ES-(2,2):2D5 - GroupNames

G = 2_-¹⁺⁴⋊₂D₅ order 320 = 2⁶·5

1^st semidirect product of 2_-¹⁺⁴ and D₅ acting via D₅/C₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2_-¹⁺⁴⋊₂D₅, (C₅×D₄).₃₄D₄, (C₂×C₂₀).₂₂D₄, (C₅×Q₈).₃₄D₄, D₄⋊D₁₀⋊₆C₂, C₄○D₄.₁₁D₁₀, C₂₀.₂₁₉(C₂×D₄), C₅⋊₅(D₄.₈D₄), (C₂×Q₈).₇₂D₁₀, C₁₀.₈₄C₂²≀C₂, C₂₀.₂₃D₄⋊₈C₂, D₄.₁₆(C₅⋊D₄), (C₂×C₂₀).₂₃C₂³, Q₈.₁₆(C₅⋊D₄), D₄⋊₂Dic₅⋊₁₃C₂, C₂₀.₁₀D₄⋊₁₁C₂, (C₅×2_-¹⁺⁴)⋊₁C₂, (Q₈×C₁₀).₉₉C₂², (C₂×D₂₀).₁₃₈C₂², C₂.₁₈(C₂⁴⋊₂D₅), (C₄×Dic₅).₆₆C₂², C₄.Dic₅.₃₀C₂², C₄.₆₆(C₂×C₅⋊D₄), (C₂×C₁₀).₄₆(C₂×D₄), (C₂×C₄).₁₃(C₅⋊D₄), (C₂×C₄).₂₃(C₂²×D₅), C₂².₁₈(C₂×C₅⋊D₄), (C₅×C₄○D₄).₂₁C₂², SmallGroup(320,872)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — 2_-¹⁺⁴⋊₂D₅

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂₀ — C₂×D₂₀ — D₄⋊D₁₀ — 2_-¹⁺⁴⋊₂D₅

Lower central

C₅ — C₁₀ — C₂×C₂₀ — 2_-¹⁺⁴⋊₂D₅

Upper central

C₁ — C₂ — C₂×C₄ — 2_-¹⁺⁴

Generators and relations for 2_-¹⁺⁴⋊₂D₅
G = < a,b,c,d,e,f | a⁴=b²=e⁵=f²=1, c²=d²=a², bab=faf=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=ab, dcd^-1=fcf=a²c, ce=ec, de=ed, fdf=a²cd, fef=e^-1 >

Subgroups: 510 in 146 conjugacy classes, 43 normal (17 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, M₄(2), D₈, SD₁₆, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, Dic₅, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₄.₁₀D₄, C₄≀C₂, C₄.₄D₄, C₈⋊C₂², 2_-¹⁺⁴, C₅⋊₂C₈, D₂₀, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×D₄, C₅×Q₈, C₅×Q₈, C₂²×D₅, D₄.₈D₄, C₄.Dic₅, C₄×Dic₅, D₁₀⋊C₄, D₄⋊D₅, Q₈⋊D₅, C₂×D₂₀, Q₈×C₁₀, Q₈×C₁₀, C₅×C₄○D₄, C₅×C₄○D₄, C₂₀.₁₀D₄, D₄⋊₂Dic₅, C₂₀.₂₃D₄, D₄⋊D₁₀, C₅×2_-¹⁺⁴, 2_-¹⁺⁴⋊₂D₅
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, D₁₀, C₂²≀C₂, C₅⋊D₄, C₂²×D₅, D₄.₈D₄, C₂×C₅⋊D₄, C₂⁴⋊₂D₅, 2_-¹⁺⁴⋊₂D₅

Smallest permutation representation of 2_-¹⁺⁴⋊₂D₅
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

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

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

50 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	5A	5B	8A	8B	10A	10B	10C	···	10L	20A	···	20T
order	1	2	2	2	2	2	4	4	4	4	4	4	4	4	5	5	8	8	10	10	10	···	10	20	···	20
size	1	1	2	4	4	40	2	2	4	4	4	4	20	20	2	2	40	40	2	2	4	···	4	4	···	4

50 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

D₁₀

C₅⋊D₄

D₄.₈D₄

2_-¹⁺⁴⋊₂D₅

kernel

2_-¹⁺⁴⋊₂D₅

C₂₀.₁₀D₄

D₄⋊₂Dic₅

C₂₀.₂₃D₄

D₄⋊D₁₀

C₅×2_-¹⁺⁴

C₂×C₂₀

C₅×D₄

C₅×Q₈

2_-¹⁺⁴

C₂×Q₈

C₄○D₄

C₂×C₄

D₄

Q₈

C₅

C₁

# reps

Matrix representation of 2_-¹⁺⁴⋊₂D₅ ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	1	22	10
0	0	40	0	1	9
0	0	0	0	40	1
0	0	0	0	39	1

0	40	0	0	0	0
40	0	0	0	0	0
0	0	19	0	0	25
0	0	40	0	1	9
0	0	1	1	0	32
0	0	2	0	0	22

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	19	32
0	0	40	0	40	10
0	0	0	0	1	40
0	0	0	0	2	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	7	0	0	20
0	0	32	0	32	8
0	0	9	32	0	33
0	0	18	0	0	34

3	7	0	0	0	0
7	3	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

38	7	0	0	0	0
34	3	0	0	0	0
0	0	1	0	22	0
0	0	0	40	1	40
0	0	0	0	40	0
0	0	0	0	39	1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,22,1,40,39,0,0,10,9,1,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,19,40,1,2,0,0,0,0,1,0,0,0,0,1,0,0,0,0,25,9,32,22],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,19,40,1,2,0,0,32,10,40,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,32,9,18,0,0,0,0,32,0,0,0,0,32,0,0,0,0,20,8,33,34],[3,7,0,0,0,0,7,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[38,34,0,0,0,0,7,3,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,22,1,40,39,0,0,0,40,0,1] >;

2_-¹⁺⁴⋊₂D₅ in GAP, Magma, Sage, TeX

2_-^{1+4}\rtimes_2D_5

% in TeX

G:=Group("ES-(2,2):2D5");

// GroupNames label

G:=SmallGroup(320,872);

// by ID

G=gap.SmallGroup(320,872);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,184,570,1684,851,438,102,12550]);

// Polycyclic

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

// generators/relations

G = 2- 1+4⋊2D5 order 320 = 26·5

1st semidirect product of 2- 1+4 and D5 acting via D5/C5=C2

G = 2_-¹⁺⁴⋊₂D₅ order 320 = 2⁶·5

1^st semidirect product of 2_-¹⁺⁴ and D₅ acting via D₅/C₅=C₂