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## G = C10×C3⋊S3order 180 = 22·32·5

### Direct product of C10 and C3⋊S3

Aliases: C10×C3⋊S3, C303S3, C158D6, C6⋊(C5×S3), C32(S3×C10), (C3×C6)⋊2C10, (C3×C30)⋊5C2, C323(C2×C10), (C3×C15)⋊10C22, SmallGroup(180,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C10×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C15 — C5×C3⋊S3 — C10×C3⋊S3
 Lower central C32 — C10×C3⋊S3
 Upper central C1 — C10

Generators and relations for C10×C3⋊S3
G = < a,b,c,d | a10=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 156 in 60 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C2, C3, C22, C5, S3, C6, C32, C10, C10, D6, C15, C3⋊S3, C3×C6, C2×C10, C5×S3, C30, C2×C3⋊S3, C3×C15, S3×C10, C5×C3⋊S3, C3×C30, C10×C3⋊S3
Quotients: C1, C2, C22, C5, S3, C10, D6, C3⋊S3, C2×C10, C5×S3, C2×C3⋊S3, S3×C10, C5×C3⋊S3, C10×C3⋊S3

Smallest permutation representation of C10×C3⋊S3
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 79 50)(2 80 41)(3 71 42)(4 72 43)(5 73 44)(6 74 45)(7 75 46)(8 76 47)(9 77 48)(10 78 49)(11 37 68)(12 38 69)(13 39 70)(14 40 61)(15 31 62)(16 32 63)(17 33 64)(18 34 65)(19 35 66)(20 36 67)(21 84 55)(22 85 56)(23 86 57)(24 87 58)(25 88 59)(26 89 60)(27 90 51)(28 81 52)(29 82 53)(30 83 54)
(1 26 36)(2 27 37)(3 28 38)(4 29 39)(5 30 40)(6 21 31)(7 22 32)(8 23 33)(9 24 34)(10 25 35)(11 41 51)(12 42 52)(13 43 53)(14 44 54)(15 45 55)(16 46 56)(17 47 57)(18 48 58)(19 49 59)(20 50 60)(61 73 83)(62 74 84)(63 75 85)(64 76 86)(65 77 87)(66 78 88)(67 79 89)(68 80 90)(69 71 81)(70 72 82)
(11 90)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 80)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 68)(52 69)(53 70)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)

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

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

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

C10×C3⋊S3 is a maximal subgroup of   C30.12D6  C15⋊D12  Dic15⋊S3  D30⋊S3  S32×C10

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 5A 5B 5C 5D 6A 6B 6C 6D 10A 10B 10C 10D 10E ··· 10L 15A ··· 15P 30A ··· 30P order 1 2 2 2 3 3 3 3 5 5 5 5 6 6 6 6 10 10 10 10 10 ··· 10 15 ··· 15 30 ··· 30 size 1 1 9 9 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 9 ··· 9 2 ··· 2 2 ··· 2

60 irreducible representations

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

Matrix representation of C10×C3⋊S3 in GL4(𝔽31) generated by

 27 0 0 0 0 27 0 0 0 0 4 0 0 0 0 4
,
 0 1 0 0 30 30 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 30 30 0 0 0 0 30 30 0 0 1 0
,
 1 0 0 0 30 30 0 0 0 0 1 0 0 0 30 30
G:=sub<GL(4,GF(31))| [27,0,0,0,0,27,0,0,0,0,4,0,0,0,0,4],[0,30,0,0,1,30,0,0,0,0,1,0,0,0,0,1],[0,30,0,0,1,30,0,0,0,0,30,1,0,0,30,0],[1,30,0,0,0,30,0,0,0,0,1,30,0,0,0,30] >;

C10×C3⋊S3 in GAP, Magma, Sage, TeX

C_{10}\times C_3\rtimes S_3
% in TeX

G:=Group("C10xC3:S3");
// GroupNames label

G:=SmallGroup(180,35);
// by ID

G=gap.SmallGroup(180,35);
# by ID

G:=PCGroup([5,-2,-2,-5,-3,-3,803,3004]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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