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## G = C5×C6.D6order 360 = 23·32·5

### Direct product of C5 and C6.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C5×C6.D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C30 — Dic3×C15 — C5×C6.D6
 Lower central C32 — C5×C6.D6
 Upper central C1 — C10

Generators and relations for C5×C6.D6
G = < a,b,c,d | a5=b6=d2=1, c6=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c5 >

Subgroups: 220 in 74 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, C2×C4, C32, C10, C10, Dic3, C12, D6, C15, C15, C3⋊S3, C3×C6, C20, C2×C10, C4×S3, C5×S3, C30, C30, C3×Dic3, C2×C3⋊S3, C2×C20, C3×C15, C5×Dic3, C60, S3×C10, C6.D6, C5×C3⋊S3, C3×C30, S3×C20, Dic3×C15, C10×C3⋊S3, C5×C6.D6
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C10, D6, C20, C2×C10, C4×S3, C5×S3, S32, C2×C20, S3×C10, C6.D6, S3×C20, C5×S32, C5×C6.D6

Smallest permutation representation of C5×C6.D6
On 60 points
Generators in S60
(1 44 26 20 50)(2 45 27 21 51)(3 46 28 22 52)(4 47 29 23 53)(5 48 30 24 54)(6 37 31 13 55)(7 38 32 14 56)(8 39 33 15 57)(9 40 34 16 58)(10 41 35 17 59)(11 42 36 18 60)(12 43 25 19 49)
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 15 17 19 21 23)(14 24 22 20 18 16)(25 27 29 31 33 35)(26 36 34 32 30 28)(37 39 41 43 45 47)(38 48 46 44 42 40)(49 51 53 55 57 59)(50 60 58 56 54 52)
(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 9)(3 7)(4 12)(6 10)(13 17)(14 22)(16 20)(19 23)(25 29)(26 34)(28 32)(31 35)(37 41)(38 46)(40 44)(43 47)(49 53)(50 58)(52 56)(55 59)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 5A 5B 5C 5D 6A 6B 6C 10A 10B 10C 10D 10E ··· 10L 12A 12B 12C 12D 15A ··· 15H 15I 15J 15K 15L 20A ··· 20P 30A ··· 30H 30I 30J 30K 30L 60A ··· 60P order 1 2 2 2 3 3 3 4 4 4 4 5 5 5 5 6 6 6 10 10 10 10 10 ··· 10 12 12 12 12 15 ··· 15 15 15 15 15 20 ··· 20 30 ··· 30 30 30 30 30 60 ··· 60 size 1 1 9 9 2 2 4 3 3 3 3 1 1 1 1 2 2 4 1 1 1 1 9 ··· 9 6 6 6 6 2 ··· 2 4 4 4 4 3 ··· 3 2 ··· 2 4 4 4 4 6 ··· 6

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + image C1 C2 C2 C4 C5 C10 C10 C20 S3 D6 C4×S3 C5×S3 S3×C10 S3×C20 S32 C6.D6 C5×S32 C5×C6.D6 kernel C5×C6.D6 Dic3×C15 C10×C3⋊S3 C5×C3⋊S3 C6.D6 C3×Dic3 C2×C3⋊S3 C3⋊S3 C5×Dic3 C30 C15 Dic3 C6 C3 C10 C5 C2 C1 # reps 1 2 1 4 4 8 4 16 2 2 4 8 8 16 1 1 4 4

Matrix representation of C5×C6.D6 in GL4(𝔽61) generated by

 58 0 0 0 0 58 0 0 0 0 20 0 0 0 0 20
,
 0 1 0 0 60 1 0 0 0 0 1 0 0 0 0 1
,
 50 11 0 0 0 11 0 0 0 0 0 1 0 0 60 60
,
 1 60 0 0 0 60 0 0 0 0 60 60 0 0 0 1
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,20,0,0,0,0,20],[0,60,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[50,0,0,0,11,11,0,0,0,0,0,60,0,0,1,60],[1,0,0,0,60,60,0,0,0,0,60,0,0,0,60,1] >;

C5×C6.D6 in GAP, Magma, Sage, TeX

C_5\times C_6.D_6
% in TeX

G:=Group("C5xC6.D6");
// GroupNames label

G:=SmallGroup(360,73);
// by ID

G=gap.SmallGroup(360,73);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,120,127,1210,8645]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^6=d^2=1,c^6=b^3,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^5>;
// generators/relations

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