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G = C4oD20order 80 = 24·5

Central product of C4 and D20

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4oD20, D20:5C2, C4oDic10, C4.16D10, Dic10:5C2, C10.4C23, C22.2D10, C20.16C22, D10.1C22, Dic5.2C22, (C2xC4):3D5, (C2xC20):4C2, (C4xD5):4C2, C5:1(C4oD4), C4o(C5:D4), C5:D4:3C2, C2.5(C22xD5), (C2xC10).11C22, SmallGroup(80,38)

Series: Derived Chief Lower central Upper central

C1C10 — C4oD20
C1C5C10D10C4xD5 — C4oD20
C5C10 — C4oD20
C1C4C2xC4

Generators and relations for C4oD20
 G = < a,b,c | a4=c2=1, b10=a2, ab=ba, ac=ca, cbc=a2b9 >

Subgroups: 106 in 40 conjugacy classes, 23 normal (15 characteristic)
Quotients: C1, C2, C22, C23, D5, C4oD4, D10, C22xD5, C4oD20
2C2
10C2
10C2
5C4
5C4
5C22
5C22
2C10
2D5
2D5
5C2xC4
5D4
5D4
5D4
5C2xC4
5Q8
5C4oD4

Character table of C4oD20

 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 C4oD4
ρ182-2000-2i2i00022-20000-20000-2i2i2i-2i    complex lifted from C4oD4
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 C4oD20
On 40 points
Generators in S40
(1 31 11 21)(2 32 12 22)(3 33 13 23)(4 34 14 24)(5 35 15 25)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)
(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 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)

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

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

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

C4oD20 is a maximal subgroup of
D20:4C4  D20:7C4  D20.3C4  D40:7C2  D20.2C4  C8:D10  C8.D10  D4.D10  C20.C23  D4.8D10  D4:6D10  Q8.10D10  D5xC4oD4  D4:8D10  D4.10D10  D20:5S3  D60:C2  D6.D10  Dic5.D6  D60:11C2  D100:5C2  D20:5D5  D10.9D10  Dic10:5D5  Dic5.D10  C20.50D10
C4oD20 is a maximal quotient of
C4xDic10  C20.6Q8  C42:D5  C4xD20  C4.D20  C42:2D5  C23.D10  D10.12D4  D10:D4  Dic5.5D4  Dic5.Q8  D10.13D4  D10:Q8  C4:C4:D5  C20.48D4  C23.21D10  C4xC5:D4  C23.23D10  C20:7D4  D20:5S3  D60:C2  D6.D10  Dic5.D6  D60:11C2  D100:5C2  D20:5D5  D10.9D10  Dic10:5D5  Dic5.D10  C20.50D10

Matrix representation of C4oD20 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] >;

C4oD20 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 C4oD20 in TeX
Character table of C4oD20 in TeX

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