower central Upper central

C1C10 — C23.D5
C1C5C10C2×C10C2×Dic5 — C23.D5
C5C10 — C23.D5
C1C22C23

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

2C2
2C2
2C22
2C22
10C4
10C4
2C10
2C10
5C2×C4
5C2×C4
2Dic5
2C2×C10
2C2×C10
2Dic5
5C22⋊C4

Character table of C23.D5

 class 12A2B2C2D2E4A4B4C4D5A5B10A10B10C10D10E10F10G10H10I10J10K10L10M10N
 size 111122101010102222222222222222
ρ111111111111111111111111111    trivial
ρ21111-1-11-11-111-1-1-1-1-111-1-1-11111    linear of order 2
ρ31111-1-1-11-1111-1-1-1-1-111-1-1-11111    linear of order 2
ρ4111111-1-1-1-11111111111111111    linear of order 2
ρ51-1-111-1ii-i-i11-111111-1-1-1-1-1-1-11    linear of order 4
ρ61-1-11-11i-i-ii111-1-1-1-11-1111-1-1-11    linear of order 4
ρ71-1-111-1-i-iii11-111111-1-1-1-1-1-1-11    linear of order 4
ρ81-1-11-11-iii-i111-1-1-1-11-1111-1-1-11    linear of order 4
ρ92-22-20000002200000-2-2000-222-2    orthogonal lifted from D4
ρ1022-2-20000002200000-220002-2-2-2    orthogonal lifted from D4
ρ112222-2-20000-1-5/2-1+5/21+5/21+5/21-5/21-5/21+5/2-1+5/2-1-5/21-5/21-5/21+5/2-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ122222220000-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ132222220000-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ142222-2-20000-1+5/2-1-5/21-5/21-5/21+5/21+5/21-5/2-1-5/2-1+5/21+5/21+5/21-5/2-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ152-2-22-220000-1+5/2-1-5/2-1+5/21-5/21+5/21+5/21-5/2-1-5/21-5/2-1-5/2-1-5/2-1+5/21+5/21-5/21+5/2-1+5/2    symplectic lifted from Dic5, Schur index 2
ρ162-2-222-20000-1+5/2-1-5/21-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1-5/21-5/21+5/21+5/21-5/21+5/21-5/21+5/2-1+5/2    symplectic lifted from Dic5, Schur index 2
ρ172-2-22-220000-1-5/2-1+5/2-1-5/21+5/21-5/21-5/21+5/2-1+5/21+5/2-1+5/2-1+5/2-1-5/21-5/21+5/21-5/2-1-5/2    symplectic lifted from Dic5, Schur index 2
ρ182-2-222-20000-1-5/2-1+5/21+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/21+5/21-5/21-5/21+5/21-5/21+5/21-5/2-1-5/2    symplectic lifted from Dic5, Schur index 2
ρ192-22-2000000-1-5/2-1+5/253525352ζ545545ζ53521-5/21+5/2ζ545545ζ53521-5/2-1-5/2-1+5/21+5/2    complex lifted from C5⋊D4
ρ2022-2-2000000-1+5/2-1-5/2ζ5455455352ζ5352ζ5451+5/2-1+5/2ζ53525352545-1-5/21-5/21+5/21-5/2    complex lifted from C5⋊D4
ρ2122-2-2000000-1-5/2-1+5/2ζ53525352ζ545545ζ53521-5/2-1-5/2545ζ5455352-1+5/21+5/21-5/21+5/2    complex lifted from C5⋊D4
ρ2222-2-2000000-1+5/2-1-5/2545ζ545ζ535253525451+5/2-1+5/25352ζ5352ζ545-1-5/21-5/21+5/21-5/2    complex lifted from C5⋊D4
ρ2322-2-2000000-1-5/2-1+5/25352ζ5352545ζ54553521-5/2-1-5/2ζ545545ζ5352-1+5/21+5/21-5/21+5/2    complex lifted from C5⋊D4
ρ242-22-2000000-1-5/2-1+5/2ζ5352ζ5352545ζ54553521-5/21+5/2545ζ54553521-5/2-1-5/2-1+5/21+5/2    complex lifted from C5⋊D4
ρ252-22-2000000-1+5/2-1-5/25455455352ζ5352ζ5451+5/21-5/25352ζ5352ζ5451+5/2-1+5/2-1-5/21-5/2    complex lifted from C5⋊D4
ρ262-22-2000000-1+5/2-1-5/2ζ545ζ545ζ535253525451+5/21-5/2ζ535253525451+5/2-1+5/2-1-5/21-5/2    complex lifted from C5⋊D4

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

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

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

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

C23.D5 is a maximal subgroup of
C23.1D10  C23⋊Dic5  C23.11D10  Dic5.14D4  C23.D10  D5×C22⋊C4  D10.12D4  Dic5.5D4  C20.48D4  C23.21D10  C4×C5⋊D4  C23.23D10  D4×Dic5  C23.18D10  C20.17D4  C23⋊D10  C202D4  Dic5⋊D4  C242D5  D6⋊Dic5  C30.38D4  A4⋊Dic5  C23.D25  D10⋊Dic5  C10211C4  D10.D10
C23.D5 is a maximal quotient of
C20.55D4  C10.10C42  D4⋊Dic5  C20.D4  C23⋊Dic5  Q8⋊Dic5  C20.10D4  D42Dic5  D6⋊Dic5  C30.38D4  C23.D25  D10⋊Dic5  C10211C4  D10.D10

Matrix representation of C23.D5 in GL3(𝔽41) generated by

4000
010
0040
,
4000
010
001
,
100
0400
0040
,
100
0370
0010
,
3200
0010
0370
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,40],[40,0,0,0,1,0,0,0,1],[1,0,0,0,40,0,0,0,40],[1,0,0,0,37,0,0,0,10],[32,0,0,0,0,37,0,10,0] >;

C23.D5 in GAP, Magma, Sage, TeX

C_2^3.D_5
% in TeX

G:=Group("C2^3.D5");
// GroupNames label

G:=SmallGroup(80,19);
// by ID

G=gap.SmallGroup(80,19);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,1604]);
// Polycyclic

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

Export

Subgroup lattice of C23.D5 in TeX
Character table of C23.D5 in TeX

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