Dic5xF5 - GroupNames

G = Dic₅×F₅ order 400 = 2⁴·5²

Direct product of Dic₅ and F₅

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

Aliases: Dic₅×F₅, C₅²⋊₂C₄², D₁₀.₅D₁₀, C₅⋊₆(C₄×F₅), (C₅×F₅)⋊₂C₄, C₂.₁(D₅×F₅), D₅.D₅⋊₂C₄, D₅.₁(C₄×D₅), C₅⋊₁(C₄×Dic₅), C₁₀.₁(C₄×D₅), D₅.(C₂×Dic₅), (C₂×F₅).₂D₅, C₅²⋊₆C₄⋊₁C₄, (C₅×Dic₅)⋊₃C₄, (C₁₀×F₅).₃C₂, C₁₀.₂₈(C₂×F₅), (D₅×Dic₅).₆C₂, (D₅×C₁₀).₅C₂², (C₅×C₁₀).₁(C₂×C₄), (C₅×D₅).₂(C₂×C₄), (C₂×D₅.D₅).₁C₂, SmallGroup(400,117)

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, C₅²⋊₆C₄, 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

G = Dic5×F5 order 400 = 24·52

Direct product of Dic5 and F5

G = Dic₅×F₅ order 400 = 2⁴·5²

Direct product of Dic₅ and F₅