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G = C3×C5⋊F5order 300 = 22·3·52

Direct product of C3 and C5⋊F5

direct product, metabelian, supersoluble, monomial, A-group

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

C1C52 — C3×C5⋊F5
C1C5C52C5⋊D5C3×C5⋊D5 — C3×C5⋊F5
C52 — C3×C5⋊F5
C1C3

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 >

25C2
25C4
25C6
5D5
5D5
5D5
5D5
5D5
5D5
25C12
5F5
5F5
5F5
5F5
5F5
5F5
5C3×D5
5C3×D5
5C3×D5
5C3×D5
5C3×D5
5C3×D5
5C3×F5
5C3×F5
5C3×F5
5C3×F5
5C3×F5
5C3×F5

Character table of C3×C5⋊F5

 class 123A3B4A4B5A5B5C5D5E5F6A6B12A12B12C12D15A15B15C15D15E15F15G15H15I15J15K15L
 size 125112525444444252525252525444444444444
ρ1111111111111111111111111111111    trivial
ρ21111-1-111111111-1-1-1-1111111111111    linear of order 2
ρ311ζ3ζ32-1-1111111ζ3ζ32ζ6ζ65ζ65ζ6ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 6
ρ411ζ3ζ3211111111ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ511ζ32ζ3-1-1111111ζ32ζ3ζ65ζ6ζ6ζ65ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 6
ρ611ζ32ζ311111111ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ71-111-ii111111-1-1-ii-ii111111111111    linear of order 4
ρ81-111i-i111111-1-1i-ii-i111111111111    linear of order 4
ρ91-1ζ3ζ32-ii111111ζ65ζ6ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 12
ρ101-1ζ32ζ3i-i111111ζ6ζ65ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 12
ρ111-1ζ3ζ32i-i111111ζ65ζ6ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ3ζ32ζ32    linear of order 12
ρ121-1ζ32ζ3-ii111111ζ6ζ65ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ32ζ3ζ3    linear of order 12
ρ134044004-1-1-1-1-10000004-1-1-14-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ14404400-1-1-14-1-1000000-14-1-1-14-1-1-1-1-1-1    orthogonal lifted from F5
ρ15404400-1-1-1-1-14000000-1-1-14-1-1-14-1-1-1-1    orthogonal lifted from F5
ρ16404400-1-1-1-14-1000000-1-14-1-1-14-1-1-1-1-1    orthogonal lifted from F5
ρ17404400-1-14-1-1-1000000-1-1-1-1-1-1-1-1-14-14    orthogonal lifted from F5
ρ18404400-14-1-1-1-1000000-1-1-1-1-1-1-1-14-14-1    orthogonal lifted from F5
ρ1940-2-2-3-2+2-300-1-1-14-1-1000000ζ65-2+2-3ζ65ζ65ζ6-2-2-3ζ6ζ6ζ6ζ6ζ65ζ65    complex lifted from C3×F5
ρ2040-2-2-3-2+2-300-1-1-1-14-1000000ζ65ζ65-2+2-3ζ65ζ6ζ6-2-2-3ζ6ζ6ζ6ζ65ζ65    complex lifted from C3×F5
ρ2140-2+2-3-2-2-300-14-1-1-1-1000000ζ6ζ6ζ6ζ6ζ65ζ65ζ65ζ65-2+2-3ζ65-2-2-3ζ6    complex lifted from C3×F5
ρ2240-2+2-3-2-2-300-1-14-1-1-1000000ζ6ζ6ζ6ζ6ζ65ζ65ζ65ζ65ζ65-2+2-3ζ6-2-2-3    complex lifted from C3×F5
ρ2340-2-2-3-2+2-300-1-14-1-1-1000000ζ65ζ65ζ65ζ65ζ6ζ6ζ6ζ6ζ6-2-2-3ζ65-2+2-3    complex lifted from C3×F5
ρ2440-2+2-3-2-2-300-1-1-14-1-1000000ζ6-2-2-3ζ6ζ6ζ65-2+2-3ζ65ζ65ζ65ζ65ζ6ζ6    complex lifted from C3×F5
ρ2540-2-2-3-2+2-300-1-1-1-1-14000000ζ65ζ65ζ65-2+2-3ζ6ζ6ζ6-2-2-3ζ6ζ6ζ65ζ65    complex lifted from C3×F5
ρ2640-2+2-3-2-2-3004-1-1-1-1-1000000-2-2-3ζ6ζ6ζ6-2+2-3ζ65ζ65ζ65ζ65ζ65ζ6ζ6    complex lifted from C3×F5
ρ2740-2-2-3-2+2-300-14-1-1-1-1000000ζ65ζ65ζ65ζ65ζ6ζ6ζ6ζ6-2-2-3ζ6-2+2-3ζ65    complex lifted from C3×F5
ρ2840-2+2-3-2-2-300-1-1-1-1-14000000ζ6ζ6ζ6-2-2-3ζ65ζ65ζ65-2+2-3ζ65ζ65ζ6ζ6    complex lifted from C3×F5
ρ2940-2-2-3-2+2-3004-1-1-1-1-1000000-2+2-3ζ65ζ65ζ65-2-2-3ζ6ζ6ζ6ζ6ζ6ζ65ζ65    complex lifted from C3×F5
ρ3040-2+2-3-2-2-300-1-1-1-14-1000000ζ6ζ6-2-2-3ζ6ζ65ζ65-2+2-3ζ65ζ65ζ65ζ6ζ6    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)

470000000
047000000
004700000
000470000
000047000
000004700
000000470
000000047
,
10000000
01000000
00100000
00010000
000060100
000060010
000060001
000060000
,
000600000
100600000
010600000
001600000
000000060
000010060
000001060
000000160
,
00100000
10000000
00010000
01000000
000060100
0000601060
0000600160
000000160

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

Subgroup lattice of C3×C5⋊F5 in TeX
Character table of C3×C5⋊F5 in TeX

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