C5xC2^2:F5 - GroupNames

G = C₅×C₂²⋊F₅ order 400 = 2⁴·5²

Direct product of C₅ and C₂²⋊F₅

direct product, metabelian, supersoluble, monomial

Aliases: C₅×C₂²⋊F₅, D₁₀⋊₂C₂₀, C₁₀²⋊₁C₄, (C₂×F₅)⋊C₁₀, C₂²⋊(C₅×F₅), (C₂×C₁₀)⋊₅F₅, (C₂×C₁₀)⋊₁C₂₀, (D₅×C₁₀)⋊₈C₄, (C₁₀×F₅)⋊₂C₂, (C₅×D₅).₅D₄, D₅.₂(C₅×D₄), C₂.₇(C₁₀×F₅), C₁₀.₇(C₂×C₂₀), C₁₀.₄₈(C₂×F₅), D₁₀.₆(C₂×C₁₀), C₅²⋊₄(C₂²⋊C₄), (C₂²×D₅).₂C₁₀, (D₅×C₁₀).₁₉C₂², C₅⋊(C₅×C₂²⋊C₄), (D₅×C₂×C₁₀).₄C₂, (C₅×C₁₀).₁₉(C₂×C₄), SmallGroup(400,141)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₅×C₂²⋊F₅

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — D₅×C₁₀ — C₁₀×F₅ — C₅×C₂²⋊F₅

Lower central

C₅ — C₁₀ — C₅×C₂²⋊F₅

Upper central

C₁ — C₁₀ — C₂×C₁₀

Generators and relations for C₅×C₂²⋊F₅
G = < a,b,c,d,e | a⁵=b²=c²=d⁵=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, ebe^-1=bc=cb, bd=db, cd=dc, ce=ec, ede^-1=d³ >

Subgroups: 256 in 73 conjugacy classes, 28 normal (20 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₅, C₂×C₄, C₂³, D₅, D₅, C₁₀, C₁₀, C₂²⋊C₄, C₂₀, F₅, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₅², C₂×C₂₀, C₂×F₅, C₂²×D₅, C₂²×C₁₀, C₅×D₅, C₅×D₅, C₅×C₁₀, C₅×C₁₀, C₅×C₂²⋊C₄, C₂²⋊F₅, C₅×F₅, D₅×C₁₀, D₅×C₁₀, C₁₀², C₁₀×F₅, D₅×C₂×C₁₀, C₅×C₂²⋊F₅
Quotients: C₁, C₂, C₄, C₂², C₅, C₂×C₄, D₄, C₁₀, C₂²⋊C₄, C₂₀, F₅, C₂×C₁₀, C₂×C₂₀, C₅×D₄, C₂×F₅, C₅×C₂²⋊C₄, C₂²⋊F₅, C₅×F₅, C₁₀×F₅, C₅×C₂²⋊F₅

Smallest permutation representation of C₅×C₂²⋊F₅
►On 40 points

Generators in S₄₀

(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)

(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)

(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)

(1 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)(21 24 22 25 23)(26 29 27 30 28)(31 33 35 32 34)(36 38 40 37 39)

(1 31 11 21)(2 32 12 22)(3 33 13 23)(4 34 14 24)(5 35 15 25)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)

G:=sub<Sym(40)| (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), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20)(21,24,22,25,23)(26,29,27,30,28)(31,33,35,32,34)(36,38,40,37,39), (1,31,11,21)(2,32,12,22)(3,33,13,23)(4,34,14,24)(5,35,15,25)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)>;

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), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20)(21,24,22,25,23)(26,29,27,30,28)(31,33,35,32,34)(36,38,40,37,39), (1,31,11,21)(2,32,12,22)(3,33,13,23)(4,34,14,24)(5,35,15,25)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30) );

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)], [(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,5,4,3,2),(6,10,9,8,7),(11,12,13,14,15),(16,17,18,19,20),(21,24,22,25,23),(26,29,27,30,28),(31,33,35,32,34),(36,38,40,37,39)], [(1,31,11,21),(2,32,12,22),(3,33,13,23),(4,34,14,24),(5,35,15,25),(6,36,16,26),(7,37,17,27),(8,38,18,28),(9,39,19,29),(10,40,20,30)]])

70 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	5A	5B	5C	5D	5E	···	5I	10A	10B	10C	10D	10E	10F	10G	10H	10I	···	10W	10X	···	10AE	10AF	10AG	10AH	10AI	20A	···	20P
order	1	2	2	2	2	2	4	4	4	4	5	5	5	5	5	···	5	10	10	10	10	10	10	10	10	10	···	10	10	···	10	10	10	10	10	20	···	20
size	1	1	2	5	5	10	10	10	10	10	1	1	1	1	4	···	4	1	1	1	1	2	2	2	2	4	···	4	5	···	5	10	10	10	10	10	···	10

70 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	4	4	4	4	4	4
type	+	+	+								+		+	+	+
image	C₁	C₂	C₂	C₄	C₄	C₅	C₁₀	C₁₀	C₂₀	C₂₀	D₄	C₅×D₄	F₅	C₂×F₅	C₂²⋊F₅	C₅×F₅	C₁₀×F₅	C₅×C₂²⋊F₅
kernel	C₅×C₂²⋊F₅	C₁₀×F₅	D₅×C₂×C₁₀	D₅×C₁₀	C₁₀²	C₂²⋊F₅	C₂×F₅	C₂²×D₅	D₁₀	C₂×C₁₀	C₅×D₅	D₅	C₂×C₁₀	C₁₀	C₅	C₂²	C₂	C₁
# reps	1	2	1	2	2	4	8	4	8	8	2	8	1	1	2	4	4	8

Matrix representation of C₅×C₂²⋊F₅ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	10	0	0	0
0	0	0	10	0	0
0	0	0	0	10	0
0	0	0	0	0	10

1	0	0	0	0	0
0	40	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

40	0	0	0	0	0
0	40	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	37	0	0	0
0	0	0	10	0	0
0	0	0	0	16	0
0	0	0	0	0	18

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	1	0	0	0

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,10,0,0,0,0,0,0,16,0,0,0,0,0,0,18],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₅×C₂²⋊F₅ in GAP, Magma, Sage, TeX

C_5\times C_2^2\rtimes F_5

% in TeX

G:=Group("C5xC2^2:F5");

// GroupNames label

G:=SmallGroup(400,141);

// by ID

G=gap.SmallGroup(400,141);

# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,505,5765,599]);

// Polycyclic

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

// generators/relations

G = C5×C22⋊F5 order 400 = 24·52

Direct product of C5 and C22⋊F5

G = C₅×C₂²⋊F₅ order 400 = 2⁴·5²

Direct product of C₅ and C₂²⋊F₅