D4.3D20 - GroupNames

G = D₄.₃D₂₀ order 320 = 2⁶·5

3^rd non-split extension by D₄ of D₂₀ acting via D₂₀/C₂₀=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄.₃D₂₀, C₄₀.₈₇D₄, Q₈.₃D₂₀, M₄(2).₃₃D₁₀, C₈○D₄⋊₁D₅, (C₅×D₄).₂₀D₄, C₄.₁₉(C₂×D₂₀), (C₂×C₈).₈₀D₁₀, C₂₀.₄₂(C₂×D₄), (C₅×Q₈).₂₀D₄, C₄○D₄.₃₁D₁₀, C₅⋊₄(D₄.₃D₄), C₈.₄₄(C₅⋊D₄), C₄₀.₆C₄⋊₁₄C₂, D₄⋊D₁₀.₁C₂, D₄.₉D₁₀⋊₃C₂, (C₂×C₄₀).₆₆C₂², C₂₀.₄₆D₄⋊₁₃C₂, C₄.₁₂D₂₀⋊₁₃C₂, C₂.₂₄(C₂₀⋊₇D₄), C₁₀.₇₆(C₄⋊D₄), (C₂×C₂₀).₄₂₁C₂³, C₂².₈(C₄○D₂₀), (C₂×D₂₀).₁₁₅C₂², C₄.Dic₅.₁₆C₂², (C₅×M₄(2)).₃₆C₂², (C₂×Dic₁₀).₁₂₁C₂², (C₅×C₈○D₄)⋊₁C₂, (C₂×C₄₀⋊C₂)⋊₃C₂, C₄.₁₁₇(C₂×C₅⋊D₄), (C₂×C₁₀).₆(C₄○D₄), (C₅×C₄○D₄).₃₆C₂², (C₂×C₄).₁₂₃(C₂²×D₅), SmallGroup(320,768)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — D₄.₃D₂₀

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂×C₂₀ — C₂×D₂₀ — C₂×C₄₀⋊C₂ — D₄.₃D₂₀

Lower central

C₅ — C₁₀ — C₂×C₂₀ — D₄.₃D₂₀

Upper central

C₁ — C₂ — C₂×C₄ — C₈○D₄

Generators and relations for D₄.₃D₂₀
G = < a,b,c,d | a⁴=b²=1, c²⁰=d²=a², bab=dad^-1=a^-1, ac=ca, bc=cb, dbd^-1=ab, dcd^-1=c¹⁹ >

Subgroups: 446 in 104 conjugacy classes, 39 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, D₅, C₁₀, C₁₀, C₂×C₈, C₂×C₈, M₄(2), M₄(2), D₈, SD₁₆, Q₁₆, C₂×D₄, C₂×Q₈, C₄○D₄, Dic₅, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₄.D₄, C₄.₁₀D₄, C₈.C₄, C₈○D₄, C₂×SD₁₆, C₈⋊C₂², C₈.C₂², C₅⋊₂C₈, C₄₀, C₄₀, Dic₁₀, D₂₀, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×D₄, C₅×Q₈, C₂²×D₅, D₄.₃D₄, C₄₀⋊C₂, C₄.Dic₅, D₄⋊D₅, D₄.D₅, Q₈⋊D₅, C₅⋊Q₁₆, C₂×C₄₀, C₂×C₄₀, C₅×M₄(2), C₅×M₄(2), C₂×Dic₁₀, C₂×D₂₀, C₅×C₄○D₄, C₄₀.₆C₄, C₂₀.₄₆D₄, C₄.₁₂D₂₀, C₂×C₄₀⋊C₂, D₄⋊D₁₀, D₄.₉D₁₀, C₅×C₈○D₄, D₄.₃D₂₀
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₄○D₄, D₁₀, C₄⋊D₄, D₂₀, C₅⋊D₄, C₂²×D₅, D₄.₃D₄, C₂×D₂₀, C₄○D₂₀, C₂×C₅⋊D₄, C₂₀⋊₇D₄, D₄.₃D₂₀

Smallest permutation representation of D₄.₃D₂₀
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+		+	+	+		+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₄	D₅	C₄○D₄	D₁₀	D₁₀	D₁₀	C₅⋊D₄	D₂₀	D₂₀	C₄○D₂₀	D₄.₃D₄	D₄.₃D₂₀
kernel	D₄.₃D₂₀	C₄₀.₆C₄	C₂₀.₄₆D₄	C₄.₁₂D₂₀	C₂×C₄₀⋊C₂	D₄⋊D₁₀	D₄.₉D₁₀	C₅×C₈○D₄	C₄₀	C₅×D₄	C₅×Q₈	C₈○D₄	C₂×C₁₀	C₂×C₈	M₄(2)	C₄○D₄	C₈	D₄	Q₈	C₂²	C₅	C₁
# reps	1	1	1	1	1	1	1	1	2	1	1	2	2	2	2	2	8	4	4	8	2	8

Matrix representation of D₄.₃D₂₀ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	1	0	0
0	0	39	40	0	0
0	0	0	0	40	40
0	0	0	0	2	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	40	40
0	0	0	0	2	1
0	0	1	1	0	0
0	0	39	40	0	0

19	27	0	0	0	0
27	19	0	0	0	0
0	0	11	26	0	0
0	0	30	0	0	0
0	0	0	0	11	26
0	0	0	0	30	0

9	30	0	0	0	0
11	32	0	0	0	0
0	0	0	26	0	0
0	0	11	0	0	0
0	0	0	0	30	15
0	0	0	0	11	11

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,39,0,0,0,0,1,40,0,0,0,0,0,0,40,2,0,0,0,0,40,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,39,0,0,0,0,1,40,0,0,40,2,0,0,0,0,40,1,0,0],[19,27,0,0,0,0,27,19,0,0,0,0,0,0,11,30,0,0,0,0,26,0,0,0,0,0,0,0,11,30,0,0,0,0,26,0],[9,11,0,0,0,0,30,32,0,0,0,0,0,0,0,11,0,0,0,0,26,0,0,0,0,0,0,0,30,11,0,0,0,0,15,11] >;

D₄.₃D₂₀ in GAP, Magma, Sage, TeX

D_4._3D_{20}

% in TeX

G:=Group("D4.3D20");

// GroupNames label

G:=SmallGroup(320,768);

// by ID

G=gap.SmallGroup(320,768);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,1123,297,136,1684,102,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^20=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^19>;

// generators/relations

G = D4.3D20 order 320 = 26·5

3rd non-split extension by D4 of D20 acting via D20/C20=C2

G = D₄.₃D₂₀ order 320 = 2⁶·5

3^rd non-split extension by D₄ of D₂₀ acting via D₂₀/C₂₀=C₂