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G = D5×Dic3order 120 = 23·3·5

Direct product of D5 and Dic3

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D5×Dic3, C10.1D6, C6.1D10, D10.2S3, Dic152C2, C30.1C22, C33(C4×D5), C154(C2×C4), (C3×D5)⋊1C4, C2.1(S3×D5), C52(C2×Dic3), (C6×D5).1C2, (C5×Dic3)⋊1C2, SmallGroup(120,8)

Series: Derived Chief Lower central Upper central

C1C15 — D5×Dic3
C1C5C15C30C6×D5 — D5×Dic3
C15 — D5×Dic3
C1C2

Generators and relations for D5×Dic3
 G = < a,b,c,d | a5=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

5C2
5C2
3C4
5C22
15C4
5C6
5C6
15C2×C4
5Dic3
5C2×C6
3C20
3Dic5
5C2×Dic3
3C4×D5

Character table of D5×Dic3

 class 12A2B2C34A4B4C4D5A5B6A6B6C10A10B15A15B20A20B20C20D30A30B
 size 1155233151522210102244666644
ρ1111111111111111111111111    trivial
ρ211-1-1111-1-1111-1-11111111111    linear of order 2
ρ311-1-11-1-111111-1-11111-1-1-1-111    linear of order 2
ρ411111-1-1-1-1111111111-1-1-1-111    linear of order 2
ρ51-1-111-iii-i11-1-11-1-111i-i-ii-1-1    linear of order 4
ρ61-11-11-ii-ii11-11-1-1-111i-i-ii-1-1    linear of order 4
ρ71-11-11i-ii-i11-11-1-1-111-iii-i-1-1    linear of order 4
ρ81-1-111i-i-ii11-1-11-1-111-iii-i-1-1    linear of order 4
ρ922-2-2-1000022-11122-1-10000-1-1    orthogonal lifted from D6
ρ1022002-2-200-1+5/2-1-5/2200-1+5/2-1-5/2-1+5/2-1-5/21-5/21-5/21+5/21+5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ11220022200-1-5/2-1+5/2200-1-5/2-1+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
ρ122222-1000022-1-1-122-1-10000-1-1    orthogonal lifted from S3
ρ1322002-2-200-1-5/2-1+5/2200-1-5/2-1+5/2-1-5/2-1+5/21+5/21+5/21-5/21-5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ14220022200-1+5/2-1-5/2200-1+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
ρ152-2-22-100002211-1-2-2-1-1000011    symplectic lifted from Dic3, Schur index 2
ρ162-22-2-10000221-11-2-2-1-1000011    symplectic lifted from Dic3, Schur index 2
ρ172-20022i-2i00-1-5/2-1+5/2-2001+5/21-5/2-1-5/2-1+5/2ζ43ζ5343ζ52ζ4ζ534ζ52ζ4ζ544ζ5ζ43ζ5443ζ51+5/21-5/2    complex lifted from C4×D5
ρ182-2002-2i2i00-1+5/2-1-5/2-2001-5/21+5/2-1+5/2-1-5/2ζ4ζ544ζ5ζ43ζ5443ζ5ζ43ζ5343ζ52ζ4ζ534ζ521-5/21+5/2    complex lifted from C4×D5
ρ192-20022i-2i00-1+5/2-1-5/2-2001-5/21+5/2-1+5/2-1-5/2ζ43ζ5443ζ5ζ4ζ544ζ5ζ4ζ534ζ52ζ43ζ5343ζ521-5/21+5/2    complex lifted from C4×D5
ρ202-2002-2i2i00-1-5/2-1+5/2-2001+5/21-5/2-1-5/2-1+5/2ζ4ζ534ζ52ζ43ζ5343ζ52ζ43ζ5443ζ5ζ4ζ544ζ51+5/21-5/2    complex lifted from C4×D5
ρ214400-20000-1-5-1+5-200-1-5-1+51+5/21-5/200001+5/21-5/2    orthogonal lifted from S3×D5
ρ224400-20000-1+5-1-5-200-1+5-1-51-5/21+5/200001-5/21+5/2    orthogonal lifted from S3×D5
ρ234-400-20000-1-5-1+52001+51-51+5/21-5/20000-1-5/2-1+5/2    symplectic faithful, Schur index 2
ρ244-400-20000-1+5-1-52001-51+51-5/21+5/20000-1+5/2-1-5/2    symplectic faithful, Schur index 2

Smallest permutation representation of D5×Dic3
On 60 points
Generators in S60
(1 34 51 37 47)(2 35 52 38 48)(3 36 53 39 43)(4 31 54 40 44)(5 32 49 41 45)(6 33 50 42 46)(7 21 17 59 30)(8 22 18 60 25)(9 23 13 55 26)(10 24 14 56 27)(11 19 15 57 28)(12 20 16 58 29)
(1 47)(2 48)(3 43)(4 44)(5 45)(6 46)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 28)(20 29)(21 30)(22 25)(23 26)(24 27)(31 40)(32 41)(33 42)(34 37)(35 38)(36 39)
(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 41 42)(43 44 45 46 47 48)(49 50 51 52 53 54)(55 56 57 58 59 60)
(1 59 4 56)(2 58 5 55)(3 57 6 60)(7 54 10 51)(8 53 11 50)(9 52 12 49)(13 48 16 45)(14 47 17 44)(15 46 18 43)(19 42 22 39)(20 41 23 38)(21 40 24 37)(25 36 28 33)(26 35 29 32)(27 34 30 31)

G:=sub<Sym(60)| (1,34,51,37,47)(2,35,52,38,48)(3,36,53,39,43)(4,31,54,40,44)(5,32,49,41,45)(6,33,50,42,46)(7,21,17,59,30)(8,22,18,60,25)(9,23,13,55,26)(10,24,14,56,27)(11,19,15,57,28)(12,20,16,58,29), (1,47)(2,48)(3,43)(4,44)(5,45)(6,46)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27)(31,40)(32,41)(33,42)(34,37)(35,38)(36,39), (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)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60), (1,59,4,56)(2,58,5,55)(3,57,6,60)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,48,16,45)(14,47,17,44)(15,46,18,43)(19,42,22,39)(20,41,23,38)(21,40,24,37)(25,36,28,33)(26,35,29,32)(27,34,30,31)>;

G:=Group( (1,34,51,37,47)(2,35,52,38,48)(3,36,53,39,43)(4,31,54,40,44)(5,32,49,41,45)(6,33,50,42,46)(7,21,17,59,30)(8,22,18,60,25)(9,23,13,55,26)(10,24,14,56,27)(11,19,15,57,28)(12,20,16,58,29), (1,47)(2,48)(3,43)(4,44)(5,45)(6,46)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27)(31,40)(32,41)(33,42)(34,37)(35,38)(36,39), (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)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60), (1,59,4,56)(2,58,5,55)(3,57,6,60)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,48,16,45)(14,47,17,44)(15,46,18,43)(19,42,22,39)(20,41,23,38)(21,40,24,37)(25,36,28,33)(26,35,29,32)(27,34,30,31) );

G=PermutationGroup([(1,34,51,37,47),(2,35,52,38,48),(3,36,53,39,43),(4,31,54,40,44),(5,32,49,41,45),(6,33,50,42,46),(7,21,17,59,30),(8,22,18,60,25),(9,23,13,55,26),(10,24,14,56,27),(11,19,15,57,28),(12,20,16,58,29)], [(1,47),(2,48),(3,43),(4,44),(5,45),(6,46),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,28),(20,29),(21,30),(22,25),(23,26),(24,27),(31,40),(32,41),(33,42),(34,37),(35,38),(36,39)], [(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),(37,38,39,40,41,42),(43,44,45,46,47,48),(49,50,51,52,53,54),(55,56,57,58,59,60)], [(1,59,4,56),(2,58,5,55),(3,57,6,60),(7,54,10,51),(8,53,11,50),(9,52,12,49),(13,48,16,45),(14,47,17,44),(15,46,18,43),(19,42,22,39),(20,41,23,38),(21,40,24,37),(25,36,28,33),(26,35,29,32),(27,34,30,31)])

D5×Dic3 is a maximal subgroup of   Dic3⋊F5  D205S3  D20⋊S3  C4×S3×D5  Dic5.D6  C30.C23  D30.S3
D5×Dic3 is a maximal quotient of   C20.32D6  D10⋊Dic3  C30.Q8  D30.S3

Matrix representation of D5×Dic3 in GL4(𝔽61) generated by

0100
601700
0010
0001
,
0100
1000
0010
0001
,
60000
06000
00601
00600
,
50000
05000
001523
003846
G:=sub<GL(4,GF(61))| [0,60,0,0,1,17,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,60,0,0,1,0],[50,0,0,0,0,50,0,0,0,0,15,38,0,0,23,46] >;

D5×Dic3 in GAP, Magma, Sage, TeX

D_5\times {\rm Dic}_3
% in TeX

G:=Group("D5xDic3");
// GroupNames label

G:=SmallGroup(120,8);
// by ID

G=gap.SmallGroup(120,8);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5,26,168,2404]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of D5×Dic3 in TeX
Character table of D5×Dic3 in TeX

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