C4^2:6F5 - GroupNames

G = C₄²:₆F₅ order 320 = 2⁶·5

3^rd semidirect product of C₄² and F₅ acting via F₅/D₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄²:₆F₅, C₂₀.₈C₄², C₄:F₅:₄C₄, (C₄xC₂₀):₇C₄, C₄.F₅:₄C₄, C₄.₁(C₄xF₅), D₅.₁C₄wrC₂, (C₄xD₅).₈₆D₄, C₄.₂₅(C₄:F₅), C₂₀.₂₅(C₄:C₄), (C₄xD₅).₂₈Q₈, C₅:₁(C₄²:₆C₄), (C₄xDic₅):₂₄C₄, D₁₀.₁₉(C₄:C₄), D₅:M₄(2).₇C₂, (D₅xC₄²).₂₃C₂, Dic₅.₂₁(C₄:C₄), (C₂xDic₅).₂₅₃D₄, (C₂²xD₅).₁₄₀D₄, D₁₀.₂₈(C₂²:C₄), C₂.₃(D₁₀.₃Q₈), C₂².₁₂(C₂²:F₅), C₁₀.₁(C₂.C₄²), Dic₅.₂₈(C₂²:C₄), D₁₀.C₂³.₇C₂, (C₄xD₅).₆₁(C₂xC₄), (C₂xC₄).₁₁₅(C₂xF₅), (C₂xC₂₀).₁₃₆(C₂xC₄), (C₂xC₄xD₅).₄₀₁C₂², (C₂xC₁₀).₁₂(C₂²:C₄), SmallGroup(320,200)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — C₄²:₆F₅

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₂²xD₅ — C₂xC₄xD₅ — D₁₀.C₂³ — C₄²:₆F₅

Lower central

C₅ — C₁₀ — C₂₀ — C₄²:₆F₅

Upper central

C₁ — C₄ — C₂xC₄ — C₄²

Generators and relations for C₄²:₆F₅
G = < a,b,c,d | a⁴=b⁴=c⁵=d⁴=1, dad^-1=ab=ba, ac=ca, bc=cb, dbd^-1=b^-1, dcd^-1=c³ >

Subgroups: 450 in 110 conjugacy classes, 36 normal (30 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂xC₄, C₂xC₄, C₂³, D₅, D₅, C₁₀, C₁₀, C₄², C₄², C₂²:C₄, C₄:C₄, C₂xC₈, M₄(2), C₂²xC₄, Dic₅, Dic₅, C₂₀, C₂₀, F₅, D₁₀, D₁₀, C₂xC₁₀, C₂xC₄², C₄²:C₂, C₂xM₄(2), C₅:C₈, C₄xD₅, C₄xD₅, C₂xDic₅, C₂xDic₅, C₂xC₂₀, C₂xC₂₀, C₂xF₅, C₂²xD₅, C₄²:₆C₄, C₄xDic₅, C₄xDic₅, C₄xC₂₀, D₅:C₈, C₄.F₅, C₄xF₅, C₄:F₅, C₂².F₅, C₂²:F₅, C₂xC₄xD₅, C₂xC₄xD₅, D₅xC₄², D₅:M₄(2), D₁₀.C₂³, C₄²:₆F₅
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, Q₈, C₄², C₂²:C₄, C₄:C₄, F₅, C₂.C₄², C₄wrC₂, C₂xF₅, C₄²:₆C₄, C₄xF₅, C₄:F₅, C₂²:F₅, D₁₀.₃Q₈, C₄²:₆F₅

Smallest permutation representation of C₄²:₆F₅
►On 40 points

Generators in S₄₀

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	···	4G	4H	4I	4J	···	4N	4O	4P	4Q	4R	5	8A	8B	8C	8D	10A	10B	10C	20A	···	20L
order	1	2	2	2	2	2	4	4	4	···	4	4	4	4	···	4	4	4	4	4	5	8	8	8	8	10	10	10	20	···	20
size	1	1	2	5	5	10	1	1	2	···	2	5	5	10	···	10	20	20	20	20	4	20	20	20	20	4	4	4	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+					+	-	+	+		+	+			+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	Q₈	D₄	D₄	C₄wrC₂	F₅	C₂xF₅	C₄xF₅	C₄:F₅	C₂²:F₅	C₄²:₆F₅
kernel	C₄²:₆F₅	D₅xC₄²	D₅:M₄(2)	D₁₀.C₂³	C₄xDic₅	C₄xC₂₀	C₄.F₅	C₄:F₅	C₄xD₅	C₄xD₅	C₂xDic₅	C₂²xD₅	D₅	C₄²	C₂xC₄	C₄	C₄	C₂²	C₁
# reps	1	1	1	1	2	2	4	4	1	1	1	1	8	1	1	2	2	2	8

Matrix representation of C₄²:₆F₅ ►in GL₆(F₄₁)

40	0	0	0	0	0
0	9	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	40	0
0	0	0	0	0	40

32	0	0	0	0	0
0	9	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	40	0
0	0	0	0	0	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	34	40	0	0
0	0	1	0	0	0
0	0	0	0	7	7
0	0	0	0	34	40

0	9	0	0	0	0
9	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	34	40	0	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,7,34,0,0,0,0,7,40],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₄²:₆F₅ in GAP, Magma, Sage, TeX

C_4^2\rtimes_6F_5

% in TeX

G:=Group("C4^2:6F5");

// GroupNames label

G:=SmallGroup(320,200);

// by ID

G=gap.SmallGroup(320,200);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,1123,1684,102,6278,3156]);

// Polycyclic

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

// generators/relations

G = C42:6F5 order 320 = 26·5

3rd semidirect product of C42 and F5 acting via F5/D5=C2

G = C₄²:₆F₅ order 320 = 2⁶·5

3^rd semidirect product of C₄² and F₅ acting via F₅/D₅=C₂