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## G = C10×He3order 270 = 2·33·5

### Direct product of C10 and He3

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Aliases: C10×He3, C322C30, C30.1C32, (C3×C6)⋊C15, (C3×C30)⋊C3, (C3×C15)⋊4C6, C15.4(C3×C6), C3.1(C3×C30), C6.1(C3×C15), SmallGroup(270,21)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C10×He3
 Chief series C1 — C3 — C15 — C3×C15 — C5×He3 — C10×He3
 Lower central C1 — C3 — C10×He3
 Upper central C1 — C30 — C10×He3

Generators and relations for C10×He3
G = < a,b,c,d | a10=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Smallest permutation representation of C10×He3
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)
(11 65 24)(12 66 25)(13 67 26)(14 68 27)(15 69 28)(16 70 29)(17 61 30)(18 62 21)(19 63 22)(20 64 23)(41 53 72)(42 54 73)(43 55 74)(44 56 75)(45 57 76)(46 58 77)(47 59 78)(48 60 79)(49 51 80)(50 52 71)
(1 83 33)(2 84 34)(3 85 35)(4 86 36)(5 87 37)(6 88 38)(7 89 39)(8 90 40)(9 81 31)(10 82 32)(11 65 24)(12 66 25)(13 67 26)(14 68 27)(15 69 28)(16 70 29)(17 61 30)(18 62 21)(19 63 22)(20 64 23)(41 72 53)(42 73 54)(43 74 55)(44 75 56)(45 76 57)(46 77 58)(47 78 59)(48 79 60)(49 80 51)(50 71 52)
(1 74 14)(2 75 15)(3 76 16)(4 77 17)(5 78 18)(6 79 19)(7 80 20)(8 71 11)(9 72 12)(10 73 13)(21 37 47)(22 38 48)(23 39 49)(24 40 50)(25 31 41)(26 32 42)(27 33 43)(28 34 44)(29 35 45)(30 36 46)(51 64 89)(52 65 90)(53 66 81)(54 67 82)(55 68 83)(56 69 84)(57 70 85)(58 61 86)(59 62 87)(60 63 88)

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

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

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

110 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 5A 5B 5C 5D 6A 6B 6C ··· 6J 10A 10B 10C 10D 15A ··· 15H 15I ··· 15AN 30A ··· 30H 30I ··· 30AN order 1 2 3 3 3 ··· 3 5 5 5 5 6 6 6 ··· 6 10 10 10 10 15 ··· 15 15 ··· 15 30 ··· 30 30 ··· 30 size 1 1 1 1 3 ··· 3 1 1 1 1 1 1 3 ··· 3 1 1 1 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C5 C6 C10 C15 C30 He3 C2×He3 C5×He3 C10×He3 kernel C10×He3 C5×He3 C3×C30 C2×He3 C3×C15 He3 C3×C6 C32 C10 C5 C2 C1 # reps 1 1 8 4 8 4 32 32 2 2 8 8

Matrix representation of C10×He3 in GL4(𝔽31) generated by

 30 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 25 0 0 0 0 1 0 0 0 7 5 0 0 14 0 25
,
 1 0 0 0 0 5 0 0 0 0 5 0 0 0 0 5
,
 25 0 0 0 0 7 4 0 0 9 24 1 0 22 8 0
G:=sub<GL(4,GF(31))| [30,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[25,0,0,0,0,1,7,14,0,0,5,0,0,0,0,25],[1,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[25,0,0,0,0,7,9,22,0,4,24,8,0,0,1,0] >;

C10×He3 in GAP, Magma, Sage, TeX

C_{10}\times {\rm He}_3
% in TeX

G:=Group("C10xHe3");
// GroupNames label

G:=SmallGroup(270,21);
// by ID

G=gap.SmallGroup(270,21);
# by ID

G:=PCGroup([5,-2,-3,-3,-5,-3,727]);
// Polycyclic

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

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