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G = F5xC3:D4  order 480 = 25·3·5

Direct product of F5 and C3:D4

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

Aliases: F5xC3:D4, C3:5(D4xF5), C15:3(C4xD4), D6:F5:3C2, D6:3(C2xF5), (C3xF5):2D4, D10:1(C4xS3), D30:3(C2xC4), C3:D20:1C4, C15:D4:1C4, C15:7D4:1C4, C22:2(S3xF5), Dic3:F5:3C2, (Dic3xF5):3C2, Dic3:1(C2xF5), (C2xF5).13D6, (C22xF5):3S3, Dic15:1(C2xC4), C6.29(C22xF5), D5.4(C4oD12), C30.29(C22xC4), (C22xD5).38D6, (C6xD5).33C23, (C6xF5).13C22, D10.D6:3C2, D10.36(C22xS3), (D5xDic3).7C22, C5:(C4xC3:D4), (C2xC6xF5):2C2, (C2xS3xF5):3C2, (C2xC6):5(C2xF5), (C2xC10):4(C4xS3), (C2xC30):1(C2xC4), C2.29(C2xS3xF5), (C5xC3:D4):1C4, C10.29(S3xC2xC4), (S3xC10):3(C2xC4), (C6xD5):11(C2xC4), (C3xD5).6(C2xD4), D5.2(C2xC3:D4), (D5xC3:D4).1C2, (C2xS3xD5).3C22, (C2xC3:F5).6C22, (C5xDic3):1(C2xC4), (D5xC2xC6).70C22, (C3xD5).8(C4oD4), SmallGroup(480,1010)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — F5xC3:D4
Chief series
C1 — C5 — C15 — C3xD5 — C6xD5 — C6xF5 — C2xS3xF5 — F5xC3:D4
Lower central
C15 — C30 — F5xC3:D4
Upper central
C1 — C2 — C22

Generators and relations for F5xC3:D4
 G = < a,b,c,d,e | a5=b4=c3=d4=e2=1, bab-1=a3, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1044 in 188 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2xC4, D4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, D6, D6, C2xC6, C2xC6, C15, C42, C22:C4, C4:C4, C22xC4, C2xD4, Dic5, C20, F5, F5, D10, D10, C2xC10, C2xC10, C4xS3, C2xDic3, C3:D4, C3:D4, C2xC12, C22xS3, C22xC6, C5xS3, C3xD5, C3xD5, D15, C30, C30, C4xD4, C4xD5, D20, C5:D4, C5xD4, C2xF5, C2xF5, C22xD5, C22xD5, C4xDic3, Dic3:C4, D6:C4, C6.D4, S3xC2xC4, C2xC3:D4, C22xC12, C5xDic3, Dic15, C3xF5, C3xF5, C3:F5, S3xD5, C6xD5, C6xD5, S3xC10, D30, C2xC30, C4xF5, C4:F5, C22:F5, D4xD5, C22xF5, C22xF5, C4xC3:D4, D5xDic3, C15:D4, C3:D20, C5xC3:D4, C15:7D4, S3xF5, C6xF5, C6xF5, C2xC3:F5, C2xS3xD5, D5xC2xC6, D4xF5, Dic3xF5, D6:F5, Dic3:F5, D10.D6, D5xC3:D4, C2xS3xF5, C2xC6xF5, F5xC3:D4
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, C22xC4, C2xD4, C4oD4, F5, C4xS3, C3:D4, C22xS3, C4xD4, C2xF5, S3xC2xC4, C4oD12, C2xC3:D4, C22xF5, C4xC3:D4, S3xF5, D4xF5, C2xS3xF5, F5xC3:D4

Smallest permutation representation of F5xC3:D4
►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 16)(2 18 5 19)(3 20 4 17)(6 21)(7 23 10 24)(8 25 9 22)(11 26)(12 28 15 29)(13 30 14 27)(31 46)(32 48 35 49)(33 50 34 47)(36 51)(37 53 40 54)(38 55 39 52)(41 56)(42 58 45 59)(43 60 44 57)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H···4L 5 6A6B6C6D6E6F6G10A10B10C12A···12H 15  20 30A30B30C
order12222222344444444···45666666610101012···121520303030
size1125561030255556101030···30422210101010482410···10824888

45 irreducible representations

dim11111111111122222222244448888
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4S3D4D6D6C4oD4C3:D4C4xS3C4xS3C4oD12F5C2xF5C2xF5C2xF5S3xF5D4xF5C2xS3xF5F5xC3:D4
kernelF5xC3:D4Dic3xF5D6:F5Dic3:F5D10.D6D5xC3:D4C2xS3xF5C2xC6xF5C15:D4C3:D20C5xC3:D4C15:7D4C22xF5C3xF5C2xF5C22xD5C3xD5F5D10C2xC10D5C3:D4Dic3D6C2xC6C22C3C2C1
# reps11111111222212212422411111112

Matrix representation of F5xC3:D4 ►in GL8(F61)

10000000
01000000
00100000
00010000
000000060
000010060
000001060
000000160
,
10000000
01000000
005000000
000500000
00000010
00001000
00000001
00000100
,
6060000000
10000000
00100000
00010000
00001000
00000100
00000010
00000001
,
600000000
11000000
00120000
0060600000
000060000
000006000
000000600
000000060
,
10000000
6060000000
00100000
0060600000
000060000
000006000
000000600
000000060

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60,60,60],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,0,2,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[1,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60] >;
 

F5xC3:D4 in GAP, Magma, Sage, TeX 

F_5\times C_3\rtimes D_4
 
% in TeX
 
G:=Group("F5xC3:D4");
 
// GroupNames label
 
G:=SmallGroup(480,1010);
 
// by ID
 
G=gap.SmallGroup(480,1010);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,1356,9414,2379]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^5=b^4=c^3=d^4=e^2=1,b*a*b^-1=a^3,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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