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

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

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

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

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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