(C2xC20).D4 - GroupNames

G = (C₂×C₂₀).D₄ order 320 = 2⁶·5

2^nd non-split extension by C₂×C₂₀ of D₄ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×C₂₀).₂D₄, (C₂×C₄).₂D₂₀, (C₂×D₄).₂D₁₀, C₂³⋊C₄.₁D₅, C₂³.D₅⋊₂C₄, C₂³.₂(C₄×D₅), (C₂²×C₁₀).₉D₄, C₂₀.D₄.₁C₂, (C₂²×Dic₅)⋊₁C₄, (D₄×C₁₀).₂C₂², C₂³.₂(C₅⋊D₄), C₅⋊₃(C₂³.D₄), C₁₀.₂₉(C₂³⋊C₄), C₂².₉(D₁₀⋊C₄), C₂³.₁₈D₁₀.₁C₂, C₂.₉(C₂³.₁D₁₀), (C₅×C₂³⋊C₄).₁C₂, (C₂²×C₁₀).₂(C₂×C₄), (C₂×C₁₀).₆₆(C₂²⋊C₄), SmallGroup(320,30)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₀ — (C₂×C₂₀).D₄

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — D₄×C₁₀ — C₂³.₁₈D₁₀ — (C₂×C₂₀).D₄

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — (C₂×C₂₀).D₄

Upper central

C₁ — C₂ — C₂² — C₂×D₄ — C₂³⋊C₄

Generators and relations for (C₂×C₂₀).D₄
G = < a,b,c,d | a²=b⁴=c²⁰=1, d²=b^-1, ab=ba, cac^-1=dad^-1=ab², cbc^-1=ab^-1, bd=db, dcd^-1=bc^-1 >

Subgroups: 286 in 68 conjugacy classes, 21 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₁₀, C₁₀, C₂²⋊C₄, C₄⋊C₄, M₄(2), C₂²×C₄, C₂×D₄, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂³⋊C₄, C₄.D₄, C₂².D₄, C₅⋊₂C₈, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂³.D₄, C₄.Dic₅, C₁₀.D₄, C₂³.D₅, C₂³.D₅, C₅×C₂²⋊C₄, C₂²×Dic₅, D₄×C₁₀, C₂₀.D₄, C₅×C₂³⋊C₄, C₂³.₁₈D₁₀, (C₂×C₂₀).D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₅, C₂²⋊C₄, D₁₀, C₂³⋊C₄, C₄×D₅, D₂₀, C₅⋊D₄, C₂³.D₄, D₁₀⋊C₄, C₂³.₁D₁₀, (C₂×C₂₀).D₄

Smallest permutation representation of (C₂×C₂₀).D₄
►On 80 points

Generators in S₈₀

(1 70)(3 72)(5 74)(7 76)(9 78)(11 80)(13 62)(15 64)(17 66)(19 68)(22 47)(24 49)(26 51)(28 53)(30 55)(32 57)(34 59)(36 41)(38 43)(40 45)

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

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

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

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

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

35 conjugacy classes

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

35 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	8
type	+	+	+	+			+	+	+	+	+			+			-
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₄	D₅	D₁₀	D₂₀	C₄×D₅	C₅⋊D₄	C₂³⋊C₄	C₂³.D₄	C₂³.₁D₁₀	(C₂×C₂₀).D₄
kernel	(C₂×C₂₀).D₄	C₂₀.D₄	C₅×C₂³⋊C₄	C₂³.₁₈D₁₀	C₂³.D₅	C₂²×Dic₅	C₂×C₂₀	C₂²×C₁₀	C₂³⋊C₄	C₂×D₄	C₂×C₄	C₂³	C₂³	C₁₀	C₅	C₂	C₁
# reps	1	1	1	1	2	2	1	1	2	2	4	4	4	1	2	4	2

Matrix representation of (C₂×C₂₀).D₄ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	20	20	1	0
0	0	19	19	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	9	0	0	0
0	0	0	32	0	0
0	0	25	0	32	0
0	0	0	7	0	9

0	40	0	0	0	0
1	7	0	0	0	0
0	0	8	8	9	0
0	0	24	24	0	9
0	0	8	7	33	33
0	0	25	24	17	17

0	40	0	0	0	0
40	0	0	0	0	0
0	0	11	11	0	40
0	0	31	31	40	0
0	0	31	22	10	10
0	0	20	11	30	30

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,20,19,0,0,0,40,20,19,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,25,0,0,0,0,32,0,7,0,0,0,0,32,0,0,0,0,0,0,9],[0,1,0,0,0,0,40,7,0,0,0,0,0,0,8,24,8,25,0,0,8,24,7,24,0,0,9,0,33,17,0,0,0,9,33,17],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,11,31,31,20,0,0,11,31,22,11,0,0,0,40,10,30,0,0,40,0,10,30] >;

(C₂×C₂₀).D₄ in GAP, Magma, Sage, TeX

(C_2\times C_{20}).D_4

% in TeX

G:=Group("(C2xC20).D4");

// GroupNames label

G:=SmallGroup(320,30);

// by ID

G=gap.SmallGroup(320,30);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,422,346,297,851,12550]);

// Polycyclic

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

// generators/relations

G = (C2×C20).D4 order 320 = 26·5

2nd non-split extension by C2×C20 of D4 acting faithfully

G = (C₂×C₂₀).D₄ order 320 = 2⁶·5

2^nd non-split extension by C₂×C₂₀ of D₄ acting faithfully