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## G = C3×D5×A4order 360 = 23·32·5

### Direct product of C3, D5 and A4

Aliases: C3×D5×A4, C5⋊(C6×A4), (C5×A4)⋊2C6, C152(C2×A4), (C2×C30)⋊2C6, (A4×C15)⋊5C2, C22⋊(C32×D5), (C22×D5)⋊C32, (D5×C2×C6)⋊C3, (C2×C10)⋊(C3×C6), (C2×C6)⋊2(C3×D5), SmallGroup(360,142)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×D5×A4
 Chief series C1 — C5 — C2×C10 — C2×C30 — A4×C15 — C3×D5×A4
 Lower central C2×C10 — C3×D5×A4
 Upper central C1 — C3

Generators and relations for C3×D5×A4
G = < a,b,c,d,e,f | a3=b5=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 312 in 64 conjugacy classes, 24 normal (15 characteristic)
C1, C2, C3, C3, C22, C22, C5, C6, C23, C32, D5, D5, C10, A4, C2×C6, C2×C6, C15, C15, C3×C6, D10, C2×C10, C2×A4, C22×C6, C3×D5, C3×D5, C30, C3×A4, C22×D5, C3×C15, C5×A4, C6×D5, C2×C30, C6×A4, C32×D5, D5×A4, D5×C2×C6, A4×C15, C3×D5×A4
Quotients: C1, C2, C3, C6, C32, D5, A4, C3×C6, C2×A4, C3×D5, C3×A4, C6×A4, C32×D5, D5×A4, C3×D5×A4

Smallest permutation representation of C3×D5×A4
On 60 points
Generators in S60
(1 44 24)(2 45 25)(3 41 21)(4 42 22)(5 43 23)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)
(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)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(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 44 24)(2 45 25)(3 41 21)(4 42 22)(5 43 23)(6 51 36)(7 52 37)(8 53 38)(9 54 39)(10 55 40)(11 56 26)(12 57 27)(13 58 28)(14 59 29)(15 60 30)(16 46 31)(17 47 32)(18 48 33)(19 49 34)(20 50 35)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 5A 5B 6A 6B 6C 6D 6E 6F 6G ··· 6L 10A 10B 15A 15B 15C 15D 15E ··· 15P 30A 30B 30C 30D order 1 2 2 2 3 3 3 ··· 3 5 5 6 6 6 6 6 6 6 ··· 6 10 10 15 15 15 15 15 ··· 15 30 30 30 30 size 1 3 5 15 1 1 4 ··· 4 2 2 3 3 5 5 15 15 20 ··· 20 6 6 2 2 2 2 8 ··· 8 6 6 6 6

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 3 3 6 6 type + + + + + + image C1 C2 C3 C3 C6 C6 D5 C3×D5 C3×D5 A4 C2×A4 C3×A4 C6×A4 D5×A4 C3×D5×A4 kernel C3×D5×A4 A4×C15 D5×A4 D5×C2×C6 C5×A4 C2×C30 C3×A4 A4 C2×C6 C3×D5 C15 D5 C5 C3 C1 # reps 1 1 6 2 6 2 2 12 4 1 1 2 2 2 4

Matrix representation of C3×D5×A4 in GL5(𝔽31)

 1 0 0 0 0 0 1 0 0 0 0 0 25 0 0 0 0 0 25 0 0 0 0 0 25
,
 12 1 0 0 0 30 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 28 12 12 0 0 10 1 2 0 0 10 2 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 19 0 0 21 30 29 0 0 0 0 30
,
 5 0 0 0 0 0 5 0 0 0 0 0 25 0 10 0 0 0 0 6 0 0 0 25 6

G:=sub<GL(5,GF(31))| [1,0,0,0,0,0,1,0,0,0,0,0,25,0,0,0,0,0,25,0,0,0,0,0,25],[12,30,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,28,10,10,0,0,12,1,2,0,0,12,2,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,21,0,0,0,0,30,0,0,0,19,29,30],[5,0,0,0,0,0,5,0,0,0,0,0,25,0,0,0,0,0,0,25,0,0,10,6,6] >;

C3×D5×A4 in GAP, Magma, Sage, TeX

C_3\times D_5\times A_4
% in TeX

G:=Group("C3xD5xA4");
// GroupNames label

G:=SmallGroup(360,142);
// by ID

G=gap.SmallGroup(360,142);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,2,-5,657,280,10373]);
// Polycyclic

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

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