Copied to
clipboard

G = C3xD4xF5order 480 = 25·3·5

Direct product of C3, D4 and F5

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

Aliases: C3xD4xF5, D20:3C12, C5:(D4xC12), C20:(C2xC12), C5:D4:C12, C4:F5:2C6, C4:1(C6xF5), C60:6(C2xC4), D10:(C2xC12), C15:24(C4xD4), (C4xF5):3C6, C12:6(C2xF5), (C5xD4):3C12, (C3xD20):6C4, (D4xC15):6C4, Dic5:(C2xC12), (C12xF5):8C2, (D4xD5).3C6, D5.2(C6xD4), C22:F5:3C6, C22:2(C6xF5), (C22xF5):2C6, C6.52(C22xF5), C30.90(C22xC4), C10.8(C22xC12), (C6xD5).70C23, (C6xF5).17C22, (D5xC12).87C22, D10.11(C22xC6), (C2xC6xF5):4C2, C2.9(C2xC6xF5), (C3xC4:F5):6C2, (C2xC6):4(C2xF5), (C3xD4xD5).6C2, (C2xC30):6(C2xC4), (C3xC5:D4):2C4, (C2xC10):2(C2xC12), (C6xD5):16(C2xC4), D5.2(C3xC4oD4), (C3xC22:F5):7C2, (C2xF5).3(C2xC6), (C3xD5).12(C2xD4), (C4xD5).12(C2xC6), (D5xC2xC6).91C22, (C3xDic5):12(C2xC4), (C3xD5).13(C4oD4), (C22xD5).18(C2xC6), SmallGroup(480,1054)

Series: Derived Chief Lower central Upper central

C1C10 — C3xD4xF5
C1C5C10D10C6xD5C6xF5C2xC6xF5 — C3xD4xF5
C5C10 — C3xD4xF5
C1C6C3xD4

Generators and relations for C3xD4xF5
 G = < a,b,c,d,e | a3=b4=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 664 in 188 conjugacy classes, 76 normal (40 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2xC4, D4, D4, C23, D5, D5, C10, C10, C12, C12, C2xC6, C2xC6, C15, C42, C22:C4, C4:C4, C22xC4, C2xD4, Dic5, C20, F5, F5, D10, D10, D10, C2xC10, C2xC12, C3xD4, C3xD4, C22xC6, C3xD5, C3xD5, C30, C30, C4xD4, C4xD5, D20, C5:D4, C5xD4, C2xF5, C2xF5, C2xF5, C22xD5, C4xC12, C3xC22:C4, C3xC4:C4, C22xC12, C6xD4, C3xDic5, C60, C3xF5, C3xF5, C6xD5, C6xD5, C6xD5, C2xC30, C4xF5, C4:F5, C22:F5, D4xD5, C22xF5, D4xC12, D5xC12, C3xD20, C3xC5:D4, D4xC15, C6xF5, C6xF5, C6xF5, D5xC2xC6, D4xF5, C12xF5, C3xC4:F5, C3xC22:F5, C3xD4xD5, C2xC6xF5, C3xD4xF5
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, D4, C23, C12, C2xC6, C22xC4, C2xD4, C4oD4, F5, C2xC12, C3xD4, C22xC6, C4xD4, C2xF5, C22xC12, C6xD4, C3xC4oD4, C3xF5, C22xF5, D4xC12, C6xF5, D4xF5, C2xC6xF5, C3xD4xF5

Smallest permutation representation of C3xD4xF5
On 60 points
Generators in S60
(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(1 31 6 36)(2 32 7 37)(3 33 8 38)(4 34 9 39)(5 35 10 40)(11 41 16 46)(12 42 17 47)(13 43 18 48)(14 44 19 49)(15 45 20 50)(21 51 26 56)(22 52 27 57)(23 53 28 58)(24 54 29 59)(25 55 30 60)
(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 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)
(2 3 5 4)(7 8 10 9)(12 13 15 14)(17 18 20 19)(22 23 25 24)(27 28 30 29)(32 33 35 34)(37 38 40 39)(42 43 45 44)(47 48 50 49)(52 53 55 54)(57 58 60 59)

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

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

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

75 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E4F···4L 5 6A6B6C6D6E6F6G6H6I6J6K6L6M6N10A10B10C12A12B12C···12J12K···12X15A15B 20 30A30B30C30D30E30F60A60B
order1222222233444444···4566666666666666101010121212···1212···121515203030303030306060
size1122551010112555510···104112222555510101010488225···510···1044844888888

75 irreducible representations

dim111111111111111111222244444488
type+++++++++++
imageC1C2C2C2C2C2C3C4C4C4C6C6C6C6C6C12C12C12D4C4oD4C3xD4C3xC4oD4F5C2xF5C2xF5C3xF5C6xF5C6xF5D4xF5C3xD4xF5
kernelC3xD4xF5C12xF5C3xC4:F5C3xC22:F5C3xD4xD5C2xC6xF5D4xF5C3xD20C3xC5:D4D4xC15C4xF5C4:F5C22:F5D4xD5C22xF5D20C5:D4C5xD4C3xF5C3xD5F5D5C3xD4C12C2xC6D4C4C22C3C1
# reps111212224222424484224411222412

Matrix representation of C3xD4xF5 in GL6(F61)

4700000
0470000
001000
000100
000010
000001
,
0600000
100000
0060000
0006000
0000600
0000060
,
100000
0600000
0060000
0006000
0000600
0000060
,
100000
010000
0060606060
001000
000100
000010
,
100000
010000
001000
000001
000100
0060606060

G:=sub<GL(6,GF(61))| [47,0,0,0,0,0,0,47,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,1,0,0,0,60,0,0,1,0,0,60,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,1,0,60] >;

C3xD4xF5 in GAP, Magma, Sage, TeX

C_3\times D_4\times F_5
% in TeX

G:=Group("C3xD4xF5");
// GroupNames label

G:=SmallGroup(480,1054);
// by ID

G=gap.SmallGroup(480,1054);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,555,9414,818]);
// Polycyclic

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

׿
x
:
Z
F
o
wr
Q
<