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G = C4.Dic5order 80 = 24·5

The non-split extension by C4 of Dic5 acting via Dic5/C10=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C4.Dic5, C20.4C4, C54M4(2), C4.15D10, C22.Dic5, C20.15C22, C52C85C2, (C2×C4).2D5, (C2×C20).5C2, (C2×C10).5C4, C10.14(C2×C4), C2.3(C2×Dic5), SmallGroup(80,10)

Series: Derived Chief Lower central Upper central

C1C10 — C4.Dic5
C1C5C10C20C52C8 — C4.Dic5
C5C10 — C4.Dic5
C1C4C2×C4

Generators and relations for C4.Dic5
 G = < a,b,c | a4=1, b10=a2, c2=b5, ab=ba, cac-1=a-1, cbc-1=b9 >

2C2
2C10
5C8
5C8
5M4(2)

Character table of C4.Dic5

 class 12A2B4A4B4C5A5B8A8B8C8D10A10B10C10D10E10F20A20B20C20D20E20F20G20H
 size 112112221010101022222222222222
ρ111111111111111111111111111    trivial
ρ211-111-111-11-111-1-1-1-111-1-1-1111-1    linear of order 2
ρ311-111-1111-11-11-1-1-1-111-1-1-1111-1    linear of order 2
ρ411111111-1-1-1-111111111111111    linear of order 2
ρ5111-1-1-111ii-i-i111111-1-1-1-1-1-1-1-1    linear of order 4
ρ611-1-1-1111-iii-i1-1-1-1-11-1111-1-1-11    linear of order 4
ρ711-1-1-1111i-i-ii1-1-1-1-11-1111-1-1-11    linear of order 4
ρ8111-1-1-111-i-iii111111-1-1-1-1-1-1-1-1    linear of order 4
ρ922-222-2-1-5/2-1+5/20000-1+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+5/21-5/2    orthogonal lifted from D10
ρ10222222-1-5/2-1+5/20000-1+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
ρ11222222-1+5/2-1-5/20000-1-5/2-1-5/2-1+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
ρ1222-222-2-1+5/2-1-5/20000-1-5/21+5/21-5/21-5/21+5/2-1+5/2-1+5/21-5/21-5/21+5/2-1+5/2-1-5/2-1-5/21+5/2    orthogonal lifted from D10
ρ13222-2-2-2-1+5/2-1-5/20000-1-5/2-1-5/2-1+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    symplectic lifted from Dic5, Schur index 2
ρ1422-2-2-22-1-5/2-1+5/20000-1+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
ρ1522-2-2-22-1+5/2-1-5/20000-1-5/21+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
ρ16222-2-2-2-1-5/2-1+5/20000-1+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    symplectic lifted from Dic5, Schur index 2
ρ172-202i-2i0220000-20000-2-2i0002i2i-2i0    complex lifted from M4(2)
ρ182-20-2i2i0220000-20000-22i000-2i-2i2i0    complex lifted from M4(2)
ρ192-20-2i2i0-1-5/2-1+5/200001-5/2ζ545ζ535253525451+5/2ζ4ζ534ζ524ζ534ζ52ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5343ζ52ζ43ζ5443ζ5ζ4ζ544ζ5ζ43ζ5443ζ5    complex faithful
ρ202-20-2i2i0-1+5/2-1-5/200001+5/25352ζ545545ζ53521-5/2ζ4ζ544ζ5ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ43ζ5443ζ5ζ43ζ5343ζ52ζ4ζ534ζ52ζ4ζ534ζ52    complex faithful
ρ212-202i-2i0-1-5/2-1+5/200001-5/2ζ545ζ535253525451+5/2ζ43ζ5343ζ52ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ5ζ4ζ534ζ52ζ4ζ544ζ5ζ43ζ5443ζ543ζ5443ζ5    complex faithful
ρ222-20-2i2i0-1+5/2-1-5/200001+5/2ζ5352545ζ54553521-5/2ζ4ζ544ζ543ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ52ζ43ζ5443ζ5ζ43ζ5343ζ52ζ4ζ534ζ524ζ534ζ52    complex faithful
ρ232-20-2i2i0-1-5/2-1+5/200001-5/25455352ζ5352ζ5451+5/2ζ4ζ534ζ52ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ5ζ43ζ5343ζ52ζ43ζ5443ζ5ζ4ζ544ζ543ζ5443ζ5    complex faithful
ρ242-202i-2i0-1+5/2-1-5/200001+5/2ζ5352545ζ54553521-5/2ζ43ζ5443ζ5ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ4ζ544ζ5ζ4ζ534ζ52ζ43ζ5343ζ52ζ4ζ534ζ52    complex faithful
ρ252-202i-2i0-1+5/2-1-5/200001+5/25352ζ545545ζ53521-5/2ζ43ζ5443ζ543ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ52ζ4ζ544ζ5ζ4ζ534ζ52ζ43ζ5343ζ524ζ534ζ52    complex faithful
ρ262-202i-2i0-1-5/2-1+5/200001-5/25455352ζ5352ζ5451+5/2ζ43ζ5343ζ524ζ534ζ52ζ4ζ534ζ5243ζ5443ζ5ζ4ζ534ζ52ζ4ζ544ζ5ζ43ζ5443ζ5ζ43ζ5443ζ5    complex faithful

Smallest permutation representation of C4.Dic5
On 40 points
Generators in S40
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 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 26 6 31 11 36 16 21)(2 35 7 40 12 25 17 30)(3 24 8 29 13 34 18 39)(4 33 9 38 14 23 19 28)(5 22 10 27 15 32 20 37)

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

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

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

C4.Dic5 is a maximal subgroup of
D204C4  C40.6C4  C20.53D4  C20.46D4  C4.12D20  C20.D4  C20.10D4  D42Dic5  D20.3C4  D5×M4(2)  D4.D10  C20.C23  D4.Dic5  D4⋊D10  D4.9D10  D6.Dic5  C60.7C4  C4.Dic25  C20.30D10  C20.59D10  C20.12F5  C102.C4
C4.Dic5 is a maximal quotient of
C42.D5  C203C8  C20.55D4  D6.Dic5  C60.7C4  C4.Dic25  C20.30D10  C20.59D10  C20.12F5  C102.C4

Matrix representation of C4.Dic5 in GL2(𝔽41) generated by

320
09
,
360
033
,
01
320
G:=sub<GL(2,GF(41))| [32,0,0,9],[36,0,0,33],[0,32,1,0] >;

C4.Dic5 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_5
% in TeX

G:=Group("C4.Dic5");
// GroupNames label

G:=SmallGroup(80,10);
// by ID

G=gap.SmallGroup(80,10);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,42,1604]);
// Polycyclic

G:=Group<a,b,c|a^4=1,b^10=a^2,c^2=b^5,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^9>;
// generators/relations

Export

Subgroup lattice of C4.Dic5 in TeX
Character table of C4.Dic5 in TeX

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