C2xC20.D4 - GroupNames

G = C₂×C₂₀.D₄ order 320 = 2⁶·5

Direct product of C₂ and C₂₀.D₄

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

Aliases: C₂×C₂₀.D₄, C₂⁴.₂Dic₅, (D₄×C₁₀).₂₃C₄, (C₂×C₂₀).₁₉₀D₄, C₂₀.₂₀₄(C₂×D₄), (C₂³×C₁₀).₄C₄, (C₂×D₄).₇Dic₅, (C₂²×D₄).₃D₅, (C₂×D₄).₁₉₇D₁₀, C₁₀⋊₃(C₄.D₄), C₄.₉(C₂³.D₅), C₂₀.₇₉(C₂²⋊C₄), (C₂×C₂₀).₄₇₁C₂³, (C₂²×C₄).₁₄₈D₁₀, C₄.Dic₅⋊₂₂C₂², C₂³.₃₂(C₂×Dic₅), (D₄×C₁₀).₂₃₉C₂², C₂².₅(C₂²×Dic₅), (C₂²×C₂₀).₁₉₆C₂², C₂².₃₄(C₂³.D₅), (D₄×C₂×C₁₀).₃C₂, C₅⋊₅(C₂×C₄.D₄), C₄.₉₀(C₂×C₅⋊D₄), (C₂×C₂₀).₂₉₀(C₂×C₄), C₂.₉(C₂×C₂³.D₅), (C₂×C₄.Dic₅)⋊₁₈C₂, (C₂×C₄).₂₄(C₂×Dic₅), (C₂×C₄).₁₉₇(C₅⋊D₄), C₁₀.₁₁₄(C₂×C₂²⋊C₄), (C₂²×C₁₀).₄₀(C₂×C₄), (C₂×C₄).₁₂₉(C₂²×D₅), (C₂×C₁₀).₂₉₅(C₂²×C₄), (C₂×C₁₀).₁₇₆(C₂²⋊C₄), SmallGroup(320,843)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₂×C₂₀.D₄

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂×C₂₀ — C₄.Dic₅ — C₂×C₄.Dic₅ — C₂×C₂₀.D₄

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂₀.D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×D₄

Generators and relations for C₂×C₂₀.D₄
G = < a,b,c,d | a²=b²⁰=1, c⁴=b¹⁰, d²=b⁵, ab=ba, ac=ca, ad=da, cbc^-1=b^-1, dbd^-1=b⁹, dcd^-1=b¹⁵c³ >

Subgroups: 478 in 186 conjugacy classes, 71 normal (21 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₂×C₈, M₄(2), C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₄.D₄, C₂×M₄(2), C₂²×D₄, C₅⋊₂C₈, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂×C₄.D₄, C₂×C₅⋊₂C₈, C₄.Dic₅, C₄.Dic₅, C₂²×C₂₀, D₄×C₁₀, D₄×C₁₀, C₂³×C₁₀, C₂₀.D₄, C₂×C₄.Dic₅, D₄×C₂×C₁₀, C₂×C₂₀.D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, D₅, C₂²⋊C₄, C₂²×C₄, C₂×D₄, Dic₅, D₁₀, C₄.D₄, C₂×C₂²⋊C₄, C₂×Dic₅, C₅⋊D₄, C₂²×D₅, C₂×C₄.D₄, C₂³.D₅, C₂²×Dic₅, C₂×C₅⋊D₄, C₂₀.D₄, C₂×C₂³.D₅, C₂×C₂₀.D₄

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

Generators in S₈₀

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

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4
type	+	+	+	+			+	+	+	-	+	-		+
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₅	D₁₀	Dic₅	D₁₀	Dic₅	C₅⋊D₄	C₄.D₄	C₂₀.D₄
kernel	C₂×C₂₀.D₄	C₂₀.D₄	C₂×C₄.Dic₅	D₄×C₂×C₁₀	D₄×C₁₀	C₂³×C₁₀	C₂×C₂₀	C₂²×D₄	C₂²×C₄	C₂×D₄	C₂×D₄	C₂⁴	C₂×C₄	C₁₀	C₂
# reps	1	4	2	1	4	4	4	2	2	4	4	4	16	2	8

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

40	0	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

38	34	0	0	0	0
34	38	0	0	0	0
0	0	0	40	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	40	0

23	35	0	0	0	0
6	18	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	40	0	0
0	0	1	0	0	0

35	23	0	0	0	0
18	6	0	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0
0	0	1	0	0	0
0	0	0	40	0	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[38,34,0,0,0,0,34,38,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[23,6,0,0,0,0,35,18,0,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,1,0,0],[35,18,0,0,0,0,23,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,1,0,0,0] >;

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(320,843);

// by ID

G=gap.SmallGroup(320,843);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,297,136,1684,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=1,c^4=b^10,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^9,d*c*d^-1=b^15*c^3>;

// generators/relations

G = C2×C20.D4 order 320 = 26·5

Direct product of C2 and C20.D4

G = C₂×C₂₀.D₄ order 320 = 2⁶·5

Direct product of C₂ and C₂₀.D₄