C2^2:C4xF5 - GroupNames

G = C₂²⋊C₄×F₅ order 320 = 2⁶·5

Direct product of C₂²⋊C₄ and F₅

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂²⋊C₄×F₅, D₁₀⋊C₄², (C₂×C₁₀)⋊C₄², C₂.₂(D₄×F₅), D₅.₂(C₄×D₄), C₁₀.₅(C₄×D₄), C₂²⋊F₅⋊₃C₄, (C₂×F₅).₈D₄, C₂²⋊₂(C₄×F₅), (C₂²×F₅)⋊₂C₄, C₂³.D₅⋊₈C₄, D₁₀⋊C₄⋊₃C₄, D₁₀.₅₉(C₂×D₄), C₁₀.₈(C₂×C₄²), (C₂³×F₅).₁C₂, C₂³.₂₇(C₂×F₅), D₁₀.₃Q₈⋊₇C₂, D₁₀.₄₂(C₄○D₄), D₁₀.₂₉(C₂²×C₄), C₂².₃₅(C₂²×F₅), D₅.₂(C₄²⋊C₂), (C₂³×D₅).₈₃C₂², (C₂²×F₅).₂₂C₂², (C₂²×D₅).₂₆₈C₂³, (C₂×C₄×F₅)⋊₇C₂, C₅⋊₁(C₄×C₂²⋊C₄), (C₂×C₄)⋊₇(C₂×F₅), (C₂×C₂₀)⋊₉(C₂×C₄), C₂.₁₀(C₂×C₄×F₅), (C₂×F₅)⋊₁(C₂×C₄), (C₅×C₂²⋊C₄)⋊₆C₄, (D₅×C₂²⋊C₄).₇C₂, D₅.₁(C₂×C₂²⋊C₄), (C₂×C₂²⋊F₅).₃C₂, (C₂×Dic₅)⋊₁₄(C₂×C₄), (C₂×C₄×D₅).₂₈₈C₂², (C₂²×C₁₀).₁₉(C₂×C₄), (C₂×C₁₀).₃₆(C₂²×C₄), (C₂²×D₅).₄₃(C₂×C₄), SmallGroup(320,1036)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₂²⋊C₄×F₅

Chief series

C₁ — C₅ — D₅ — D₁₀ — C₂²×D₅ — C₂²×F₅ — C₂³×F₅ — C₂²⋊C₄×F₅

Lower central

C₅ — C₁₀ — C₂²⋊C₄×F₅

Upper central

C₁ — C₂² — C₂²⋊C₄

Generators and relations for C₂²⋊C₄×F₅
G = < a,b,c,d,e | a²=b²=c⁴=d⁵=e⁴=1, cac^-1=ab=ba, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d³ >

Subgroups: 1018 in 258 conjugacy classes, 84 normal (24 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, C₂³, C₂³, D₅, D₅, D₅, C₁₀, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₂²⋊C₄, C₂²×C₄, C₂⁴, Dic₅, C₂₀, F₅, F₅, D₁₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂³×C₄, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂×F₅, C₂×F₅, C₂²×D₅, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₄×C₂²⋊C₄, D₁₀⋊C₄, C₂³.D₅, C₅×C₂²⋊C₄, C₄×F₅, C₂²⋊F₅, C₂×C₄×D₅, C₂²×F₅, C₂²×F₅, C₂²×F₅, C₂³×D₅, D₁₀.₃Q₈, D₅×C₂²⋊C₄, C₂×C₄×F₅, C₂×C₂²⋊F₅, C₂³×F₅, C₂²⋊C₄×F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₄², C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄○D₄, F₅, C₂×C₄², C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₂×F₅, C₄×C₂²⋊C₄, C₄×F₅, C₂²×F₅, C₂×C₄×F₅, D₄×F₅, C₂²⋊C₄×F₅

Smallest permutation representation of C₂²⋊C₄×F₅
►On 40 points

Generators in S₄₀

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

(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 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)

(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)

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	2	2	4	4	4	4	8
type	+	+	+	+	+	+						+		+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	C₄	D₄	C₄○D₄	F₅	C₂×F₅	C₂×F₅	C₄×F₅	D₄×F₅
kernel	C₂²⋊C₄×F₅	D₁₀.₃Q₈	D₅×C₂²⋊C₄	C₂×C₄×F₅	C₂×C₂²⋊F₅	C₂³×F₅	D₁₀⋊C₄	C₂³.D₅	C₅×C₂²⋊C₄	C₂²⋊F₅	C₂²×F₅	C₂×F₅	D₁₀	C₂²⋊C₄	C₂×C₄	C₂³	C₂²	C₂
# reps	1	2	1	2	1	1	4	2	2	8	8	4	4	1	2	1	4	2

Matrix representation of C₂²⋊C₄×F₅ ►in GL₆(𝔽₄₁)

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	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

0	1	0	0	0	0
40	0	0	0	0	0
0	0	32	0	0	0
0	0	0	32	0	0
0	0	0	0	32	0
0	0	0	0	0	32

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	1	0	0
0	0	0	0	1	0

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	40	0
0	0	40	0	0	0
0	0	0	0	0	40
0	0	0	40	0	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,0,40,0] >;

C₂²⋊C₄×F₅ in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times F_5

% in TeX

G:=Group("C2^2:C4xF5");

// GroupNames label

G:=SmallGroup(320,1036);

// by ID

G=gap.SmallGroup(320,1036);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,6278,1595]);

// Polycyclic

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

// generators/relations

G = C22⋊C4×F5 order 320 = 26·5

Direct product of C22⋊C4 and F5

G = C₂²⋊C₄×F₅ order 320 = 2⁶·5

Direct product of C₂²⋊C₄ and F₅