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## G = C5×C3⋊D12order 360 = 23·32·5

### Direct product of C5 and C3⋊D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C5×C3⋊D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C30 — S3×C30 — C5×C3⋊D12
 Lower central C32 — C3×C6 — C5×C3⋊D12
 Upper central C1 — C10

Generators and relations for C5×C3⋊D12
G = < a,b,c,d | a5=b3=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 252 in 74 conjugacy classes, 28 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C15, C3×S3, C3⋊S3, C3×C6, C20, C2×C10, D12, C3⋊D4, C5×S3, C30, C30, C3×Dic3, S3×C6, C2×C3⋊S3, C5×D4, C3×C15, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C3⋊D12, S3×C15, C5×C3⋊S3, C3×C30, C5×D12, C5×C3⋊D4, Dic3×C15, S3×C30, C10×C3⋊S3, C5×C3⋊D12
Quotients: C1, C2, C22, C5, S3, D4, C10, D6, C2×C10, D12, C3⋊D4, C5×S3, S32, C5×D4, S3×C10, C3⋊D12, C5×D12, C5×C3⋊D4, C5×S32, C5×C3⋊D12

Smallest permutation representation of C5×C3⋊D12
On 60 points
Generators in S60
(1 28 41 13 59)(2 29 42 14 60)(3 30 43 15 49)(4 31 44 16 50)(5 32 45 17 51)(6 33 46 18 52)(7 34 47 19 53)(8 35 48 20 54)(9 36 37 21 55)(10 25 38 22 56)(11 26 39 23 57)(12 27 40 24 58)
(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 17 21)(14 22 18)(15 19 23)(16 24 20)(25 33 29)(26 30 34)(27 35 31)(28 32 36)(37 41 45)(38 46 42)(39 43 47)(40 48 44)(49 53 57)(50 58 54)(51 55 59)(52 60 56)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 33)(26 32)(27 31)(28 30)(34 36)(37 47)(38 46)(39 45)(40 44)(41 43)(49 59)(50 58)(51 57)(52 56)(53 55)

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

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

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

75 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4 5A 5B 5C 5D 6A 6B 6C 6D 6E 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 12A 12B 15A ··· 15H 15I 15J 15K 15L 20A 20B 20C 20D 30A ··· 30H 30I 30J 30K 30L 30M ··· 30T 60A ··· 60H order 1 2 2 2 3 3 3 4 5 5 5 5 6 6 6 6 6 10 10 10 10 10 10 10 10 10 10 10 10 12 12 15 ··· 15 15 15 15 15 20 20 20 20 30 ··· 30 30 30 30 30 30 ··· 30 60 ··· 60 size 1 1 6 18 2 2 4 6 1 1 1 1 2 2 4 6 6 1 1 1 1 6 6 6 6 18 18 18 18 6 6 2 ··· 2 4 4 4 4 6 6 6 6 2 ··· 2 4 4 4 4 6 ··· 6 6 ··· 6

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 S3 S3 D4 D6 D12 C3⋊D4 C5×S3 C5×S3 C5×D4 S3×C10 C5×D12 C5×C3⋊D4 S32 C3⋊D12 C5×S32 C5×C3⋊D12 kernel C5×C3⋊D12 Dic3×C15 S3×C30 C10×C3⋊S3 C3⋊D12 C3×Dic3 S3×C6 C2×C3⋊S3 C5×Dic3 S3×C10 C3×C15 C30 C15 C15 Dic3 D6 C32 C6 C3 C3 C10 C5 C2 C1 # reps 1 1 1 1 4 4 4 4 1 1 1 2 2 2 4 4 4 8 8 8 1 1 4 4

Matrix representation of C5×C3⋊D12 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 58 0 0 0 0 0 0 58 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 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 0 60 0 0 0 0 1 60
,
 0 1 0 0 0 0 60 0 0 0 0 0 0 0 1 1 0 0 0 0 60 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 60 0 0 0 0 0 0 1 0 0 0 0 0 0 60 60 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C5×C3⋊D12 in GAP, Magma, Sage, TeX

C_5\times C_3\rtimes D_{12}
% in TeX

G:=Group("C5xC3:D12");
// GroupNames label

G:=SmallGroup(360,75);
// by ID

G=gap.SmallGroup(360,75);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,265,127,1210,8645]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^3=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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