D10:(C4:C4) - GroupNames

G = D₁₀⋊(C₄⋊C₄) order 320 = 2⁶·5

The semidirect product of D₁₀ and C₄⋊C₄ acting via C₄⋊C₄/C₂²=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₀⋊(C₄⋊C₄), C₂.₉(D₄×F₅), C₁₀.₆(C₄×D₄), C₂²⋊C₄⋊₄F₅, (C₂×F₅).₃D₄, C₂²⋊₂(C₄⋊F₅), C₂³.D₅⋊₉C₄, D₁₀⋊C₄⋊₁C₄, C₅⋊(C₂³.₈Q₈), D₁₀.₂₇(C₂×D₄), D₅.₁C₂²≀C₂, D₁₀.₁₀(C₂×Q₈), (C₂³×F₅).₂C₂, C₂³.₂₈(C₂×F₅), D₁₀.₃Q₈⋊₆C₂, (C₂²×D₅).₆₂D₄, D₅.₁(C₂²⋊Q₈), D₁₀.₄₃(C₄○D₄), (C₂²×D₅).₁₀Q₈, (C₂²×F₅).₄C₂², C₂².₇₄(C₂²×F₅), (C₂³×D₅).₈₄C₂², D₅.₁(C₂².D₄), (C₂²×D₅).₂₆₉C₂³, (C₂×C₁₀)⋊(C₄⋊C₄), (C₂×C₄⋊F₅)⋊₆C₂, (C₂×C₄)⋊₂(C₂×F₅), (C₂×C₂₀)⋊₈(C₂×C₄), C₁₀.₈(C₂×C₄⋊C₄), C₂.₁₁(C₂×C₄⋊F₅), (C₅×C₂²⋊C₄)⋊₄C₄, (D₅×C₂²⋊C₄).₅C₂, (C₂×C₂²⋊F₅).₄C₂, (C₂×Dic₅)⋊₁₅(C₂×C₄), (C₂×C₄×D₅).₂₇₅C₂², (C₂²×C₁₀).₂₀(C₂×C₄), (C₂×C₁₀).₃₇(C₂²×C₄), (C₂²×D₅).₄₄(C₂×C₄), SmallGroup(320,1037)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — D₁₀⋊(C₄⋊C₄)

Chief series

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

Lower central

C₅ — C₂×C₁₀ — D₁₀⋊(C₄⋊C₄)

Upper central

C₁ — C₂² — C₂²⋊C₄

Generators and relations for D₁₀⋊(C₄⋊C₄)
G = < a,b,c,d | a¹⁰=b²=c⁴=d⁴=1, bab=cac^-1=a^-1, dad^-1=a³, cbc^-1=a³b, dbd^-1=a²b, dcd^-1=c^-1 >

Subgroups: 1018 in 234 conjugacy classes, 64 normal (24 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, C₂³, C₂³, D₅, D₅, D₅, C₁₀, C₁₀, C₁₀, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂⁴, Dic₅, C₂₀, F₅, D₁₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³×C₄, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂×F₅, C₂×F₅, C₂²×D₅, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂³.₈Q₈, 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, D₁₀⋊(C₄⋊C₄)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, F₅, C₂×C₄⋊C₄, C₄×D₄, C₂²≀C₂, C₂²⋊Q₈, C₂².D₄, C₂×F₅, C₂³.₈Q₈, C₄⋊F₅, C₂²×F₅, C₂×C₄⋊F₅, D₄×F₅, D₁₀⋊(C₄⋊C₄)

Smallest permutation representation of D₁₀⋊(C₄⋊C₄)
►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)

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

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

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

Matrix representation of D₁₀⋊(C₄⋊C₄) ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	40	1
0	0	0	0	40	0
0	0	1	0	40	0
0	0	0	1	40	0

40	0	0	0	0	0
0	1	0	0	0	0
0	0	0	40	1	0
0	0	40	0	1	0
0	0	0	0	1	0
0	0	0	0	1	40

0	40	0	0	0	0
40	0	0	0	0	0
0	0	27	0	14	34
0	0	27	14	7	0
0	0	0	7	14	27
0	0	34	14	0	27

9	0	0	0	0	0
0	9	0	0	0	0
0	0	0	0	1	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	1	0	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40,0,0,1,0,0,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,1,1,1,1,0,0,0,0,0,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,27,27,0,34,0,0,0,14,7,14,0,0,14,7,14,0,0,0,34,0,27,27],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

D₁₀⋊(C₄⋊C₄) in GAP, Magma, Sage, TeX

D_{10}\rtimes (C_4\rtimes C_4)

% in TeX

G:=Group("D10:(C4:C4)");

// GroupNames label

G:=SmallGroup(320,1037);

// by ID

G=gap.SmallGroup(320,1037);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,100,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^4=d^4=1,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^3,c*b*c^-1=a^3*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;

// generators/relations

G = D10⋊(C4⋊C4) order 320 = 26·5

The semidirect product of D10 and C4⋊C4 acting via C4⋊C4/C22=C22

G = D₁₀⋊(C₄⋊C₄) order 320 = 2⁶·5

The semidirect product of D₁₀ and C₄⋊C₄ acting via C₄⋊C₄/C₂²=C₂²