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## G = C3×He5order 375 = 3·53

### Direct product of C3 and He5

direct product, metabelian, nilpotent (class 2), monomial, 5-elementary

Aliases: C3×He5, C52⋊C15, C15.1C52, (C5×C15)⋊C5, C5.1(C5×C15), SmallGroup(375,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C3×He5
 Chief series C1 — C5 — C52 — He5 — C3×He5
 Lower central C1 — C5 — C3×He5
 Upper central C1 — C15 — C3×He5

Generators and relations for C3×He5
G = < a,b,c,d | a3=b5=c5=d5=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Smallest permutation representation of C3×He5
On 75 points
Generators in S75
(1 11 57)(2 12 58)(3 13 59)(4 14 60)(5 15 56)(6 51 33)(7 52 34)(8 53 35)(9 54 31)(10 55 32)(16 63 38)(17 64 39)(18 65 40)(19 61 36)(20 62 37)(21 69 44)(22 70 45)(23 66 41)(24 67 42)(25 68 43)(26 75 50)(27 71 46)(28 72 47)(29 73 48)(30 74 49)
(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)
(1 43 29 39 33)(2 44 30 40 34)(3 45 26 36 35)(4 41 27 37 31)(5 42 28 38 32)(6 11 25 73 17)(7 12 21 74 18)(8 13 22 75 19)(9 14 23 71 20)(10 15 24 72 16)(46 62 54 60 66)(47 63 55 56 67)(48 64 51 57 68)(49 65 52 58 69)(50 61 53 59 70)
(1 2 35 27 32)(3 37 5 43 44)(4 28 39 40 26)(6 7 19 23 16)(8 71 10 11 12)(9 24 73 74 22)(13 20 15 25 21)(14 72 17 18 75)(29 30 45 31 42)(33 34 36 41 38)(46 55 57 58 53)(47 64 65 50 60)(48 49 70 54 67)(51 52 61 66 63)(56 68 69 59 62)

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

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

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

87 conjugacy classes

 class 1 3A 3B 5A 5B 5C 5D 5E ··· 5AB 15A ··· 15H 15I ··· 15BD order 1 3 3 5 5 5 5 5 ··· 5 15 ··· 15 15 ··· 15 size 1 1 1 1 1 1 1 5 ··· 5 1 ··· 1 5 ··· 5

87 irreducible representations

 dim 1 1 1 1 5 5 type + image C1 C3 C5 C15 He5 C3×He5 kernel C3×He5 He5 C5×C15 C52 C3 C1 # reps 1 2 24 48 4 8

Matrix representation of C3×He5 in GL5(𝔽31)

 5 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 5
,
 2 1 20 26 15 0 0 2 0 0 0 0 0 16 0 0 0 0 0 4 7 15 23 27 29
,
 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 2 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 7 15 23 27 29

G:=sub<GL(5,GF(31))| [5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5],[2,0,0,0,7,1,0,0,0,15,20,2,0,0,23,26,0,16,0,27,15,0,0,4,29],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[2,0,0,0,7,0,0,0,0,15,0,1,0,0,23,0,0,1,0,27,0,0,0,1,29] >;

C3×He5 in GAP, Magma, Sage, TeX

C_3\times {\rm He}_5
% in TeX

G:=Group("C3xHe5");
// GroupNames label

G:=SmallGroup(375,4);
// by ID

G=gap.SmallGroup(375,4);
# by ID

G:=PCGroup([4,-3,-5,-5,-5,366]);
// Polycyclic

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

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