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
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
(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 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4L | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 10A | 10B | 10C | 12A | 12B | 12C | ··· | 12J | 12K | ··· | 12X | 15A | 15B | 20 | 30A | 30B | 30C | 30D | 30E | 30F | 60A | 60B |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
size | 1 | 1 | 2 | 2 | 5 | 5 | 10 | 10 | 1 | 1 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 4 | 8 | 8 | 2 | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 4 | 4 | 8 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
75 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||||
image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | D4 | C4oD4 | C3xD4 | C3xC4oD4 | F5 | C2xF5 | C2xF5 | C3xF5 | C6xF5 | C6xF5 | D4xF5 | C3xD4xF5 |
kernel | C3xD4xF5 | C12xF5 | C3xC4:F5 | C3xC22:F5 | C3xD4xD5 | C2xC6xF5 | D4xF5 | C3xD20 | C3xC5:D4 | D4xC15 | C4xF5 | C4:F5 | C22:F5 | D4xD5 | C22xF5 | D20 | C5:D4 | C5xD4 | C3xF5 | C3xD5 | F5 | D5 | C3xD4 | C12 | C2xC6 | D4 | C4 | C22 | C3 | C1 |
# reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 |
Matrix representation of C3xD4xF5 ►in GL6(F61)
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 | 60 | 0 | 0 | 0 | 0 |
1 | 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 | 60 | 60 | 60 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 60 | 60 | 60 | 60 |
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