D5xES-(2,2) - GroupNames

G = D₅×2_-¹⁺⁴ order 320 = 2⁶·5

Direct product of D₅ and 2_-¹⁺⁴

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

Aliases: D₅×2_-¹⁺⁴, C₁₀.₁₇C₂⁵, C₂₀.₅₂C₂⁴, D₂₀.₃₈C₂³, D₁₀.₂₆C₂⁴, Dic₅.₁₂C₂⁴, Dic₁₀.₃₉C₂³, C₄○D₄⋊₁₂D₁₀, (C₂×Q₈)⋊₂₄D₁₀, (C₂×C₁₀).₈C₂⁴, (Q₈×D₅)⋊₁₇C₂², C₄.₄₉(C₂³×D₅), C₂.₁₈(D₅×C₂⁴), C₅⋊D₄.₄C₂³, C₅⋊₃(C₂×2_-¹⁺⁴), C₄○D₂₀⋊₁₄C₂², (Q₈×C₁₀)⋊₂₄C₂², D₄.₃₂(C₂²×D₅), (C₅×D₄).₃₂C₂³, (D₄×D₅).₂₀C₂², (C₄×D₅).₂₁C₂³, Q₈.₃₃(C₂²×D₅), (C₅×Q₈).₃₃C₂³, D₄⋊₂D₅⋊₁₇C₂², C₂².₅(C₂³×D₅), (C₂×C₂₀).₁₂₃C₂³, Q₈.₁₀D₁₀⋊₇C₂, Q₈⋊₂D₅⋊₁₆C₂², (C₅×2_-¹⁺⁴)⋊₄C₂, D₄.₁₀D₁₀⋊₁₂C₂, (C₂×Dic₁₀)⋊₄₅C₂², (C₂×Dic₅).₁₇₀C₂³, (C₂²×D₅).₂₆₁C₂³, (C₂×Q₈×D₅)⋊₂₂C₂, (D₅×C₄○D₄)⋊₉C₂, (C₅×C₄○D₄)⋊₁₂C₂², (C₂×C₄×D₅).₁₈₃C₂², (C₂×C₄).₁₀₇(C₂²×D₅), SmallGroup(320,1624)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — D₅×2_-¹⁺⁴

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₂²×D₅ — C₂×C₄×D₅ — C₂×Q₈×D₅ — D₅×2_-¹⁺⁴

Lower central

C₅ — C₁₀ — D₅×2_-¹⁺⁴

Upper central

C₁ — C₂ — 2_-¹⁺⁴

Generators and relations for D₅×2_-¹⁺⁴
G = < a,b,c,d,e,f | a⁵=b²=c⁴=d²=1, e²=f²=c², bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef^-1=c²e >

Subgroups: 2190 in 794 conjugacy classes, 445 normal (8 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, D₅, D₅, C₁₀, C₁₀, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, Dic₅, C₂₀, D₁₀, D₁₀, D₁₀, C₂×C₁₀, C₂²×Q₈, C₂×C₄○D₄, 2_-¹⁺⁴, 2_-¹⁺⁴, Dic₁₀, C₄×D₅, D₂₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₅×Q₈, C₂²×D₅, C₂×2_-¹⁺⁴, C₂×Dic₁₀, C₂×C₄×D₅, C₄○D₂₀, D₄×D₅, D₄⋊₂D₅, Q₈×D₅, Q₈⋊₂D₅, Q₈×C₁₀, C₅×C₄○D₄, C₂×Q₈×D₅, Q₈.₁₀D₁₀, D₅×C₄○D₄, D₄.₁₀D₁₀, C₅×2_-¹⁺⁴, D₅×2_-¹⁺⁴
Quotients: C₁, C₂, C₂², C₂³, D₅, C₂⁴, D₁₀, 2_-¹⁺⁴, C₂⁵, C₂²×D₅, C₂×2_-¹⁺⁴, C₂³×D₅, D₅×C₂⁴, D₅×2_-¹⁺⁴

Smallest permutation representation of D₅×2_-¹⁺⁴
►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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(62 65)(63 64)(67 70)(68 69)(72 75)(73 74)(77 80)(78 79)

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

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

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

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

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

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

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

68 conjugacy classes

class	1	2A	2B	···	2F	2G	2H	2I	···	2M	4A	···	4J	4K	···	4T	5A	5B	10A	10B	10C	···	10L	20A	···	20T
order	1	2	2	···	2	2	2	2	···	2	4	···	4	4	···	4	5	5	10	10	10	···	10	20	···	20
size	1	1	2	···	2	5	5	10	···	10	2	···	2	10	···	10	2	2	2	2	4	···	4	4	···	4

68 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4	8
type	+	+	+	+	+	+	+	+	+	-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	D₅	D₁₀	D₁₀	2_-¹⁺⁴	D₅×2_-¹⁺⁴
kernel	D₅×2_-¹⁺⁴	C₂×Q₈×D₅	Q₈.₁₀D₁₀	D₅×C₄○D₄	D₄.₁₀D₁₀	C₅×2_-¹⁺⁴	2_-¹⁺⁴	C₂×Q₈	C₄○D₄	D₅	C₁
# reps	1	5	5	10	10	1	2	10	20	2	2

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

40	1	0	0	0	0
5	35	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

1	0	0	0	0	0
36	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

40	0	0	0	0	0
0	40	0	0	0	0
0	0	32	9	9	39
0	0	0	0	9	0
0	0	0	9	0	0
0	0	0	0	0	9

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	40	0
0	0	40	1	1	18
0	0	40	0	0	0
0	0	32	0	9	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	9	32	32	2
0	0	0	0	9	0
0	0	0	9	0	0
0	0	0	40	40	32

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	9	0
0	0	32	9	9	39
0	0	9	0	0	0
0	0	0	0	0	32

G:=sub<GL(6,GF(41))| [40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,36,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,9,0,9,0,0,0,9,9,0,0,0,0,39,0,0,9],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,40,32,0,0,0,1,0,0,0,0,40,1,0,9,0,0,0,18,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,32,0,9,40,0,0,32,9,0,40,0,0,2,0,0,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,9,0,0,0,0,9,0,0,0,0,9,9,0,0,0,0,0,39,0,32] >;

D₅×2_-¹⁺⁴ in GAP, Magma, Sage, TeX

D_5\times 2_-^{1+4}

% in TeX

G:=Group("D5xES-(2,2)");

// GroupNames label

G:=SmallGroup(320,1624);

// by ID

G=gap.SmallGroup(320,1624);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,297,136,851,12550]);

// Polycyclic

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

// generators/relations

G = D5×2- 1+4 order 320 = 26·5

Direct product of D5 and 2- 1+4

G = D₅×2_-¹⁺⁴ order 320 = 2⁶·5

Direct product of D₅ and 2_-¹⁺⁴