C2xC8:D10 - GroupNames

G = C₂×C₈⋊D₁₀ order 320 = 2⁶·5

Direct product of C₂ and C₈⋊D₁₀

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

Aliases: C₂×C₈⋊D₁₀, C₄₀⋊₂C₂³, D₄₀⋊₉C₂², D₂₀⋊₅C₂³, C₂₀.₅₈C₂⁴, C₂³.₅₂D₂₀, M₄(2)⋊₁₇D₁₀, Dic₁₀⋊₅C₂³, (C₂×C₈)⋊₄D₁₀, C₈⋊₂(C₂²×D₅), (C₂×D₄₀)⋊₁₄C₂, (C₂×C₄₀)⋊₇C₂², (C₂×C₄).₅₇D₂₀, C₄.₄₈(C₂×D₂₀), C₁₀⋊₁(C₈⋊C₂²), (C₂×C₂₀).₂₀₃D₄, C₂₀.₂₃₈(C₂×D₄), C₄₀⋊C₂⋊₈C₂², (C₂×M₄(2))⋊₃D₅, C₄.₅₅(C₂³×D₅), C₄○D₂₀⋊₁₈C₂², (C₂×D₂₀)⋊₄₉C₂², (C₂²×D₂₀)⋊₁₇C₂, (C₁₀×M₄(2))⋊₃C₂, C₂².₇₃(C₂×D₂₀), C₂.₂₇(C₂²×D₂₀), C₁₀.₂₅(C₂²×D₄), (C₂×C₂₀).₅₁₁C₂³, (C₂²×C₄).₂₆₅D₁₀, (C₂²×C₁₀).₁₁₈D₄, (C₂×Dic₁₀)⋊₅₇C₂², (C₅×M₄(2))⋊₁₉C₂², (C₂²×C₂₀).₂₆₆C₂², C₅⋊₁(C₂×C₈⋊C₂²), (C₂×C₄₀⋊C₂)⋊₄C₂, (C₂×C₄○D₂₀)⋊₂₆C₂, (C₂×C₁₀).₆₂(C₂×D₄), (C₂×C₄).₂₂₃(C₂²×D₅), SmallGroup(320,1418)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — C₂×C₈⋊D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — D₂₀ — C₂×D₂₀ — C₂²×D₂₀ — C₂×C₈⋊D₁₀

Lower central

C₅ — C₁₀ — C₂₀ — C₂×C₈⋊D₁₀

Upper central

C₁ — C₂² — C₂²×C₄ — C₂×M₄(2)

Generators and relations for C₂×C₈⋊D₁₀
G = < a,b,c,d | a²=b⁸=c¹⁰=d²=1, ab=ba, ac=ca, ad=da, cbc^-1=b⁵, dbd=b^-1, dcd=c^-1 >

Subgroups: 1438 in 298 conjugacy classes, 111 normal (25 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, D₅, C₁₀, C₁₀, C₁₀, C₂×C₈, M₄(2), D₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, Dic₅, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×M₄(2), C₂×D₈, C₂×SD₁₆, C₈⋊C₂², C₂²×D₄, C₂×C₄○D₄, C₄₀, Dic₁₀, Dic₁₀, C₄×D₅, D₂₀, D₂₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₂²×D₅, C₂²×C₁₀, C₂×C₈⋊C₂², C₄₀⋊C₂, D₄₀, C₂×C₄₀, C₅×M₄(2), C₂×Dic₁₀, C₂×C₄×D₅, C₂×D₂₀, C₂×D₂₀, C₂×D₂₀, C₄○D₂₀, C₄○D₂₀, C₂×C₅⋊D₄, C₂²×C₂₀, C₂³×D₅, C₂×C₄₀⋊C₂, C₂×D₄₀, C₈⋊D₁₀, C₁₀×M₄(2), C₂²×D₂₀, C₂×C₄○D₂₀, C₂×C₈⋊D₁₀
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₂⁴, D₁₀, C₈⋊C₂², C₂²×D₄, D₂₀, C₂²×D₅, C₂×C₈⋊C₂², C₂×D₂₀, C₂³×D₅, C₈⋊D₁₀, C₂²×D₂₀, C₂×C₈⋊D₁₀

Smallest permutation representation of C₂×C₈⋊D₁₀
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

D₁₀

D₂₀

C₈⋊C₂²

C₈⋊D₁₀

kernel

C₂×C₈⋊D₁₀

C₂×C₄₀⋊C₂

C₂×D₄₀

C₈⋊D₁₀

C₁₀×M₄(2)

C₂²×D₂₀

C₂×C₄○D₂₀

C₂×C₂₀

C₂²×C₁₀

C₂×M₄(2)

C₂×C₈

M₄(2)

C₂²×C₄

C₂×C₄

C₂³

C₁₀

C₂

# reps

Matrix representation of C₂×C₈⋊D₁₀ ►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

0	40	0	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	11	9	0	0
0	0	32	30	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	34	0	0
0	0	7	7	0	0
0	0	0	0	1	7
0	0	0	0	34	34

40	0	0	0	0	0
0	1	0	0	0	0
0	0	40	34	0	0
0	0	0	1	0	0
0	0	0	0	30	14
0	0	0	0	9	11

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],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,7,0,0,0,0,34,7,0,0,0,0,0,0,1,34,0,0,0,0,7,34],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,34,1,0,0,0,0,0,0,30,9,0,0,0,0,14,11] >;

C₂×C₈⋊D₁₀ in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes D_{10}

% in TeX

G:=Group("C2xC8:D10");

// GroupNames label

G:=SmallGroup(320,1418);

// by ID

G=gap.SmallGroup(320,1418);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,297,80,1684,102,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^10=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;

// generators/relations

G = C2×C8⋊D10 order 320 = 26·5

Direct product of C2 and C8⋊D10

G = C₂×C₈⋊D₁₀ order 320 = 2⁶·5

Direct product of C₂ and C₈⋊D₁₀