C4oD4xF5 - GroupNames

G = C₄○D₄×F₅ order 320 = 2⁶·5

Direct product of C₄○D₄ and F₅

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

Aliases: C₄○D₄×F₅, D₁₀.₁₈C₂⁴, (D₄×F₅)⋊₅C₂, D₄⋊₈(C₂×F₅), Q₈⋊₇(C₂×F₅), (Q₈×F₅)⋊₅C₂, C₄○D₂₀⋊₅C₄, D₂₀⋊₉(C₂×C₄), D₄⋊₂D₅⋊₁₀C₄, Q₈⋊₂D₅⋊₁₀C₄, Dic₁₀⋊₈(C₂×C₄), C₄⋊F₅.₁₃C₂², (C₂×F₅).₇C₂³, C₄.₃₃(C₂²×F₅), C₂.₁₈(C₂³×F₅), C₂₀.₃₄(C₂²×C₄), C₁₀.₁₇(C₂³×C₄), (D₄×D₅).₁₈C₂², (C₄×D₅).₅₇C₂³, D₁₀.₈(C₂²×C₄), (C₄×F₅).₂₁C₂², (Q₈×D₅).₁₆C₂², C₂²⋊F₅.₄C₂², C₂².₅(C₂²×F₅), Dic₅.₄₉(C₂²×C₄), (C₂²×F₅).₂₀C₂², D₁₀.C₂³⋊₉C₂, (C₂²×D₅).₁₅₃C₂³, C₅⋊(C₄×C₄○D₄), (C₂×C₄×F₅)⋊₆C₂, (C₂×C₄)⋊₉(C₂×F₅), (C₂×C₂₀)⋊₆(C₂×C₄), (C₅×C₄○D₄)⋊₅C₄, (C₅×D₄)⋊₉(C₂×C₄), (C₄×D₅)⋊₉(C₂×C₄), C₅⋊D₄⋊₃(C₂×C₄), (C₅×Q₈)⋊₈(C₂×C₄), D₅.₄(C₂×C₄○D₄), (D₅×C₄○D₄).₁₁C₂, (C₂×Dic₅)⋊₁₉(C₂×C₄), (C₂×C₁₀).₆(C₂²×C₄), (C₂×C₄×D₅).₂₂₄C₂², SmallGroup(320,1603)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₄○D₄×F₅

Chief series

C₁ — C₅ — D₅ — D₁₀ — C₂×F₅ — C₂²×F₅ — C₂×C₄×F₅ — C₄○D₄×F₅

Lower central

C₅ — C₁₀ — C₄○D₄×F₅

Upper central

C₁ — C₄ — C₄○D₄

Generators and relations for C₄○D₄×F₅
G = < a,b,c,d,e | a⁴=c²=d⁵=e⁴=1, b²=a², ab=ba, ac=ca, ad=da, ae=ea, cbc=a²b, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d³ >

Subgroups: 1018 in 310 conjugacy classes, 142 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, D₅, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, Dic₅, Dic₅, C₂₀, C₂₀, F₅, F₅, D₁₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₄², C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂×C₄○D₄, Dic₁₀, C₄×D₅, C₄×D₅, D₂₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₅×Q₈, C₂×F₅, C₂×F₅, C₂²×D₅, C₄×C₄○D₄, C₄×F₅, C₄×F₅, C₄⋊F₅, C₂²⋊F₅, C₂×C₄×D₅, C₄○D₂₀, D₄×D₅, D₄⋊₂D₅, Q₈×D₅, Q₈⋊₂D₅, C₅×C₄○D₄, C₂²×F₅, C₂×C₄×F₅, D₁₀.C₂³, D₄×F₅, Q₈×F₅, D₅×C₄○D₄, C₄○D₄×F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₄○D₄, C₂⁴, F₅, C₂³×C₄, C₂×C₄○D₄, C₂×F₅, C₄×C₄○D₄, C₂²×F₅, C₂³×F₅, C₄○D₄×F₅

Smallest permutation representation of C₄○D₄×F₅
►On 40 points

Generators in S₄₀

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

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

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

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

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

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

50 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	···	4O	4P	···	4AD	5	10A	10B	10C	10D	20A	20B	20C	20D	20E
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	···	4	4	···	4	5	10	10	10	10	20	20	20	20	20
size	1	1	2	2	2	5	5	10	10	10	1	1	2	2	2	5	···	5	10	···	10	4	4	8	8	8	4	4	8	8	8

50 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	4	4	4	4	8
type	+	+	+	+	+	+						+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	C₄○D₄	F₅	C₂×F₅	C₂×F₅	C₂×F₅	C₄○D₄×F₅
kernel	C₄○D₄×F₅	C₂×C₄×F₅	D₁₀.C₂³	D₄×F₅	Q₈×F₅	D₅×C₄○D₄	C₄○D₂₀	D₄⋊₂D₅	Q₈⋊₂D₅	C₅×C₄○D₄	F₅	C₄○D₄	C₂×C₄	D₄	Q₈	C₁
# reps	1	3	3	6	2	1	6	6	2	2	8	1	3	3	1	2

Matrix representation of C₄○D₄×F₅ ►in GL₆(𝔽₄₁)

32	0	0	0	0	0
0	32	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

32	0	0	0	0	0
21	9	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	5	0	0	0	0
0	1	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	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	1	0	0	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	40	40	40	40

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,21,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,5,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

C₄○D₄×F₅ in GAP, Magma, Sage, TeX

C_4\circ D_4\times F_5

% in TeX

G:=Group("C4oD4xF5");

// GroupNames label

G:=SmallGroup(320,1603);

// by ID

G=gap.SmallGroup(320,1603);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,570,6278,818]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^2=d^5=e^4=1,b^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^2*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 = C4○D4×F5 order 320 = 26·5

Direct product of C4○D4 and F5

G = C₄○D₄×F₅ order 320 = 2⁶·5

Direct product of C₄○D₄ and F₅