D10:9M4(2) - GroupNames

G = D₁₀⋊₉M₄(2) order 320 = 2⁶·5

3^rd semidirect product of D₁₀ and M₄(2) acting via M₄(2)/C₂×C₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₀⋊₉M₄(2), D₁₀⋊C₈⋊₅C₂, (C₂²×C₄).₁₆F₅, C₂³.₄₂(C₂×F₅), (C₂²×C₂₀).₂₀C₄, C₅⋊₁(C₂⁴.₄C₄), (C₂³×D₅).₁₅C₄, Dic₅.₁₀₅(C₂×D₄), (C₂×Dic₅).₂₅₈D₄, C₁₀.₂₁(C₂×M₄(2)), C₂².₈₄(C₂²×F₅), C₂.₁₅(D₅⋊M₄(2)), C₂².₂₄(C₂²⋊F₅), Dic₅.₄₁(C₂²⋊C₄), (C₂×Dic₅).₃₄₆C₂³, (C₂²×Dic₅).₂₇₄C₂², (C₂×C₅⋊C₈)⋊₁C₂², (C₂×C₄×D₅).₃₄C₄, C₂.₈(C₂×C₂²⋊F₅), C₁₀.₅(C₂×C₂²⋊C₄), (C₂×C₄).₁₀₉(C₂×F₅), (D₅×C₂²×C₄).₂₂C₂, (C₂×C₂².F₅)⋊₃C₂, (C₂×C₂₀).₁₀₈(C₂×C₄), (C₂×C₄×D₅).₃₆₄C₂², (C₂²×C₁₀).₆₂(C₂×C₄), (C₂×C₁₀).₆₂(C₂²×C₄), (C₂×C₁₀).₄₉(C₂²⋊C₄), (C₂×Dic₅).₁₈₄(C₂×C₄), (C₂²×D₅).₁₂₅(C₂×C₄), SmallGroup(320,1093)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — D₁₀⋊₉M₄(2)

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₂×Dic₅ — C₂×C₅⋊C₈ — D₁₀⋊C₈ — D₁₀⋊₉M₄(2)

Lower central

C₅ — C₂×C₁₀ — D₁₀⋊₉M₄(2)

Upper central

C₁ — C₂² — C₂²×C₄

Generators and relations for D₁₀⋊₉M₄(2)
G = < a,b,c,d | a¹⁰=b²=c⁸=d²=1, bab=a^-1, cac^-1=a³, ad=da, cbc^-1=a⁷b, bd=db, dcd=c⁵ >

Subgroups: 762 in 190 conjugacy classes, 56 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, D₅, C₁₀, C₁₀, C₁₀, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂⁴, Dic₅, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂²⋊C₈, C₂×M₄(2), C₂³×C₄, C₅⋊C₈, C₄×D₅, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂⁴.₄C₄, C₂×C₅⋊C₈, C₂².F₅, C₂×C₄×D₅, C₂×C₄×D₅, C₂²×Dic₅, C₂²×C₂₀, C₂³×D₅, D₁₀⋊C₈, C₂×C₂².F₅, D₅×C₂²×C₄, D₁₀⋊₉M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, M₄(2), C₂²×C₄, C₂×D₄, F₅, C₂×C₂²⋊C₄, C₂×M₄(2), C₂×F₅, C₂⁴.₄C₄, C₂²⋊F₅, C₂²×F₅, D₅⋊M₄(2), C₂×C₂²⋊F₅, D₁₀⋊₉M₄(2)

Smallest permutation representation of D₁₀⋊₉M₄(2)
►On 80 points

Generators in S₈₀

(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)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)

(1 10)(2 9)(3 8)(4 7)(5 6)(12 20)(13 19)(14 18)(15 17)(21 26)(22 25)(23 24)(27 30)(28 29)(31 32)(33 40)(34 39)(35 38)(36 37)(41 50)(42 49)(43 48)(44 47)(45 46)(52 60)(53 59)(54 58)(55 57)(61 63)(64 70)(65 69)(66 68)(71 75)(72 74)(76 80)(77 79)

(1 76 41 70 24 19 32 54)(2 73 50 63 25 16 31 57)(3 80 49 66 26 13 40 60)(4 77 48 69 27 20 39 53)(5 74 47 62 28 17 38 56)(6 71 46 65 29 14 37 59)(7 78 45 68 30 11 36 52)(8 75 44 61 21 18 35 55)(9 72 43 64 22 15 34 58)(10 79 42 67 23 12 33 51)

(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 21)(9 22)(10 23)(31 50)(32 41)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 49)

G:=sub<Sym(80)| (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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,10)(2,9)(3,8)(4,7)(5,6)(12,20)(13,19)(14,18)(15,17)(21,26)(22,25)(23,24)(27,30)(28,29)(31,32)(33,40)(34,39)(35,38)(36,37)(41,50)(42,49)(43,48)(44,47)(45,46)(52,60)(53,59)(54,58)(55,57)(61,63)(64,70)(65,69)(66,68)(71,75)(72,74)(76,80)(77,79), (1,76,41,70,24,19,32,54)(2,73,50,63,25,16,31,57)(3,80,49,66,26,13,40,60)(4,77,48,69,27,20,39,53)(5,74,47,62,28,17,38,56)(6,71,46,65,29,14,37,59)(7,78,45,68,30,11,36,52)(8,75,44,61,21,18,35,55)(9,72,43,64,22,15,34,58)(10,79,42,67,23,12,33,51), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,21)(9,22)(10,23)(31,50)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)>;

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)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,10)(2,9)(3,8)(4,7)(5,6)(12,20)(13,19)(14,18)(15,17)(21,26)(22,25)(23,24)(27,30)(28,29)(31,32)(33,40)(34,39)(35,38)(36,37)(41,50)(42,49)(43,48)(44,47)(45,46)(52,60)(53,59)(54,58)(55,57)(61,63)(64,70)(65,69)(66,68)(71,75)(72,74)(76,80)(77,79), (1,76,41,70,24,19,32,54)(2,73,50,63,25,16,31,57)(3,80,49,66,26,13,40,60)(4,77,48,69,27,20,39,53)(5,74,47,62,28,17,38,56)(6,71,46,65,29,14,37,59)(7,78,45,68,30,11,36,52)(8,75,44,61,21,18,35,55)(9,72,43,64,22,15,34,58)(10,79,42,67,23,12,33,51), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,21)(9,22)(10,23)(31,50)(32,41)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49) );

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),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80)], [(1,10),(2,9),(3,8),(4,7),(5,6),(12,20),(13,19),(14,18),(15,17),(21,26),(22,25),(23,24),(27,30),(28,29),(31,32),(33,40),(34,39),(35,38),(36,37),(41,50),(42,49),(43,48),(44,47),(45,46),(52,60),(53,59),(54,58),(55,57),(61,63),(64,70),(65,69),(66,68),(71,75),(72,74),(76,80),(77,79)], [(1,76,41,70,24,19,32,54),(2,73,50,63,25,16,31,57),(3,80,49,66,26,13,40,60),(4,77,48,69,27,20,39,53),(5,74,47,62,28,17,38,56),(6,71,46,65,29,14,37,59),(7,78,45,68,30,11,36,52),(8,75,44,61,21,18,35,55),(9,72,43,64,22,15,34,58),(10,79,42,67,23,12,33,51)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,21),(9,22),(10,23),(31,50),(32,41),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,49)]])

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	1	2	2	4	4	4	4	4
type	+	+	+	+				+		+	+	+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	D₄	M₄(2)	F₅	C₂×F₅	C₂×F₅	C₂²⋊F₅	D₅⋊M₄(2)
kernel	D₁₀⋊₉M₄(2)	D₁₀⋊C₈	C₂×C₂².F₅	D₅×C₂²×C₄	C₂×C₄×D₅	C₂²×C₂₀	C₂³×D₅	C₂×Dic₅	D₁₀	C₂²×C₄	C₂×C₄	C₂³	C₂²	C₂
# reps	1	4	2	1	4	2	2	4	8	1	2	1	4	8

Matrix representation of D₁₀⋊₉M₄(2) ►in GL₆(𝔽₄₁)

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

40	0	0	0	0	0
33	1	0	0	0	0
0	0	0	0	0	40
0	0	0	0	40	0
0	0	0	40	0	0
0	0	40	0	0	0

33	2	0	0	0	0
34	8	0	0	0	0
0	0	6	12	11	1
0	0	40	30	29	35
0	0	40	5	11	10
0	0	6	5	36	35

1	0	0	0	0	0
8	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

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,40,0,0,0,0,1,0,40,0,0,0,1,0,0,0,0,40,1,0,0],[40,33,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0],[33,34,0,0,0,0,2,8,0,0,0,0,0,0,6,40,40,6,0,0,12,30,5,5,0,0,11,29,11,36,0,0,1,35,10,35],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D₁₀⋊₉M₄(2) in GAP, Magma, Sage, TeX

D_{10}\rtimes_9M_4(2)

% in TeX

G:=Group("D10:9M4(2)");

// GroupNames label

G:=SmallGroup(320,1093);

// by ID

G=gap.SmallGroup(320,1093);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,232,758,136,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d=c^5>;

// generators/relations

G = D10⋊9M4(2) order 320 = 26·5

3rd semidirect product of D10 and M4(2) acting via M4(2)/C2×C4=C2

G = D₁₀⋊₉M₄(2) order 320 = 2⁶·5

3^rd semidirect product of D₁₀ and M₄(2) acting via M₄(2)/C₂×C₄=C₂