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## G = C6×C52⋊C3order 450 = 2·32·52

### Direct product of C6 and C52⋊C3

Aliases: C6×C52⋊C3, (C5×C30)⋊C3, (C5×C10)⋊C32, (C5×C15)⋊4C6, C522(C3×C6), SmallGroup(450,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C6×C52⋊C3
 Chief series C1 — C52 — C5×C15 — C3×C52⋊C3 — C6×C52⋊C3
 Lower central C52 — C6×C52⋊C3
 Upper central C1 — C6

Generators and relations for C6×C52⋊C3
G = < a,b,c,d | a6=b5=c5=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3c3, dcd-1=b-1c >

Smallest permutation representation of C6×C52⋊C3
On 90 points
Generators in S90
(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)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)
(1 36 79 37 20)(2 31 80 38 21)(3 32 81 39 22)(4 33 82 40 23)(5 34 83 41 24)(6 35 84 42 19)(13 87 55 66 27)(14 88 56 61 28)(15 89 57 62 29)(16 90 58 63 30)(17 85 59 64 25)(18 86 60 65 26)
(1 20 37 79 36)(2 21 38 80 31)(3 22 39 81 32)(4 23 40 82 33)(5 24 41 83 34)(6 19 42 84 35)(7 71 46 49 73)(8 72 47 50 74)(9 67 48 51 75)(10 68 43 52 76)(11 69 44 53 77)(12 70 45 54 78)(13 55 27 87 66)(14 56 28 88 61)(15 57 29 89 62)(16 58 30 90 63)(17 59 25 85 64)(18 60 26 86 65)
(1 68 89)(2 69 90)(3 70 85)(4 71 86)(5 72 87)(6 67 88)(7 26 82)(8 27 83)(9 28 84)(10 29 79)(11 30 80)(12 25 81)(13 34 50)(14 35 51)(15 36 52)(16 31 53)(17 32 54)(18 33 49)(19 75 56)(20 76 57)(21 77 58)(22 78 59)(23 73 60)(24 74 55)(37 43 62)(38 44 63)(39 45 64)(40 46 65)(41 47 66)(42 48 61)

G:=sub<Sym(90)| (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)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90), (1,36,79,37,20)(2,31,80,38,21)(3,32,81,39,22)(4,33,82,40,23)(5,34,83,41,24)(6,35,84,42,19)(13,87,55,66,27)(14,88,56,61,28)(15,89,57,62,29)(16,90,58,63,30)(17,85,59,64,25)(18,86,60,65,26), (1,20,37,79,36)(2,21,38,80,31)(3,22,39,81,32)(4,23,40,82,33)(5,24,41,83,34)(6,19,42,84,35)(7,71,46,49,73)(8,72,47,50,74)(9,67,48,51,75)(10,68,43,52,76)(11,69,44,53,77)(12,70,45,54,78)(13,55,27,87,66)(14,56,28,88,61)(15,57,29,89,62)(16,58,30,90,63)(17,59,25,85,64)(18,60,26,86,65), (1,68,89)(2,69,90)(3,70,85)(4,71,86)(5,72,87)(6,67,88)(7,26,82)(8,27,83)(9,28,84)(10,29,79)(11,30,80)(12,25,81)(13,34,50)(14,35,51)(15,36,52)(16,31,53)(17,32,54)(18,33,49)(19,75,56)(20,76,57)(21,77,58)(22,78,59)(23,73,60)(24,74,55)(37,43,62)(38,44,63)(39,45,64)(40,46,65)(41,47,66)(42,48,61)>;

G:=Group( (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)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90), (1,36,79,37,20)(2,31,80,38,21)(3,32,81,39,22)(4,33,82,40,23)(5,34,83,41,24)(6,35,84,42,19)(13,87,55,66,27)(14,88,56,61,28)(15,89,57,62,29)(16,90,58,63,30)(17,85,59,64,25)(18,86,60,65,26), (1,20,37,79,36)(2,21,38,80,31)(3,22,39,81,32)(4,23,40,82,33)(5,24,41,83,34)(6,19,42,84,35)(7,71,46,49,73)(8,72,47,50,74)(9,67,48,51,75)(10,68,43,52,76)(11,69,44,53,77)(12,70,45,54,78)(13,55,27,87,66)(14,56,28,88,61)(15,57,29,89,62)(16,58,30,90,63)(17,59,25,85,64)(18,60,26,86,65), (1,68,89)(2,69,90)(3,70,85)(4,71,86)(5,72,87)(6,67,88)(7,26,82)(8,27,83)(9,28,84)(10,29,79)(11,30,80)(12,25,81)(13,34,50)(14,35,51)(15,36,52)(16,31,53)(17,32,54)(18,33,49)(19,75,56)(20,76,57)(21,77,58)(22,78,59)(23,73,60)(24,74,55)(37,43,62)(38,44,63)(39,45,64)(40,46,65)(41,47,66)(42,48,61) );

G=PermutationGroup([[(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),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90)], [(1,36,79,37,20),(2,31,80,38,21),(3,32,81,39,22),(4,33,82,40,23),(5,34,83,41,24),(6,35,84,42,19),(13,87,55,66,27),(14,88,56,61,28),(15,89,57,62,29),(16,90,58,63,30),(17,85,59,64,25),(18,86,60,65,26)], [(1,20,37,79,36),(2,21,38,80,31),(3,22,39,81,32),(4,23,40,82,33),(5,24,41,83,34),(6,19,42,84,35),(7,71,46,49,73),(8,72,47,50,74),(9,67,48,51,75),(10,68,43,52,76),(11,69,44,53,77),(12,70,45,54,78),(13,55,27,87,66),(14,56,28,88,61),(15,57,29,89,62),(16,58,30,90,63),(17,59,25,85,64),(18,60,26,86,65)], [(1,68,89),(2,69,90),(3,70,85),(4,71,86),(5,72,87),(6,67,88),(7,26,82),(8,27,83),(9,28,84),(10,29,79),(11,30,80),(12,25,81),(13,34,50),(14,35,51),(15,36,52),(16,31,53),(17,32,54),(18,33,49),(19,75,56),(20,76,57),(21,77,58),(22,78,59),(23,73,60),(24,74,55),(37,43,62),(38,44,63),(39,45,64),(40,46,65),(41,47,66),(42,48,61)]])

66 conjugacy classes

 class 1 2 3A 3B 3C ··· 3H 5A ··· 5H 6A 6B 6C ··· 6H 10A ··· 10H 15A ··· 15P 30A ··· 30P order 1 2 3 3 3 ··· 3 5 ··· 5 6 6 6 ··· 6 10 ··· 10 15 ··· 15 30 ··· 30 size 1 1 1 1 25 ··· 25 3 ··· 3 1 1 25 ··· 25 3 ··· 3 3 ··· 3 3 ··· 3

66 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C6 C6 C52⋊C3 C2×C52⋊C3 C3×C52⋊C3 C6×C52⋊C3 kernel C6×C52⋊C3 C3×C52⋊C3 C2×C52⋊C3 C5×C30 C52⋊C3 C5×C15 C6 C3 C2 C1 # reps 1 1 6 2 6 2 8 8 16 16

Matrix representation of C6×C52⋊C3 in GL4(𝔽31) generated by

 25 0 0 0 0 30 0 0 0 0 30 0 0 0 0 30
,
 1 0 0 0 0 8 0 0 0 0 4 0 0 0 0 1
,
 1 0 0 0 0 4 0 0 0 0 16 0 0 0 0 16
,
 25 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(31))| [25,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[1,0,0,0,0,8,0,0,0,0,4,0,0,0,0,1],[1,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[25,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

C6×C52⋊C3 in GAP, Magma, Sage, TeX

C_6\times C_5^2\rtimes C_3
% in TeX

G:=Group("C6xC5^2:C3");
// GroupNames label

G:=SmallGroup(450,25);
// by ID

G=gap.SmallGroup(450,25);
# by ID

G:=PCGroup([5,-2,-3,-3,-5,5,2888,4284]);
// Polycyclic

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

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