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G = C5×D4order 40 = 23·5

Direct product of C5 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×D4, C4⋊C10, C203C2, C22⋊C10, C10.6C22, (C2×C10)⋊1C2, C2.1(C2×C10), SmallGroup(40,10)

Series: Derived Chief Lower central Upper central

C1C2 — C5×D4
C1C2C10C2×C10 — C5×D4
C1C2 — C5×D4
C1C10 — C5×D4

Generators and relations for C5×D4
 G = < a,b,c | a5=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C10
2C10

Character table of C5×D4

 class 12A2B2C45A5B5C5D10A10B10C10D10E10F10G10H10I10J10K10L20A20B20C20D
 size 1122211111111222222222222
ρ11111111111111111111111111    trivial
ρ2111-1-111111111111-1-1-1-11-1-1-1-1    linear of order 2
ρ311-11-111111111-1-1-11111-1-1-1-1-1    linear of order 2
ρ411-1-1111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ511-11-1ζ5ζ54ζ52ζ53ζ52ζ54ζ53ζ555253ζ53ζ5ζ54ζ52545355452    linear of order 10
ρ6111-1-1ζ5ζ54ζ52ζ53ζ52ζ54ζ53ζ5ζ5ζ52ζ535355452ζ545355452    linear of order 10
ρ711111ζ54ζ5ζ53ζ52ζ53ζ5ζ52ζ54ζ54ζ53ζ52ζ52ζ54ζ5ζ53ζ5ζ52ζ54ζ5ζ53    linear of order 5
ρ811-1-11ζ54ζ5ζ53ζ52ζ53ζ5ζ52ζ5454535252545535ζ52ζ54ζ5ζ53    linear of order 10
ρ9111-1-1ζ53ζ52ζ5ζ54ζ5ζ52ζ54ζ53ζ53ζ5ζ545453525ζ525453525    linear of order 10
ρ1011-1-11ζ53ζ52ζ5ζ54ζ5ζ52ζ54ζ5353554545352552ζ54ζ53ζ52ζ5    linear of order 10
ρ1111111ζ53ζ52ζ5ζ54ζ5ζ52ζ54ζ53ζ53ζ5ζ54ζ54ζ53ζ52ζ5ζ52ζ54ζ53ζ52ζ5    linear of order 5
ρ1211-1-11ζ5ζ54ζ52ζ53ζ52ζ54ζ53ζ555253535545254ζ53ζ5ζ54ζ52    linear of order 10
ρ1311-11-1ζ53ζ52ζ5ζ54ζ5ζ52ζ54ζ5353554ζ54ζ53ζ52ζ5525453525    linear of order 10
ρ1411111ζ52ζ53ζ54ζ5ζ54ζ53ζ5ζ52ζ52ζ54ζ5ζ5ζ52ζ53ζ54ζ53ζ5ζ52ζ53ζ54    linear of order 5
ρ1511111ζ5ζ54ζ52ζ53ζ52ζ54ζ53ζ5ζ5ζ52ζ53ζ53ζ5ζ54ζ52ζ54ζ53ζ5ζ54ζ52    linear of order 5
ρ16111-1-1ζ52ζ53ζ54ζ5ζ54ζ53ζ5ζ52ζ52ζ54ζ55525354ζ535525354    linear of order 10
ρ17111-1-1ζ54ζ5ζ53ζ52ζ53ζ5ζ52ζ54ζ54ζ53ζ525254553ζ55254553    linear of order 10
ρ1811-11-1ζ52ζ53ζ54ζ5ζ54ζ53ζ5ζ5252545ζ5ζ52ζ53ζ54535525354    linear of order 10
ρ1911-11-1ζ54ζ5ζ53ζ52ζ53ζ5ζ52ζ54545352ζ52ζ54ζ5ζ5355254553    linear of order 10
ρ2011-1-11ζ52ζ53ζ54ζ5ζ54ζ53ζ5ζ5252545552535453ζ5ζ52ζ53ζ54    linear of order 10
ρ212-20002222-2-2-2-2000000000000    orthogonal lifted from D4
ρ222-20005455352-2ζ53-2ζ5-2ζ52-2ζ54000000000000    complex faithful
ρ232-20005545253-2ζ52-2ζ54-2ζ53-2ζ5000000000000    complex faithful
ρ242-20005253545-2ζ54-2ζ53-2ζ5-2ζ52000000000000    complex faithful
ρ252-20005352554-2ζ5-2ζ52-2ζ54-2ζ53000000000000    complex faithful

Permutation representations of C5×D4
On 20 points - transitive group 20T12
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 8 11 17)(2 9 12 18)(3 10 13 19)(4 6 14 20)(5 7 15 16)
(1 17)(2 18)(3 19)(4 20)(5 16)(6 14)(7 15)(8 11)(9 12)(10 13)

G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,8,11,17)(2,9,12,18)(3,10,13,19)(4,6,14,20)(5,7,15,16), (1,17)(2,18)(3,19)(4,20)(5,16)(6,14)(7,15)(8,11)(9,12)(10,13)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,8,11,17)(2,9,12,18)(3,10,13,19)(4,6,14,20)(5,7,15,16), (1,17)(2,18)(3,19)(4,20)(5,16)(6,14)(7,15)(8,11)(9,12)(10,13) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,8,11,17),(2,9,12,18),(3,10,13,19),(4,6,14,20),(5,7,15,16)], [(1,17),(2,18),(3,19),(4,20),(5,16),(6,14),(7,15),(8,11),(9,12),(10,13)])

G:=TransitiveGroup(20,12);

Matrix representation of C5×D4 in GL2(𝔽11) generated by

90
09
,
07
30
,
04
30
G:=sub<GL(2,GF(11))| [9,0,0,9],[0,3,7,0],[0,3,4,0] >;

C5×D4 in GAP, Magma, Sage, TeX

C_5\times D_4
% in TeX

G:=Group("C5xD4");
// GroupNames label

G:=SmallGroup(40,10);
// by ID

G=gap.SmallGroup(40,10);
# by ID

G:=PCGroup([4,-2,-2,-5,-2,177]);
// Polycyclic

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

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