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G = Dic3×F5order 240 = 24·3·5

Direct product of Dic3 and F5

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

Aliases: Dic3×F5, C15⋊C42, D10.5D6, Dic151C4, C3⋊F5⋊C4, (C3×F5)⋊C4, C32(C4×F5), C51(C4×Dic3), C2.1(S3×F5), C6.1(C2×F5), C10.1(C4×S3), C30.1(C2×C4), D5.1(C4×S3), D5.(C2×Dic3), (C6×F5).1C2, (C2×F5).2S3, (C5×Dic3)⋊1C4, (D5×Dic3).2C2, (C6×D5).5C22, (C3×D5).(C2×C4), (C2×C3⋊F5).1C2, SmallGroup(240,95)

Series: Derived Chief Lower central Upper central

C1C15 — Dic3×F5
C1C5C15C3×D5C6×D5C6×F5 — Dic3×F5
C15 — Dic3×F5
C1C2

Generators and relations for Dic3×F5
 G = < a,b,c,d | a6=c5=d4=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 232 in 60 conjugacy classes, 28 normal (22 characteristic)
C1, C2, C2 [×2], C3, C4 [×6], C22, C5, C6, C6 [×2], C2×C4 [×3], D5 [×2], C10, Dic3, Dic3 [×3], C12 [×2], C2×C6, C15, C42, Dic5, C20, F5 [×2], F5 [×2], D10, C2×Dic3 [×2], C2×C12, C3×D5 [×2], C30, C4×D5, C2×F5, C2×F5, C4×Dic3, C5×Dic3, Dic15, C3×F5 [×2], C3⋊F5 [×2], C6×D5, C4×F5, D5×Dic3, C6×F5, C2×C3⋊F5, Dic3×F5
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C2×C4 [×3], Dic3 [×2], D6, C42, F5, C4×S3 [×2], C2×Dic3, C2×F5, C4×Dic3, C4×F5, S3×F5, Dic3×F5

Character table of Dic3×F5

 class 12A2B2C34A4B4C4D4E4F4G4H4I4J4K4L56A6B6C1012A12B12C12D1520A20B30
 size 11552335555151515151515421010410101010812128
ρ1111111111111111111111111111111    trivial
ρ21111111-1-1-1-1-11-1-1-1111111-1-1-1-11111    linear of order 2
ρ311111-1-1-1-1-1-11-1111-111111-1-1-1-11-1-11    linear of order 2
ρ411111-1-11111-1-1-1-1-1-11111111111-1-11    linear of order 2
ρ51-1-111i-ii-i-ii-1i-111-i1-1-11-1i-i-ii1i-i-1    linear of order 4
ρ611-1-1111-i-iiii-1-i-ii-111-1-11ii-i-i1111    linear of order 4
ρ71-11-11i-i-11-11i-i-ii-ii1-11-1-11-11-11i-i-1    linear of order 4
ρ811-1-1111ii-i-i-i-1ii-i-111-1-11-i-iii1111    linear of order 4
ρ91-11-11i-i1-11-1-i-ii-iii1-11-1-1-11-111i-i-1    linear of order 4
ρ101-1-111i-i-iii-i1i1-1-1-i1-1-11-1-iii-i1i-i-1    linear of order 4
ρ1111-1-11-1-1ii-i-ii1-i-ii111-1-11-i-iii1-1-11    linear of order 4
ρ121-1-111-ii-iii-i-1-i-111i1-1-11-1-iii-i1-ii-1    linear of order 4
ρ131-11-11-ii1-11-1ii-ii-i-i1-11-1-1-11-111-ii-1    linear of order 4
ρ1411-1-11-1-1-i-iii-i1ii-i111-1-11ii-i-i1-1-11    linear of order 4
ρ151-11-11-ii-11-11-iii-ii-i1-11-1-11-11-11-ii-1    linear of order 4
ρ161-1-111-iii-i-ii1-i1-1-1i1-1-11-1i-i-ii1-ii-1    linear of order 4
ρ172222-10022220000002-1-1-12-1-1-1-1-100-1    orthogonal lifted from S3
ρ182222-100-2-2-2-20000002-1-1-121111-100-1    orthogonal lifted from D6
ρ192-22-2-1002-22-200000021-11-21-11-1-1001    symplectic lifted from Dic3, Schur index 2
ρ202-22-2-100-22-2200000021-11-2-11-11-1001    symplectic lifted from Dic3, Schur index 2
ρ2122-2-2-100-2i-2i2i2i0000002-1112-i-iii-100-1    complex lifted from C4×S3
ρ2222-2-2-1002i2i-2i-2i0000002-1112ii-i-i-100-1    complex lifted from C4×S3
ρ232-2-22-100-2i2i2i-2i000000211-1-2i-i-ii-1001    complex lifted from C4×S3
ρ242-2-22-1002i-2i-2i2i000000211-1-2-iii-i-1001    complex lifted from C4×S3
ρ2544004440000000000-1400-10000-1-1-1-1    orthogonal lifted from F5
ρ2644004-4-40000000000-1400-10000-111-1    orthogonal lifted from C2×F5
ρ274-40044i-4i0000000000-1-40010000-1-ii1    complex lifted from C4×F5
ρ284-4004-4i4i0000000000-1-40010000-1i-i1    complex lifted from C4×F5
ρ298800-4000000000000-2-400-200001001    orthogonal lifted from S3×F5
ρ308-800-4000000000000-240020000100-1    symplectic faithful, Schur index 2

Smallest permutation representation of Dic3×F5
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 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)
(1 42 53 46 34)(2 37 54 47 35)(3 38 49 48 36)(4 39 50 43 31)(5 40 51 44 32)(6 41 52 45 33)(7 14 26 55 24)(8 15 27 56 19)(9 16 28 57 20)(10 17 29 58 21)(11 18 30 59 22)(12 13 25 60 23)
(1 56 4 59)(2 57 5 60)(3 58 6 55)(7 36 17 41)(8 31 18 42)(9 32 13 37)(10 33 14 38)(11 34 15 39)(12 35 16 40)(19 50 30 46)(20 51 25 47)(21 52 26 48)(22 53 27 43)(23 54 28 44)(24 49 29 45)

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

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

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

Dic3×F5 is a maximal subgroup of   C4⋊F53S3  Dic65F5  C4×S3×F5  C22⋊F5.S3  C3⋊D4⋊F5
Dic3×F5 is a maximal quotient of   C30.C42  C30.3C42  C30.4C42  D10.20D12  C30.M4(2)

Matrix representation of Dic3×F5 in GL6(𝔽61)

0600000
110000
0060000
0006000
0000600
0000060
,
5000000
11110000
0050000
0005000
0000500
0000050
,
100000
010000
0000060
0010060
0001060
0000160
,
100000
010000
0000110
0011000
0000011
0001100

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[50,11,0,0,0,0,0,11,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,50],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,60,60,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,11,0,0,0,0,0,0,0,11,0] >;

Dic3×F5 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times F_5
% in TeX

G:=Group("Dic3xF5");
// GroupNames label

G:=SmallGroup(240,95);
// by ID

G=gap.SmallGroup(240,95);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,55,490,3461,1745]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^5=d^4=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

Export

Character table of Dic3×F5 in TeX

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