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G = D30.C2order 120 = 23·3·5

The non-split extension by D30 of C2 acting faithfully

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

Aliases: D30.C2, D152C4, C10.3D6, C6.3D10, Dic3Dic5, Dic52S3, Dic32D5, C30.3C22, C52(C4×S3), C31(C4×D5), C156(C2×C4), C2.3(S3×D5), (C5×Dic3)⋊2C2, (C3×Dic5)⋊2C2, SmallGroup(120,10)

Series: Derived Chief Lower central Upper central

C1C15 — D30.C2
C1C5C15C30C3×Dic5 — D30.C2
C15 — D30.C2
C1C2

Generators and relations for D30.C2
 G = < a,b,c | a30=b2=1, c2=a15, bab=a-1, cac-1=a19, cbc-1=a18b >

15C2
15C2
3C4
5C4
15C22
5S3
5S3
3D5
3D5
15C2×C4
5C12
5D6
3C20
3D10
5C4×S3
3C4×D5

Character table of D30.C2

 class 12A2B2C34A4B4C4D5A5B610A10B12A12B15A15B20A20B20C20D30A30B
 size 1115152335522222101044666644
ρ1111111111111111111111111    trivial
ρ211-1-11-1-111111111111-1-1-1-111    linear of order 2
ρ311-1-1111-1-111111-1-111111111    linear of order 2
ρ411111-1-1-1-111111-1-111-1-1-1-111    linear of order 2
ρ51-11-11-ii-ii11-1-1-1-ii11-iii-i-1-1    linear of order 4
ρ61-1-111-iii-i11-1-1-1i-i11-iii-i-1-1    linear of order 4
ρ71-11-11i-ii-i11-1-1-1i-i11i-i-ii-1-1    linear of order 4
ρ81-1-111i-i-ii11-1-1-1-ii11i-i-ii-1-1    linear of order 4
ρ92200-100-2-222-12211-1-10000-1-1    orthogonal lifted from D6
ρ102200-1002222-122-1-1-1-10000-1-1    orthogonal lifted from S3
ρ1122002-2-200-1+5/2-1-5/22-1-5/2-1+5/200-1-5/2-1+5/21-5/21+5/21-5/21+5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ12220022200-1-5/2-1+5/22-1+5/2-1-5/200-1+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
ρ1322002-2-200-1-5/2-1+5/22-1+5/2-1-5/200-1+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+5/2-1-5/22-1-5/2-1+5/200-1-5/2-1+5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ152-200-100-2i2i221-2-2i-i-1-1000011    complex lifted from C4×S3
ρ162-200-1002i-2i221-2-2-ii-1-1000011    complex lifted from C4×S3
ρ172-20022i-2i00-1+5/2-1-5/2-21+5/21-5/200-1-5/2-1+5/2ζ4ζ544ζ5ζ43ζ5343ζ52ζ43ζ5443ζ5ζ4ζ534ζ521-5/21+5/2    complex lifted from C4×D5
ρ182-2002-2i2i00-1+5/2-1-5/2-21+5/21-5/200-1-5/2-1+5/2ζ43ζ5443ζ5ζ4ζ534ζ52ζ4ζ544ζ5ζ43ζ5343ζ521-5/21+5/2    complex lifted from C4×D5
ρ192-2002-2i2i00-1-5/2-1+5/2-21-5/21+5/200-1+5/2-1-5/2ζ43ζ5343ζ52ζ4ζ544ζ5ζ4ζ534ζ52ζ43ζ5443ζ51+5/21-5/2    complex lifted from C4×D5
ρ202-20022i-2i00-1-5/2-1+5/2-21-5/21+5/200-1+5/2-1-5/2ζ4ζ534ζ52ζ43ζ5443ζ5ζ43ζ5343ζ52ζ4ζ544ζ51+5/21-5/2    complex lifted from C4×D5
ρ214400-20000-1+5-1-5-2-1-5-1+5001+5/21-5/200001-5/21+5/2    orthogonal lifted from S3×D5
ρ224-400-20000-1+5-1-521+51-5001+5/21-5/20000-1+5/2-1-5/2    orthogonal faithful
ρ234-400-20000-1-5-1+521-51+5001-5/21+5/20000-1-5/2-1+5/2    orthogonal faithful
ρ244400-20000-1-5-1+5-2-1+5-1-5001-5/21+5/200001+5/21-5/2    orthogonal lifted from S3×D5

Smallest permutation representation of D30.C2
On 60 points
Generators in S60
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 60)(52 59)(53 58)(54 57)(55 56)
(1 41 16 56)(2 60 17 45)(3 49 18 34)(4 38 19 53)(5 57 20 42)(6 46 21 31)(7 35 22 50)(8 54 23 39)(9 43 24 58)(10 32 25 47)(11 51 26 36)(12 40 27 55)(13 59 28 44)(14 48 29 33)(15 37 30 52)

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

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

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

D30.C2 is a maximal subgroup of
D15⋊C8  Dic3.F5  D60⋊C2  D15⋊Q8  C12.28D10  C4×S3×D5  Dic5.D6  Dic3.D10  D10⋊D6  D90.C2  C30.D6  C6.D30  D30.S3  Dic5.7S4  Dic52S4
D30.C2 is a maximal quotient of
D152C8  D30.5C4  Dic3×Dic5  D304C4  Dic155C4  D90.C2  C30.D6  C6.D30  D30.S3  Dic52S4

Matrix representation of D30.C2 in GL4(𝔽61) generated by

06000
1100
00060
00117
,
06000
60000
004460
004417
,
50000
05000
004417
00117
G:=sub<GL(4,GF(61))| [0,1,0,0,60,1,0,0,0,0,0,1,0,0,60,17],[0,60,0,0,60,0,0,0,0,0,44,44,0,0,60,17],[50,0,0,0,0,50,0,0,0,0,44,1,0,0,17,17] >;

D30.C2 in GAP, Magma, Sage, TeX

D_{30}.C_2
% in TeX

G:=Group("D30.C2");
// GroupNames label

G:=SmallGroup(120,10);
// by ID

G=gap.SmallGroup(120,10);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5,20,26,168,2404]);
// Polycyclic

G:=Group<a,b,c|a^30=b^2=1,c^2=a^15,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^18*b>;
// generators/relations

Export

Subgroup lattice of D30.C2 in TeX
Character table of D30.C2 in TeX

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