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## G = Dic5×F5order 400 = 24·52

### Direct product of Dic5 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — Dic5×F5
 Chief series C1 — C5 — C52 — C5×D5 — D5×C10 — C10×F5 — Dic5×F5
 Lower central C52 — Dic5×F5
 Upper central C1 — C2

Generators and relations for Dic5×F5
G = < a,b,c,d | a10=c5=d4=1, b2=a5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 336 in 63 conjugacy classes, 28 normal (22 characteristic)
C1, C2, C2, C4, C22, C5, C5, C2×C4, D5, C10, C10, C42, Dic5, Dic5, C20, F5, F5, D10, C2×C10, C52, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C5×D5, C5×C10, C4×Dic5, C4×F5, C5×Dic5, C526C4, C5×F5, D5.D5, D5×C10, D5×Dic5, C10×F5, C2×D5.D5, Dic5×F5
Quotients: C1, C2, C4, C22, C2×C4, D5, C42, Dic5, F5, D10, C4×D5, C2×Dic5, C2×F5, C4×Dic5, C4×F5, D5×F5, Dic5×F5

Smallest permutation representation of Dic5×F5
On 80 points
Generators in S80
(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 76 77 78 79 80)
(1 24 6 29)(2 23 7 28)(3 22 8 27)(4 21 9 26)(5 30 10 25)(11 34 16 39)(12 33 17 38)(13 32 18 37)(14 31 19 36)(15 40 20 35)(41 64 46 69)(42 63 47 68)(43 62 48 67)(44 61 49 66)(45 70 50 65)(51 74 56 79)(52 73 57 78)(53 72 58 77)(54 71 59 76)(55 80 60 75)
(1 7 3 9 5)(2 8 4 10 6)(11 15 19 13 17)(12 16 20 14 18)(21 25 29 23 27)(22 26 30 24 28)(31 37 33 39 35)(32 38 34 40 36)(41 43 45 47 49)(42 44 46 48 50)(51 59 57 55 53)(52 60 58 56 54)(61 69 67 65 63)(62 70 68 66 64)(71 73 75 77 79)(72 74 76 78 80)
(1 51 11 41)(2 52 12 42)(3 53 13 43)(4 54 14 44)(5 55 15 45)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)

G:=sub<Sym(80)| (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)(71,72,73,74,75,76,77,78,79,80), (1,24,6,29)(2,23,7,28)(3,22,8,27)(4,21,9,26)(5,30,10,25)(11,34,16,39)(12,33,17,38)(13,32,18,37)(14,31,19,36)(15,40,20,35)(41,64,46,69)(42,63,47,68)(43,62,48,67)(44,61,49,66)(45,70,50,65)(51,74,56,79)(52,73,57,78)(53,72,58,77)(54,71,59,76)(55,80,60,75), (1,7,3,9,5)(2,8,4,10,6)(11,15,19,13,17)(12,16,20,14,18)(21,25,29,23,27)(22,26,30,24,28)(31,37,33,39,35)(32,38,34,40,36)(41,43,45,47,49)(42,44,46,48,50)(51,59,57,55,53)(52,60,58,56,54)(61,69,67,65,63)(62,70,68,66,64)(71,73,75,77,79)(72,74,76,78,80), (1,51,11,41)(2,52,12,42)(3,53,13,43)(4,54,14,44)(5,55,15,45)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70)>;

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)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,24,6,29)(2,23,7,28)(3,22,8,27)(4,21,9,26)(5,30,10,25)(11,34,16,39)(12,33,17,38)(13,32,18,37)(14,31,19,36)(15,40,20,35)(41,64,46,69)(42,63,47,68)(43,62,48,67)(44,61,49,66)(45,70,50,65)(51,74,56,79)(52,73,57,78)(53,72,58,77)(54,71,59,76)(55,80,60,75), (1,7,3,9,5)(2,8,4,10,6)(11,15,19,13,17)(12,16,20,14,18)(21,25,29,23,27)(22,26,30,24,28)(31,37,33,39,35)(32,38,34,40,36)(41,43,45,47,49)(42,44,46,48,50)(51,59,57,55,53)(52,60,58,56,54)(61,69,67,65,63)(62,70,68,66,64)(71,73,75,77,79)(72,74,76,78,80), (1,51,11,41)(2,52,12,42)(3,53,13,43)(4,54,14,44)(5,55,15,45)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70) );

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),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80)], [(1,24,6,29),(2,23,7,28),(3,22,8,27),(4,21,9,26),(5,30,10,25),(11,34,16,39),(12,33,17,38),(13,32,18,37),(14,31,19,36),(15,40,20,35),(41,64,46,69),(42,63,47,68),(43,62,48,67),(44,61,49,66),(45,70,50,65),(51,74,56,79),(52,73,57,78),(53,72,58,77),(54,71,59,76),(55,80,60,75)], [(1,7,3,9,5),(2,8,4,10,6),(11,15,19,13,17),(12,16,20,14,18),(21,25,29,23,27),(22,26,30,24,28),(31,37,33,39,35),(32,38,34,40,36),(41,43,45,47,49),(42,44,46,48,50),(51,59,57,55,53),(52,60,58,56,54),(61,69,67,65,63),(62,70,68,66,64),(71,73,75,77,79),(72,74,76,78,80)], [(1,51,11,41),(2,52,12,42),(3,53,13,43),(4,54,14,44),(5,55,15,45),(6,56,16,46),(7,57,17,47),(8,58,18,48),(9,59,19,49),(10,60,20,50),(21,71,31,61),(22,72,32,62),(23,73,33,63),(24,74,34,64),(25,75,35,65),(26,76,36,66),(27,77,37,67),(28,78,38,68),(29,79,39,69),(30,80,40,70)]])

40 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G ··· 4L 5A 5B 5C 5D 5E 10A 10B 10C 10D 10E 10F 10G 10H 10I 20A ··· 20H 20I 20J order 1 2 2 2 4 ··· 4 4 ··· 4 5 5 5 5 5 10 10 10 10 10 10 10 10 10 20 ··· 20 20 20 size 1 1 5 5 5 ··· 5 25 ··· 25 2 2 4 8 8 2 2 4 8 8 10 10 10 10 10 ··· 10 20 20

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 8 8 type + + + + + - + + + + - image C1 C2 C2 C2 C4 C4 C4 C4 D5 Dic5 D10 C4×D5 C4×D5 F5 C2×F5 C4×F5 D5×F5 Dic5×F5 kernel Dic5×F5 D5×Dic5 C10×F5 C2×D5.D5 C5×Dic5 C52⋊6C4 C5×F5 D5.D5 C2×F5 F5 D10 D5 C10 Dic5 C10 C5 C2 C1 # reps 1 1 1 1 2 2 4 4 2 4 2 4 4 1 1 2 2 2

Matrix representation of Dic5×F5 in GL6(𝔽41)

 35 1 0 0 0 0 40 0 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
,
 0 9 0 0 0 0 9 0 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 40 1 0 0 0 0 40 0 0 1 0 0 40 0 0 0 0 0 40 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(6,GF(41))| [35,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0] >;

Dic5×F5 in GAP, Magma, Sage, TeX

{\rm Dic}_5\times F_5
% in TeX

G:=Group("Dic5xF5");
// GroupNames label

G:=SmallGroup(400,117);
// by ID

G=gap.SmallGroup(400,117);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,970,5765,2897]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^5=d^4=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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