(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^[×3], C₄^[×4], C₂², C₂²^[×4], C₅, C₈, C₂×C₄, C₂×C₄^[×4], D₄, C₂³^[×2], C₁₀, C₁₀^[×3], C₂²⋊C₄^[×3], C₄⋊C₄, M₄(2), C₂²×C₄, C₂×D₄, Dic₅^[×2], C₂₀^[×2], C₂×C₁₀, C₂×C₁₀^[×4], C₂³⋊C₄, C₄.D₄, C₂².D₄, C₅⋊₂C₈, C₂×Dic₅^[×3], C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×C₁₀^[×2], 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^[×3], C₄^[×2], C₂², C₂×C₄, D₄^[×2], 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 22)(3 24)(5 26)(7 28)(9 30)(11 32)(13 34)(15 36)(17 38)(19 40)(41 74)(43 76)(45 78)(47 80)(49 62)(51 64)(53 66)(55 68)(57 70)(59 72)

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

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

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

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

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

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