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## G = D4×C3⋊F5order 480 = 25·3·5

### Direct product of D4 and C3⋊F5

Series: Derived Chief Lower central Upper central

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

Generators and relations for D4×C3⋊F5
G = < a,b,c,d,e | a4=b2=c3=d5=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >

Subgroups: 988 in 188 conjugacy classes, 63 normal (35 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, D4, D4, C23, D5, D5, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, F5, D10, D10, D10, C2×C10, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, C3×D5, C3×D5, C30, C30, C4×D4, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C22×D5, C4×Dic3, C4⋊Dic3, C6.D4, C22×Dic3, C6×D4, C3×Dic5, C60, C3⋊F5, C3⋊F5, C6×D5, C6×D5, C6×D5, C2×C30, C4×F5, C4⋊F5, C22⋊F5, D4×D5, C22×F5, D4×Dic3, D5×C12, C3×D20, C3×C5⋊D4, D4×C15, C2×C3⋊F5, C2×C3⋊F5, C2×C3⋊F5, D5×C2×C6, D4×F5, C4×C3⋊F5, C60⋊C4, D10.D6, C3×D4×D5, C22×C3⋊F5, D4×C3⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22×C4, C2×D4, C4○D4, F5, C2×Dic3, C22×S3, C4×D4, C2×F5, S3×D4, D42S3, C22×Dic3, C3⋊F5, C22×F5, D4×Dic3, C2×C3⋊F5, D4×F5, C22×C3⋊F5, D4×C3⋊F5

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

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G ··· 4L 5 6A 6B 6C 6D 6E 6F 6G 10A 10B 10C 12A 12B 15A 15B 20 30A 30B 30C 30D 30E 30F 60A 60B order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 ··· 4 5 6 6 6 6 6 6 6 10 10 10 12 12 15 15 20 30 30 30 30 30 30 60 60 size 1 1 2 2 5 5 10 10 2 2 10 15 15 15 15 30 ··· 30 4 2 4 4 10 10 20 20 4 8 8 4 20 4 4 8 4 4 8 8 8 8 8 8

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 8 type + + + + + + + + + - - - + + + + + - + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D4 D6 Dic3 Dic3 Dic3 D6 C4○D4 F5 C2×F5 C2×F5 S3×D4 D4⋊2S3 C3⋊F5 C2×C3⋊F5 C2×C3⋊F5 D4×F5 D4×C3⋊F5 kernel D4×C3⋊F5 C4×C3⋊F5 C60⋊C4 D10.D6 C3×D4×D5 C22×C3⋊F5 C3×D20 C3×C5⋊D4 D4×C15 D4×D5 C3⋊F5 C4×D5 D20 C5⋊D4 C5×D4 C22×D5 C3×D5 C3×D4 C12 C2×C6 D5 D5 D4 C4 C22 C3 C1 # reps 1 1 1 2 1 2 2 4 2 1 2 1 1 2 1 2 2 1 1 2 1 1 2 2 4 1 2

Matrix representation of D4×C3⋊F5 in GL6(𝔽61)

 60 36 0 0 0 0 44 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 17 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 33 0 6 6 0 0 55 27 55 0 0 0 0 55 27 55 0 0 6 6 0 33
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 60 60 60 60
,
 50 0 0 0 0 0 0 50 0 0 0 0 0 0 0 6 34 6 0 0 28 0 55 55 0 0 55 55 0 28 0 0 6 34 6 0

G:=sub<GL(6,GF(61))| [60,44,0,0,0,0,36,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,55,0,6,0,0,0,27,55,6,0,0,6,55,27,0,0,0,6,0,55,33],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,1,0,0,60,0,0,0,1,0,60,0,0,0,0,1,60],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,0,28,55,6,0,0,6,0,55,34,0,0,34,55,0,6,0,0,6,55,28,0] >;

D4×C3⋊F5 in GAP, Magma, Sage, TeX

D_4\times C_3\rtimes F_5
% in TeX

G:=Group("D4xC3:F5");
// GroupNames label

G:=SmallGroup(480,1067);
// by ID

G=gap.SmallGroup(480,1067);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,2693,14118,2379]);
// Polycyclic

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

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