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## G = D5×C3×C6order 180 = 22·32·5

### Direct product of C3×C6 and D5

Aliases: D5×C3×C6, C5⋊C62, C302C6, C10⋊(C3×C6), C153(C2×C6), (C3×C30)⋊3C2, (C3×C15)⋊8C22, SmallGroup(180,32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C3×C6
 Chief series C1 — C5 — C15 — C3×C15 — C32×D5 — D5×C3×C6
 Lower central C5 — D5×C3×C6
 Upper central C1 — C3×C6

Generators and relations for D5×C3×C6
G = < a,b,c,d | a3=b6=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 132 in 60 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C5, C6, C6, C32, D5, C10, C2×C6, C15, C3×C6, C3×C6, D10, C3×D5, C30, C62, C3×C15, C6×D5, C32×D5, C3×C30, D5×C3×C6
Quotients: C1, C2, C3, C22, C6, C32, D5, C2×C6, C3×C6, D10, C3×D5, C62, C6×D5, C32×D5, D5×C3×C6

Smallest permutation representation of D5×C3×C6
On 90 points
Generators in S90
(1 68 89)(2 69 90)(3 70 85)(4 71 86)(5 72 87)(6 67 88)(7 61 42)(8 62 37)(9 63 38)(10 64 39)(11 65 40)(12 66 41)(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)(25 81 47)(26 82 48)(27 83 43)(28 84 44)(29 79 45)(30 80 46)
(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)(7 75 67 51 44)(8 76 68 52 45)(9 77 69 53 46)(10 78 70 54 47)(11 73 71 49 48)(12 74 72 50 43)(13 27 66 55 87)(14 28 61 56 88)(15 29 62 57 89)(16 30 63 58 90)(17 25 64 59 85)(18 26 65 60 86)
(1 23)(2 24)(3 19)(4 20)(5 21)(6 22)(7 54)(8 49)(9 50)(10 51)(11 52)(12 53)(13 63)(14 64)(15 65)(16 66)(17 61)(18 62)(25 28)(26 29)(27 30)(31 41)(32 42)(33 37)(34 38)(35 39)(36 40)(43 46)(44 47)(45 48)(55 90)(56 85)(57 86)(58 87)(59 88)(60 89)(67 78)(68 73)(69 74)(70 75)(71 76)(72 77)(79 82)(80 83)(81 84)

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

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

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

D5×C3×C6 is a maximal subgroup of   C30.12D6  C327D20

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 D5 D10 C3×D5 C6×D5 kernel D5×C3×C6 C32×D5 C3×C30 C6×D5 C3×D5 C30 C3×C6 C32 C6 C3 # reps 1 2 1 8 16 8 2 2 16 16

Matrix representation of D5×C3×C6 in GL3(𝔽31) generated by

 25 0 0 0 25 0 0 0 25
,
 6 0 0 0 5 0 0 0 5
,
 1 0 0 0 19 1 0 11 30
,
 1 0 0 0 30 30 0 0 1
G:=sub<GL(3,GF(31))| [25,0,0,0,25,0,0,0,25],[6,0,0,0,5,0,0,0,5],[1,0,0,0,19,11,0,1,30],[1,0,0,0,30,0,0,30,1] >;

D5×C3×C6 in GAP, Magma, Sage, TeX

D_5\times C_3\times C_6
% in TeX

G:=Group("D5xC3xC6");
// GroupNames label

G:=SmallGroup(180,32);
// by ID

G=gap.SmallGroup(180,32);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-5,3604]);
// Polycyclic

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

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