Copied to
clipboard

## G = C3×C5⋊F5order 300 = 22·3·52

### Direct product of C3 and C5⋊F5

Aliases: C3×C5⋊F5, C153F5, C525C12, (C5×C15)⋊7C4, C51(C3×F5), C5⋊D5.2C6, (C3×C5⋊D5).3C2, SmallGroup(300,30)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C3×C5⋊F5
 Chief series C1 — C5 — C52 — C5⋊D5 — C3×C5⋊D5 — C3×C5⋊F5
 Lower central C52 — C3×C5⋊F5
 Upper central C1 — C3

Generators and relations for C3×C5⋊F5
G = < a,b,c,d | a3=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c3 >

Character table of C3×C5⋊F5

 class 1 2 3A 3B 4A 4B 5A 5B 5C 5D 5E 5F 6A 6B 12A 12B 12C 12D 15A 15B 15C 15D 15E 15F 15G 15H 15I 15J 15K 15L size 1 25 1 1 25 25 4 4 4 4 4 4 25 25 25 25 25 25 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 ζ3 ζ32 -1 -1 1 1 1 1 1 1 ζ3 ζ32 ζ6 ζ65 ζ65 ζ6 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 6 ρ4 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ5 1 1 ζ32 ζ3 -1 -1 1 1 1 1 1 1 ζ32 ζ3 ζ65 ζ6 ζ6 ζ65 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 6 ρ6 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ7 1 -1 1 1 -i i 1 1 1 1 1 1 -1 -1 -i i -i i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 -1 1 1 i -i 1 1 1 1 1 1 -1 -1 i -i i -i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ9 1 -1 ζ3 ζ32 -i i 1 1 1 1 1 1 ζ65 ζ6 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 12 ρ10 1 -1 ζ32 ζ3 i -i 1 1 1 1 1 1 ζ6 ζ65 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 12 ρ11 1 -1 ζ3 ζ32 i -i 1 1 1 1 1 1 ζ65 ζ6 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 12 ρ12 1 -1 ζ32 ζ3 -i i 1 1 1 1 1 1 ζ6 ζ65 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 12 ρ13 4 0 4 4 0 0 4 -1 -1 -1 -1 -1 0 0 0 0 0 0 4 -1 -1 -1 4 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ14 4 0 4 4 0 0 -1 -1 -1 4 -1 -1 0 0 0 0 0 0 -1 4 -1 -1 -1 4 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ15 4 0 4 4 0 0 -1 -1 -1 -1 -1 4 0 0 0 0 0 0 -1 -1 -1 4 -1 -1 -1 4 -1 -1 -1 -1 orthogonal lifted from F5 ρ16 4 0 4 4 0 0 -1 -1 -1 -1 4 -1 0 0 0 0 0 0 -1 -1 4 -1 -1 -1 4 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ17 4 0 4 4 0 0 -1 -1 4 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 4 -1 4 orthogonal lifted from F5 ρ18 4 0 4 4 0 0 -1 4 -1 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 4 -1 4 -1 orthogonal lifted from F5 ρ19 4 0 -2-2√-3 -2+2√-3 0 0 -1 -1 -1 4 -1 -1 0 0 0 0 0 0 ζ65 -2+2√-3 ζ65 ζ65 ζ6 -2-2√-3 ζ6 ζ6 ζ6 ζ6 ζ65 ζ65 complex lifted from C3×F5 ρ20 4 0 -2-2√-3 -2+2√-3 0 0 -1 -1 -1 -1 4 -1 0 0 0 0 0 0 ζ65 ζ65 -2+2√-3 ζ65 ζ6 ζ6 -2-2√-3 ζ6 ζ6 ζ6 ζ65 ζ65 complex lifted from C3×F5 ρ21 4 0 -2+2√-3 -2-2√-3 0 0 -1 4 -1 -1 -1 -1 0 0 0 0 0 0 ζ6 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 ζ65 -2+2√-3 ζ65 -2-2√-3 ζ6 complex lifted from C3×F5 ρ22 4 0 -2+2√-3 -2-2√-3 0 0 -1 -1 4 -1 -1 -1 0 0 0 0 0 0 ζ6 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 ζ65 ζ65 -2+2√-3 ζ6 -2-2√-3 complex lifted from C3×F5 ρ23 4 0 -2-2√-3 -2+2√-3 0 0 -1 -1 4 -1 -1 -1 0 0 0 0 0 0 ζ65 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 ζ6 ζ6 -2-2√-3 ζ65 -2+2√-3 complex lifted from C3×F5 ρ24 4 0 -2+2√-3 -2-2√-3 0 0 -1 -1 -1 4 -1 -1 0 0 0 0 0 0 ζ6 -2-2√-3 ζ6 ζ6 ζ65 -2+2√-3 ζ65 ζ65 ζ65 ζ65 ζ6 ζ6 complex lifted from C3×F5 ρ25 4 0 -2-2√-3 -2+2√-3 0 0 -1 -1 -1 -1 -1 4 0 0 0 0 0 0 ζ65 ζ65 ζ65 -2+2√-3 ζ6 ζ6 ζ6 -2-2√-3 ζ6 ζ6 ζ65 ζ65 complex lifted from C3×F5 ρ26 4 0 -2+2√-3 -2-2√-3 0 0 4 -1 -1 -1 -1 -1 0 0 0 0 0 0 -2-2√-3 ζ6 ζ6 ζ6 -2+2√-3 ζ65 ζ65 ζ65 ζ65 ζ65 ζ6 ζ6 complex lifted from C3×F5 ρ27 4 0 -2-2√-3 -2+2√-3 0 0 -1 4 -1 -1 -1 -1 0 0 0 0 0 0 ζ65 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 ζ6 -2-2√-3 ζ6 -2+2√-3 ζ65 complex lifted from C3×F5 ρ28 4 0 -2+2√-3 -2-2√-3 0 0 -1 -1 -1 -1 -1 4 0 0 0 0 0 0 ζ6 ζ6 ζ6 -2-2√-3 ζ65 ζ65 ζ65 -2+2√-3 ζ65 ζ65 ζ6 ζ6 complex lifted from C3×F5 ρ29 4 0 -2-2√-3 -2+2√-3 0 0 4 -1 -1 -1 -1 -1 0 0 0 0 0 0 -2+2√-3 ζ65 ζ65 ζ65 -2-2√-3 ζ6 ζ6 ζ6 ζ6 ζ6 ζ65 ζ65 complex lifted from C3×F5 ρ30 4 0 -2+2√-3 -2-2√-3 0 0 -1 -1 -1 -1 4 -1 0 0 0 0 0 0 ζ6 ζ6 -2-2√-3 ζ6 ζ65 ζ65 -2+2√-3 ζ65 ζ65 ζ65 ζ6 ζ6 complex lifted from C3×F5

Smallest permutation representation of C3×C5⋊F5
On 75 points
Generators in S75
(1 12 56)(2 13 57)(3 14 58)(4 15 59)(5 11 60)(6 55 34)(7 51 35)(8 52 31)(9 53 32)(10 54 33)(16 65 40)(17 61 36)(18 62 37)(19 63 38)(20 64 39)(21 70 45)(22 66 41)(23 67 42)(24 68 43)(25 69 44)(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 42 32 39 26)(2 43 33 40 27)(3 44 34 36 28)(4 45 35 37 29)(5 41 31 38 30)(6 17 72 14 25)(7 18 73 15 21)(8 19 74 11 22)(9 20 75 12 23)(10 16 71 13 24)(46 57 68 54 65)(47 58 69 55 61)(48 59 70 51 62)(49 60 66 52 63)(50 56 67 53 64)
(2 3 5 4)(6 74 18 24)(7 71 17 22)(8 73 16 25)(9 75 20 23)(10 72 19 21)(11 15 13 14)(26 39 42 32)(27 36 41 35)(28 38 45 33)(29 40 44 31)(30 37 43 34)(46 61 66 51)(47 63 70 54)(48 65 69 52)(49 62 68 55)(50 64 67 53)(57 58 60 59)

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

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

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

Matrix representation of C3×C5⋊F5 in GL8(𝔽61)

 47 0 0 0 0 0 0 0 0 47 0 0 0 0 0 0 0 0 47 0 0 0 0 0 0 0 0 47 0 0 0 0 0 0 0 0 47 0 0 0 0 0 0 0 0 47 0 0 0 0 0 0 0 0 47 0 0 0 0 0 0 0 0 47
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 1 0 0 0 0 0 0 60 0 1 0 0 0 0 0 60 0 0 1 0 0 0 0 60 0 0 0
,
 0 0 0 60 0 0 0 0 1 0 0 60 0 0 0 0 0 1 0 60 0 0 0 0 0 0 1 60 0 0 0 0 0 0 0 0 0 0 0 60 0 0 0 0 1 0 0 60 0 0 0 0 0 1 0 60 0 0 0 0 0 0 1 60
,
 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 60 1 0 0 0 0 0 0 60 1 0 60 0 0 0 0 60 0 1 60 0 0 0 0 0 0 1 60

G:=sub<GL(8,GF(61))| [47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,0,47],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,60,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60,60,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60,60,60],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,60,60,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,60,60,60] >;

C3×C5⋊F5 in GAP, Magma, Sage, TeX

C_3\times C_5\rtimes F_5
% in TeX

G:=Group("C3xC5:F5");
// GroupNames label

G:=SmallGroup(300,30);
// by ID

G=gap.SmallGroup(300,30);
# by ID

G:=PCGroup([5,-2,-3,-2,-5,-5,30,483,173,3004,1014]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^5=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^3>;
// generators/relations

Export

׿
×
𝔽