C2xC5:2C8 - GroupNames

G = C₂xC₅:₂C₈ order 80 = 2⁴·5

Direct product of C₂ and C₅:₂C₈

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C₂xC₅:₂C₈, C₁₀:₂C₈, C₂₀.₆C₄, C₄.₁₄D₁₀, C₄.₃Dic₅, C₂₀.₁₄C₂², C₂².₂Dic₅, C₅:₄(C₂xC₈), C₄ o(C₅:₂C₈), (C₂xC₄).₅D₅, (C₂xC₁₀).₄C₄, (C₂xC₂₀).₆C₂, C₁₀.₁₃(C₂xC₄), C₂.₁(C₂xDic₅), SmallGroup(80,9)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅ — C₂xC₅:₂C₈

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₅:₂C₈ — C₂xC₅:₂C₈

Lower central

C₅ — C₂xC₅:₂C₈

Upper central

C₁ — C₂xC₄

Generators and relations for C₂xC₅:₂C₈
G = < a,b,c | a²=b⁵=c⁸=1, ab=ba, ac=ca, cbc^-1=b^-1 >

Subgroups: 34 in 22 conjugacy classes, 19 normal (13 characteristic)
Quotients: C₁, C₂, C₄, C₂², C₈, C₂xC₄, D₅, C₂xC₈, Dic₅, D₁₀, C₅:₂C₈, C₂xDic₅, C₂xC₅:₂C₈

C₁

C₂

C₅

C₄

C₂²

C₄

C₁₀

C₂xC₄

₅C₈

C₂xC₁₀

C₂₀

₅C₂xC₈

C₅:₂C₈

C₂xC₂₀

C₅:₂C₈

C₂xC₅:₂C₈

Smallest permutation representation of C₂xC₅:₂C₈
►Regular action on 80 points

Generators in S₈₀

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

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

(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)

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

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

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

C₂xC₅:₂C₈ is a maximal subgroup of
C₄².D₅ C₂₀:₃C₈ C₁₀.D₈ C₂₀.Q₈ D₂₀:₆C₄ C₁₀.Q₁₆ C₈xDic₅ C₂₀.₈Q₈ C₄₀:₈C₄ D₁₀:₁C₈ C₂₀.₅₃D₄ C₂₀.₅₅D₄ D₄:Dic₅ Q₈:Dic₅ C₂₀.C₈ D₅xC₂xC₈ D₂₀.₂C₄ D₄.Dic₅ D₄.₈D₁₀ C₂₀.₁₄F₅
C₂xC₅:₂C₈ is a maximal quotient of
C₂₀:₃C₈ C₂₀.₄C₈ C₂₀.₅₅D₄ C₂₀.₁₄F₅

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2
type	+	+	+				+	-	+	-
image	C₁	C₂	C₂	C₄	C₄	C₈	D₅	Dic₅	D₁₀	Dic₅	C₅:₂C₈
kernel	C₂xC₅:₂C₈	C₅:₂C₈	C₂xC₂₀	C₂₀	C₂xC₁₀	C₁₀	C₂xC₄	C₄	C₄	C₂²	C₂
# reps	1	2	1	2	2	8	2	2	2	2	8

Matrix representation of C₂xC₅:₂C₈ ►in GL₃(F₄₁) generated by

40	0	0
0	1	0
0	0	1

1	0	0
0	34	40
0	1	0

1	0	0
0	31	3
0	32	10

G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,34,1,0,40,0],[1,0,0,0,31,32,0,3,10] >;

C₂xC₅:₂C₈ in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes_2C_8

% in TeX

G:=Group("C2xC5:2C8");

// GroupNames label

G:=SmallGroup(80,9);

// by ID

G=gap.SmallGroup(80,9);

# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,20,42,1604]);

// Polycyclic

G:=Group<a,b,c|a^2=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;

// generators/relations

Export

Subgroup lattice of C₂xC₅:₂C₈ in TeX

G = C2xC5:2C8 order 80 = 24·5

Direct product of C2 and C5:2C8

G = C₂xC₅:₂C₈ order 80 = 2⁴·5

Direct product of C₂ and C₅:₂C₈