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G = C4○D20order 80 = 24·5

Central product of C4 and D20

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4D20, D205C2, C4Dic10, C4.16D10, Dic105C2, C10.4C23, C22.2D10, C20.16C22, D10.1C22, Dic5.2C22, (C2×C4)⋊3D5, (C2×C20)⋊4C2, (C4×D5)⋊4C2, C51(C4○D4), C4(C5⋊D4), C5⋊D43C2, C2.5(C22×D5), (C2×C10).11C22, SmallGroup(80,38)

Series: Derived Chief Lower central Upper central

C1C10 — C4○D20
C1C5C10D10C4×D5 — C4○D20
C5C10 — C4○D20
C1C4C2×C4

Generators and relations for C4○D20
 G = < a,b,c | a4=c2=1, b10=a2, ab=ba, ac=ca, cbc=a2b9 >

2C2
10C2
10C2
5C4
5C4
5C22
5C22
2C10
2D5
2D5
5C2×C4
5D4
5D4
5D4
5C2×C4
5Q8
5C4○D4

Character table of C4○D20

 class 12A2B2C2D4A4B4C4D4E5A5B10A10B10C10D10E10F20A20B20C20D20E20F20G20H
 size 112101011210102222222222222222
ρ111111111111111111111111111    trivial
ρ21111-1-1-1-11-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311-1-1111-11-1111-1-1-1-11-1-1-1-11111    linear of order 2
ρ411-1-1-1-1-1111111-1-1-1-111111-1-1-1-1    linear of order 2
ρ5111-11-1-1-1-1111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ6111-1-1111-1-11111111111111111    linear of order 2
ρ711-111-1-11-1-1111-1-1-1-111111-1-1-1-1    linear of order 2
ρ811-11-111-1-11111-1-1-1-11-1-1-1-11111    linear of order 2
ρ922200-2-2-200-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2-1+5/21-5/21+5/21+5/21-5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ102220022200-1-5/2-1+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
ρ112220022200-1+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
ρ1222-20022-200-1-5/2-1+5/2-1+5/21-5/21+5/21+5/21-5/2-1-5/21+5/21-5/21-5/21+5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D10
ρ1322200-2-2-200-1-5/2-1+5/2-1+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+5/21-5/21-5/2    orthogonal lifted from D10
ρ1422-200-2-2200-1+5/2-1-5/2-1-5/21+5/21-5/21-5/21+5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ1522-20022-200-1+5/2-1-5/2-1-5/21+5/21-5/21-5/21+5/2-1+5/21-5/21+5/21+5/21-5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D10
ρ1622-200-2-2200-1-5/2-1+5/2-1+5/21-5/21+5/21+5/21-5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ172-20002i-2i00022-20000-200002i-2i-2i2i    complex lifted from C4○D4
ρ182-2000-2i2i00022-20000-20000-2i2i2i-2i    complex lifted from C4○D4
ρ192-2000-2i2i000-1-5/2-1+5/21-5/2545ζ53525352ζ5451+5/2ζ4ζ534ζ52ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ43ζ5343ζ52ζ4ζ534ζ52ζ4ζ544ζ5ζ43ζ5443ζ5    complex faithful
ρ202-2000-2i2i000-1+5/2-1-5/21+5/2ζ5352ζ54554553521-5/243ζ5443ζ5ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ5ζ43ζ5443ζ5ζ4ζ544ζ5ζ4ζ534ζ52ζ43ζ5343ζ52    complex faithful
ρ212-20002i-2i000-1+5/2-1-5/21+5/25352545ζ545ζ53521-5/243ζ5443ζ5ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ5ζ4ζ544ζ5ζ43ζ5443ζ5ζ43ζ5343ζ52ζ4ζ534ζ52    complex faithful
ρ222-20002i-2i000-1-5/2-1+5/21-5/2545ζ53525352ζ5451+5/24ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ52ζ4ζ534ζ52ζ43ζ5343ζ52ζ43ζ5443ζ5ζ4ζ544ζ5    complex faithful
ρ232-2000-2i2i000-1-5/2-1+5/21-5/2ζ5455352ζ53525451+5/24ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ52ζ43ζ5343ζ52ζ4ζ534ζ52ζ4ζ544ζ5ζ43ζ5443ζ5    complex faithful
ρ242-20002i-2i000-1-5/2-1+5/21-5/2ζ5455352ζ53525451+5/2ζ4ζ534ζ52ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ4ζ534ζ52ζ43ζ5343ζ52ζ43ζ5443ζ5ζ4ζ544ζ5    complex faithful
ρ252-2000-2i2i000-1+5/2-1-5/21+5/25352545ζ545ζ53521-5/2ζ43ζ5443ζ54ζ534ζ52ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ544ζ5ζ4ζ534ζ52ζ43ζ5343ζ52    complex faithful
ρ262-20002i-2i000-1+5/2-1-5/21+5/2ζ5352ζ54554553521-5/2ζ43ζ5443ζ54ζ534ζ52ζ4ζ534ζ5243ζ5443ζ5ζ4ζ544ζ5ζ43ζ5443ζ5ζ43ζ5343ζ52ζ4ζ534ζ52    complex faithful

Smallest permutation representation of C4○D20
On 40 points
Generators in S40
(1 32 11 22)(2 33 12 23)(3 34 13 24)(4 35 14 25)(5 36 15 26)(6 37 16 27)(7 38 17 28)(8 39 18 29)(9 40 19 30)(10 21 20 31)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(38 40)

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

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

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

C4○D20 is a maximal subgroup of
D204C4  D207C4  D20.3C4  D407C2  D20.2C4  C8⋊D10  C8.D10  D4.D10  C20.C23  D4.8D10  D46D10  Q8.10D10  D5×C4○D4  D48D10  D4.10D10  D205S3  D60⋊C2  D6.D10  Dic5.D6  D6011C2  D1005C2  D205D5  D10.9D10  Dic105D5  Dic5.D10  C20.50D10
C4○D20 is a maximal quotient of
C4×Dic10  C20.6Q8  C42⋊D5  C4×D20  C4.D20  C422D5  C23.D10  D10.12D4  D10⋊D4  Dic5.5D4  Dic5.Q8  D10.13D4  D10⋊Q8  C4⋊C4⋊D5  C20.48D4  C23.21D10  C4×C5⋊D4  C23.23D10  C207D4  D205S3  D60⋊C2  D6.D10  Dic5.D6  D6011C2  D1005C2  D205D5  D10.9D10  Dic105D5  Dic5.D10  C20.50D10

Matrix representation of C4○D20 in GL2(𝔽41) generated by

90
09
,
162
3928
,
351
66
G:=sub<GL(2,GF(41))| [9,0,0,9],[16,39,2,28],[35,6,1,6] >;

C4○D20 in GAP, Magma, Sage, TeX

C_4\circ D_{20}
% in TeX

G:=Group("C4oD20");
// GroupNames label

G:=SmallGroup(80,38);
// by ID

G=gap.SmallGroup(80,38);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,46,182,1604]);
// Polycyclic

G:=Group<a,b,c|a^4=c^2=1,b^10=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^9>;
// generators/relations

Export

Subgroup lattice of C4○D20 in TeX
Character table of C4○D20 in TeX

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