D10.D10 - GroupNames

G = D₁₀.D₁₀ order 400 = 2⁴·5²

5^th non-split extension by D₁₀ of D₁₀ acting via D₁₀/C₁₀=C₂

Aliases: C₁₀²⋊₂C₄, D₁₀⋊₂Dic₅, D₁₀.₁₃D₁₀, (C₂×C₁₀)⋊₆F₅, (D₅×C₁₀)⋊₉C₄, (C₅×D₅).₆D₄, C₅⋊(C₂³.D₅), (C₂×C₁₀)⋊₁Dic₅, C₂²⋊(D₅.D₅), C₅⋊₅(C₂²⋊F₅), C₁₀.₄₁(C₂×F₅), D₅.₃(C₅⋊D₄), C₅²⋊₅(C₂²⋊C₄), C₁₀.₇(C₂×Dic₅), (C₂²×D₅).₂D₅, (D₅×C₁₀).₂₂C₂², (D₅×C₂×C₁₀).₅C₂, (C₂×D₅.D₅)⋊₂C₂, C₂.₇(C₂×D₅.D₅), (C₅×C₁₀).₂₆(C₂×C₄), SmallGroup(400,148)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅×C₁₀ — D₁₀.D₁₀

Chief series

C₁ — C₅ — C₅² — C₅×D₅ — D₅×C₁₀ — C₂×D₅.D₅ — D₁₀.D₁₀

Lower central

C₅² — C₅×C₁₀ — D₁₀.D₁₀

Upper central

C₁ — C₂ — C₂²

Generators and relations for D₁₀.D₁₀
G = < a,b,c,d | a¹⁰=b²=c¹⁰=1, d²=a⁴b, bab=a^-1, ac=ca, dad^-1=a⁷, bc=cb, dbd^-1=a⁶b, dcd^-1=a⁵c^-1 >

Subgroups: 424 in 73 conjugacy classes, 25 normal (19 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₅, C₂×C₄, C₂³, D₅, D₅, C₁₀, C₁₀, C₂²⋊C₄, Dic₅, F₅, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₅², C₂×Dic₅, C₂×F₅, C₂²×D₅, C₂²×C₁₀, C₅×D₅, C₅×D₅, C₅×C₁₀, C₅×C₁₀, C₂³.D₅, C₂²⋊F₅, D₅.D₅, D₅×C₁₀, D₅×C₁₀, C₁₀², C₂×D₅.D₅, D₅×C₂×C₁₀, D₁₀.D₁₀
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₅, C₂²⋊C₄, Dic₅, F₅, D₁₀, C₂×Dic₅, C₅⋊D₄, C₂×F₅, C₂³.D₅, C₂²⋊F₅, D₅.D₅, C₂×D₅.D₅, D₁₀.D₁₀

Smallest permutation representation of D₁₀.D₁₀
►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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 20)(8 19)(9 18)(10 17)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 40)(30 39)

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

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

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

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

46 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	5A	5B	5C	···	5G	10A	···	10F	10G	···	10U	10V	···	10AC
order	1	2	2	2	2	2	4	4	4	4	5	5	5	···	5	10	···	10	10	···	10	10	···	10
size	1	1	2	5	5	10	50	50	50	50	2	2	4	···	4	2	···	2	4	···	4	10	···	10

46 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+			+	+	-	+	-		+	+	+
image	C₁	C₂	C₂	C₄	C₄	D₄	D₅	Dic₅	D₁₀	Dic₅	C₅⋊D₄	F₅	C₂×F₅	C₂²⋊F₅	D₅.D₅	C₂×D₅.D₅	D₁₀.D₁₀
kernel	D₁₀.D₁₀	C₂×D₅.D₅	D₅×C₂×C₁₀	D₅×C₁₀	C₁₀²	C₅×D₅	C₂²×D₅	D₁₀	D₁₀	C₂×C₁₀	D₅	C₂×C₁₀	C₁₀	C₅	C₂²	C₂	C₁
# reps	1	2	1	2	2	2	2	2	2	2	8	1	1	2	4	4	8

Matrix representation of D₁₀.D₁₀ ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	10	0	0	0
0	0	4	37	0	0
0	0	31	0	16	0
0	0	14	0	0	18

40	0	0	0	0	0
0	40	0	0	0	0
0	0	37	14	0	0
0	0	37	4	0	0
0	0	18	37	0	18
0	0	35	37	16	0

1	0	0	0	0	0
0	40	0	0	0	0
0	0	16	0	0	0
0	0	0	16	0	0
0	0	24	0	18	0
0	0	24	0	0	18

0	1	0	0	0	0
40	0	0	0	0	0
0	0	23	0	19	0
0	0	0	0	23	18
0	0	24	16	18	0
0	0	24	0	18	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,10,4,31,14,0,0,0,37,0,0,0,0,0,0,16,0,0,0,0,0,0,18],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,37,37,18,35,0,0,14,4,37,37,0,0,0,0,0,16,0,0,0,0,18,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,16,0,24,24,0,0,0,16,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,23,0,24,24,0,0,0,0,16,0,0,0,19,23,18,18,0,0,0,18,0,0] >;

D₁₀.D₁₀ in GAP, Magma, Sage, TeX

D_{10}.D_{10}

% in TeX

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

// GroupNames label

G:=SmallGroup(400,148);

// by ID

G=gap.SmallGroup(400,148);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,1924,8645,2897]);

// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^10=1,d^2=a^4*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=a^5*c^-1>;

// generators/relations

G = D10.D10 order 400 = 24·52

5th non-split extension by D10 of D10 acting via D10/C10=C2

G = D₁₀.D₁₀ order 400 = 2⁴·5²

5^th non-split extension by D₁₀ of D₁₀ acting via D₁₀/C₁₀=C₂