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## G = C3×D4×D5order 240 = 24·3·5

### Direct product of C3, D4 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×D4×D5
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C2×C6 — C3×D4×D5
 Lower central C5 — C10 — C3×D4×D5
 Upper central C1 — C6 — C3×D4

Generators and relations for C3×D4×D5
G = < a,b,c,d,e | a3=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 340 in 108 conjugacy classes, 50 normal (26 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, C2×D4, Dic5, C20, D10, D10, D10, C2×C10, C2×C12, C3×D4, C3×D4, C22×C6, C3×D5, C3×D5, C30, C30, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, C6×D4, C3×Dic5, C60, C6×D5, C6×D5, C6×D5, C2×C30, D4×D5, D5×C12, C3×D20, C3×C5⋊D4, D4×C15, D5×C2×C6, C3×D4×D5
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C22×D5, C6×D4, C6×D5, D4×D5, D5×C2×C6, C3×D4×D5

Smallest permutation representation of C3×D4×D5
On 60 points
Generators in S60
(1 24 14)(2 25 15)(3 21 11)(4 22 12)(5 23 13)(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 34 9 39)(2 35 10 40)(3 31 6 36)(4 32 7 37)(5 33 8 38)(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)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)

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

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

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

C3×D4×D5 is a maximal subgroup of   D20⋊Dic3  D1210D10  D20.9D6  D2013D6  D2014D6

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 15A 15B 15C 15D 20A 20B 30A 30B 30C 30D 30E ··· 30L 60A 60B 60C 60D order 1 2 2 2 2 2 2 2 3 3 4 4 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 10 10 10 10 10 10 12 12 12 12 15 15 15 15 20 20 30 30 30 30 30 ··· 30 60 60 60 60 size 1 1 2 2 5 5 10 10 1 1 2 10 2 2 1 1 2 2 2 2 5 5 5 5 10 10 10 10 2 2 4 4 4 4 2 2 10 10 2 2 2 2 4 4 2 2 2 2 4 ··· 4 4 4 4 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D5 D10 D10 C3×D4 C3×D5 C6×D5 C6×D5 D4×D5 C3×D4×D5 kernel C3×D4×D5 D5×C12 C3×D20 C3×C5⋊D4 D4×C15 D5×C2×C6 D4×D5 C4×D5 D20 C5⋊D4 C5×D4 C22×D5 C3×D5 C3×D4 C12 C2×C6 D5 D4 C4 C22 C3 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 2 2 2 4 4 4 4 8 2 4

Matrix representation of C3×D4×D5 in GL4(𝔽61) generated by

 47 0 0 0 0 47 0 0 0 0 47 0 0 0 0 47
,
 60 0 0 0 0 60 0 0 0 0 60 20 0 0 6 1
,
 1 0 0 0 0 1 0 0 0 0 1 41 0 0 0 60
,
 0 1 0 0 60 17 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,47,0,0,0,0,47],[60,0,0,0,0,60,0,0,0,0,60,6,0,0,20,1],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,41,60],[0,60,0,0,1,17,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

C3×D4×D5 in GAP, Magma, Sage, TeX

C_3\times D_4\times D_5
% in TeX

G:=Group("C3xD4xD5");
// GroupNames label

G:=SmallGroup(240,159);
// by ID

G=gap.SmallGroup(240,159);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-5,260,6917]);
// Polycyclic

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

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