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## G = C3×D4×F5order 480 = 25·3·5

### Direct product of C3, D4 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×D4×F5
 Chief series C1 — C5 — C10 — D10 — C6×D5 — C6×F5 — C2×C6×F5 — C3×D4×F5
 Lower central C5 — C10 — C3×D4×F5
 Upper central C1 — C6 — C3×D4

Generators and relations for C3×D4×F5
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, C2×C4, D4, D4, C23, D5, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, F5, F5, D10, D10, D10, C2×C10, C2×C12, C3×D4, C3×D4, C22×C6, C3×D5, C3×D5, C30, C30, C4×D4, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C2×F5, C2×F5, C22×D5, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C3×Dic5, C60, C3×F5, C3×F5, C6×D5, C6×D5, C6×D5, C2×C30, C4×F5, C4⋊F5, C22⋊F5, D4×D5, C22×F5, D4×C12, D5×C12, C3×D20, C3×C5⋊D4, D4×C15, C6×F5, C6×F5, C6×F5, D5×C2×C6, D4×F5, C12×F5, C3×C4⋊F5, C3×C22⋊F5, C3×D4×D5, C2×C6×F5, C3×D4×F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22×C4, C2×D4, C4○D4, F5, C2×C12, C3×D4, C22×C6, C4×D4, C2×F5, C22×C12, C6×D4, C3×C4○D4, C3×F5, C22×F5, D4×C12, C6×F5, D4×F5, C2×C6×F5, C3×D4×F5

Smallest permutation representation of C3×D4×F5
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 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 C4○D4 C3×D4 C3×C4○D4 F5 C2×F5 C2×F5 C3×F5 C6×F5 C6×F5 D4×F5 C3×D4×F5 kernel C3×D4×F5 C12×F5 C3×C4⋊F5 C3×C22⋊F5 C3×D4×D5 C2×C6×F5 D4×F5 C3×D20 C3×C5⋊D4 D4×C15 C4×F5 C4⋊F5 C22⋊F5 D4×D5 C22×F5 D20 C5⋊D4 C5×D4 C3×F5 C3×D5 F5 D5 C3×D4 C12 C2×C6 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 C3×D4×F5 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] >;

C3×D4×F5 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

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