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## G = F5×C3⋊D4order 480 = 25·3·5

### Direct product of F5 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — F5×C3⋊D4
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C6×F5 — C2×S3×F5 — F5×C3⋊D4
 Lower central C15 — C30 — F5×C3⋊D4
 Upper central C1 — C2 — C22

Generators and relations for F5×C3⋊D4
G = < a,b,c,d,e | a5=b4=c3=d4=e2=1, bab-1=a3, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1044 in 188 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, F5, F5, D10, D10, C2×C10, C2×C10, C4×S3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5, C3×D5, D15, C30, C30, C4×D4, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C2×F5, C22×D5, C22×D5, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C5×Dic3, Dic15, C3×F5, C3×F5, C3⋊F5, S3×D5, C6×D5, C6×D5, S3×C10, D30, C2×C30, C4×F5, C4⋊F5, C22⋊F5, D4×D5, C22×F5, C22×F5, C4×C3⋊D4, D5×Dic3, C15⋊D4, C3⋊D20, C5×C3⋊D4, C157D4, S3×F5, C6×F5, C6×F5, C2×C3⋊F5, C2×S3×D5, D5×C2×C6, D4×F5, Dic3×F5, D6⋊F5, Dic3⋊F5, D10.D6, D5×C3⋊D4, C2×S3×F5, C2×C6×F5, F5×C3⋊D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, F5, C4×S3, C3⋊D4, C22×S3, C4×D4, C2×F5, S3×C2×C4, C4○D12, C2×C3⋊D4, C22×F5, C4×C3⋊D4, S3×F5, D4×F5, C2×S3×F5, F5×C3⋊D4

Smallest permutation representation of F5×C3⋊D4
On 60 points
Generators in S60
(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)
(1 16)(2 18 5 19)(3 20 4 17)(6 21)(7 23 10 24)(8 25 9 22)(11 26)(12 28 15 29)(13 30 14 27)(31 46)(32 48 35 49)(33 50 34 47)(36 51)(37 53 40 54)(38 55 39 52)(41 56)(42 58 45 59)(43 60 44 57)
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)
(1 31 16 46)(2 32 17 47)(3 33 18 48)(4 34 19 49)(5 35 20 50)(6 41 21 56)(7 42 22 57)(8 43 23 58)(9 44 24 59)(10 45 25 60)(11 36 26 51)(12 37 27 52)(13 38 28 53)(14 39 29 54)(15 40 30 55)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 56)(37 57)(38 58)(39 59)(40 60)(41 51)(42 52)(43 53)(44 54)(45 55)

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

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), (1,16)(2,18,5,19)(3,20,4,17)(6,21)(7,23,10,24)(8,25,9,22)(11,26)(12,28,15,29)(13,30,14,27)(31,46)(32,48,35,49)(33,50,34,47)(36,51)(37,53,40,54)(38,55,39,52)(41,56)(42,58,45,59)(43,60,44,57), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45)(46,51,56)(47,52,57)(48,53,58)(49,54,59)(50,55,60), (1,31,16,46)(2,32,17,47)(3,33,18,48)(4,34,19,49)(5,35,20,50)(6,41,21,56)(7,42,22,57)(8,43,23,58)(9,44,24,59)(10,45,25,60)(11,36,26,51)(12,37,27,52)(13,38,28,53)(14,39,29,54)(15,40,30,55), (6,11)(7,12)(8,13)(9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30)(31,46)(32,47)(33,48)(34,49)(35,50)(36,56)(37,57)(38,58)(39,59)(40,60)(41,51)(42,52)(43,53)(44,54)(45,55) );

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)], [(1,16),(2,18,5,19),(3,20,4,17),(6,21),(7,23,10,24),(8,25,9,22),(11,26),(12,28,15,29),(13,30,14,27),(31,46),(32,48,35,49),(33,50,34,47),(36,51),(37,53,40,54),(38,55,39,52),(41,56),(42,58,45,59),(43,60,44,57)], [(1,6,11),(2,7,12),(3,8,13),(4,9,14),(5,10,15),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30),(31,36,41),(32,37,42),(33,38,43),(34,39,44),(35,40,45),(46,51,56),(47,52,57),(48,53,58),(49,54,59),(50,55,60)], [(1,31,16,46),(2,32,17,47),(3,33,18,48),(4,34,19,49),(5,35,20,50),(6,41,21,56),(7,42,22,57),(8,43,23,58),(9,44,24,59),(10,45,25,60),(11,36,26,51),(12,37,27,52),(13,38,28,53),(14,39,29,54),(15,40,30,55)], [(6,11),(7,12),(8,13),(9,14),(10,15),(21,26),(22,27),(23,28),(24,29),(25,30),(31,46),(32,47),(33,48),(34,49),(35,50),(36,56),(37,57),(38,58),(39,59),(40,60),(41,51),(42,52),(43,53),(44,54),(45,55)]])

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H ··· 4L 5 6A 6B 6C 6D 6E 6F 6G 10A 10B 10C 12A ··· 12H 15 20 30A 30B 30C order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 ··· 4 5 6 6 6 6 6 6 6 10 10 10 12 ··· 12 15 20 30 30 30 size 1 1 2 5 5 6 10 30 2 5 5 5 5 6 10 10 30 ··· 30 4 2 2 2 10 10 10 10 4 8 24 10 ··· 10 8 24 8 8 8

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 8 8 8 8 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D4 D6 D6 C4○D4 C3⋊D4 C4×S3 C4×S3 C4○D12 F5 C2×F5 C2×F5 C2×F5 S3×F5 D4×F5 C2×S3×F5 F5×C3⋊D4 kernel F5×C3⋊D4 Dic3×F5 D6⋊F5 Dic3⋊F5 D10.D6 D5×C3⋊D4 C2×S3×F5 C2×C6×F5 C15⋊D4 C3⋊D20 C5×C3⋊D4 C15⋊7D4 C22×F5 C3×F5 C2×F5 C22×D5 C3×D5 F5 D10 C2×C10 D5 C3⋊D4 Dic3 D6 C2×C6 C22 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 1 2 2 1 2 4 2 2 4 1 1 1 1 1 1 1 2

Matrix representation of F5×C3⋊D4 in GL8(𝔽61)

 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 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
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 50 0 0 0 0 0 0 0 0 50 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 1 0 0 0 0 0 1 0 0
,
 60 60 0 0 0 0 0 0 1 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 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 60 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 0 0 60 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60,60,60],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,0,2,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[1,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60] >;

F5×C3⋊D4 in GAP, Magma, Sage, TeX

F_5\times C_3\rtimes D_4
% in TeX

G:=Group("F5xC3:D4");
// GroupNames label

G:=SmallGroup(480,1010);
// by ID

G=gap.SmallGroup(480,1010);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,1356,9414,2379]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=c^3=d^4=e^2=1,b*a*b^-1=a^3,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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