D20.4D4 - GroupNames

G = D₂₀.₄D₄ order 320 = 2⁶·5

4^th non-split extension by D₂₀ of D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₂₀.₄D₄, Dic₁₀.₄D₄, M₄(2).₃D₁₀, (C₂×C₄).₆D₂₀, (C₂×C₂₀).₈D₄, C₄.₈₀(D₄×D₅), C₂₀.₉₇(C₂×D₄), (C₂×Q₈).₅D₁₀, D₂₀⋊₇C₄⋊₃C₂, C₈.D₁₀⋊₇C₂, C₄.₁₀D₄⋊₁D₅, (C₂×C₂₀).₉C₂³, C₁₀.₁₇C₂²≀C₂, Dic₅⋊Q₈⋊₁C₂, C₅⋊₁(D₄.₁₀D₄), C₄○D₂₀.₅C₂², C₂².₁₂(C₂×D₂₀), (Q₈×C₁₀).₇C₂², C₂.₂₀(C₂²⋊D₂₀), (C₄×Dic₅).₇C₂², Q₈.₁₀D₁₀.₁C₂, (C₅×M₄(2)).₂C₂², (C₂×Dic₁₀).₅₁C₂², (C₂×C₁₀).₂₂(C₂×D₄), (C₂×C₄).₉(C₂²×D₅), (C₅×C₄.₁₀D₄)⋊₃C₂, SmallGroup(320,379)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — D₂₀.₄D₄

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂×C₂₀ — C₄○D₂₀ — Q₈.₁₀D₁₀ — D₂₀.₄D₄

Lower central

C₅ — C₁₀ — C₂×C₂₀ — D₂₀.₄D₄

Upper central

C₁ — C₂ — C₂×C₄ — C₄.₁₀D₄

Generators and relations for D₂₀.₄D₄
G = < a,b,c,d | a²⁰=b²=1, c⁴=a¹⁰, d²=a¹⁵, bab=a^-1, cac^-1=a¹¹, ad=da, cbc^-1=a¹⁵b, dbd^-1=a⁵b, dcd^-1=a¹⁵c³ >

Subgroups: 622 in 142 conjugacy classes, 39 normal (17 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, D₅, C₁₀, C₁₀, C₄², C₄⋊C₄, M₄(2), SD₁₆, Q₁₆, C₂×Q₈, C₂×Q₈, C₄○D₄, Dic₅, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₄.₁₀D₄, C₄≀C₂, C₄⋊Q₈, C₈.C₂², 2_-¹⁺⁴, C₄₀, Dic₁₀, Dic₁₀, C₄×D₅, D₂₀, D₂₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₅×Q₈, D₄.₁₀D₄, C₄₀⋊C₂, Dic₂₀, C₄×Dic₅, C₁₀.D₄, C₅×M₄(2), C₂×Dic₁₀, C₄○D₂₀, C₄○D₂₀, Q₈×D₅, Q₈⋊₂D₅, Q₈×C₁₀, D₂₀⋊₇C₄, C₅×C₄.₁₀D₄, C₈.D₁₀, Dic₅⋊Q₈, Q₈.₁₀D₁₀, D₂₀.₄D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, D₁₀, C₂²≀C₂, D₂₀, C₂²×D₅, D₄.₁₀D₄, C₂×D₂₀, D₄×D₅, C₂²⋊D₂₀, D₂₀.₄D₄

Smallest permutation representation of D₂₀.₄D₄
►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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 40)(19 39)(20 38)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	8
type	+	+	+	+	+	+	+	+	+	+	+	+	+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₄	D₅	D₁₀	D₁₀	D₂₀	D₄.₁₀D₄	D₄×D₅	D₂₀.₄D₄
kernel	D₂₀.₄D₄	D₂₀⋊₇C₄	C₅×C₄.₁₀D₄	C₈.D₁₀	Dic₅⋊Q₈	Q₈.₁₀D₁₀	Dic₁₀	D₂₀	C₂×C₂₀	C₄.₁₀D₄	M₄(2)	C₂×Q₈	C₂×C₄	C₅	C₄	C₁
# reps	1	2	1	2	1	1	2	2	2	2	4	2	8	2	4	2

Matrix representation of D₂₀.₄D₄ ►in GL₆(𝔽₄₁)

34	1	0	0	0	0
40	0	0	0	0	0
0	0	0	1	0	0
0	0	40	0	0	0
0	0	40	40	1	4
0	0	0	21	20	40

40	7	0	0	0	0
0	1	0	0	0	0
0	0	1	1	40	37
0	0	0	0	40	0
0	0	0	40	0	0
0	0	0	21	20	40

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	1	0
0	0	1	1	40	37
0	0	0	1	0	0
0	0	21	0	0	40

40	0	0	0	0	0
0	40	0	0	0	0
0	0	11	11	29	38
0	0	1	1	10	37
0	0	30	40	0	0
0	0	6	21	20	29

G:=sub<GL(6,GF(41))| [34,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,40,0,0,0,1,0,40,21,0,0,0,0,1,20,0,0,0,0,4,40],[40,0,0,0,0,0,7,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,40,21,0,0,40,40,0,20,0,0,37,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,21,0,0,0,1,1,0,0,0,1,40,0,0,0,0,0,37,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,11,1,30,6,0,0,11,1,40,21,0,0,29,10,0,20,0,0,38,37,0,29] >;

D₂₀.₄D₄ in GAP, Magma, Sage, TeX

D_{20}._4D_4

% in TeX

G:=Group("D20.4D4");

// GroupNames label

G:=SmallGroup(320,379);

// by ID

G=gap.SmallGroup(320,379);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,58,1123,570,136,1684,438,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^15,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^15*b,d*b*d^-1=a^5*b,d*c*d^-1=a^15*c^3>;

// generators/relations

G = D20.4D4 order 320 = 26·5

4th non-split extension by D20 of D4 acting via D4/C2=C22

G = D₂₀.₄D₄ order 320 = 2⁶·5

4^th non-split extension by D₂₀ of D₄ acting via D₄/C₂=C₂²