(C2xC20):15D4 - GroupNames

G = (C₂xC₂₀):₁₅D₄ order 320 = 2⁶·5

11^st semidirect product of C₂xC₂₀ and D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂xC₂₀):₁₅D₄, (C₂xD₄):₄₃D₁₀, (C₂xQ₈):₃₂D₁₀, D₁₀:₆(C₄oD₄), C₂₀:₂D₄:₄₆C₂, C₂₀.₂₆₅(C₂xD₄), (C₂²xC₄):₂₉D₁₀, C₂³:D₁₀:₃₄C₂, D₁₀:₃Q₈:₄₇C₂, (D₄xC₁₀):₄₇C₂², C₄:Dic₅:₈₀C₂², (Q₈xC₁₀):₃₉C₂², (C₂xC₂₀).₆₅₁C₂³, (C₂xC₁₀).₃₁₄C₂⁴, (C₂²xC₂₀):₄₂C₂², C₅:₈(C₂².₁₉C₂⁴), (C₄xDic₅):₆₀C₂², C₁₀.₁₆₄(C₂²xD₄), C₂³.D₅:₆₅C₂², C₁₀.D₄:₇₆C₂², C₂².₃₂₃(C₂³xD₅), C₂³.₁₃₇(C₂²xD₅), C₂³.₁₈D₁₀:₃₄C₂, C₂³.₂₁D₁₀:₃₈C₂, (C₂²xC₁₀).₂₄₀C₂³, (C₂xDic₅).₃₀₃C₂³, (C₂³xD₅).₁₂₇C₂², (C₂²xD₅).₂₅₃C₂³, D₁₀:C₄.₁₅₉C₂², (C₂²xDic₅).₂₅₉C₂², (C₂xC₄oD₄):₆D₅, (D₅xC₂²xC₄):₇C₂, (C₁₀xC₄oD₄):₆C₂, (C₄xC₅:D₄):₆₀C₂, (C₂xC₄):₁₄(C₅:D₄), C₂.₁₀₃(D₅xC₄oD₄), (C₂xC₁₀).₈₀(C₂xD₄), C₄.₁₀₀(C₂xC₅:D₄), C₁₀.₂₁₅(C₂xC₄oD₄), (C₂xC₄xD₅).₃₃₃C₂², C₂².₂₃(C₂xC₅:D₄), C₂.₃₇(C₂²xC₅:D₄), (C₂xC₄).₆₃₉(C₂²xD₅), (C₂xC₅:D₄).₁₄₈C₂², SmallGroup(320,1500)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₀ — (C₂xC₂₀):₁₅D₄

Chief series

C₁ — C₅ — C₁₀ — C₂xC₁₀ — C₂²xD₅ — C₂³xD₅ — D₅xC₂²xC₄ — (C₂xC₂₀):₁₅D₄

Lower central

C₅ — C₂xC₁₀ — (C₂xC₂₀):₁₅D₄

Upper central

C₁ — C₂xC₄ — C₂xC₄oD₄

Generators and relations for (C₂xC₂₀):₁₅D₄
G = < a,b,c,d | a²=b²⁰=c⁴=d²=1, ab=ba, cac^-1=ab¹⁰, ad=da, cbc^-1=dbd=b⁹, dcd=c^-1 >

Subgroups: 1118 in 330 conjugacy classes, 115 normal (25 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₅, C₂xC₄, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, C₂³, D₅, C₁₀, C₁₀, C₁₀, C₄², C₂²:C₄, C₄:C₄, C₂²xC₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, Dic₅, C₂₀, C₂₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₂xC₁₀, C₄²:C₂, C₄xD₄, C₂²wrC₂, C₄:D₄, C₂²:Q₈, C₂².D₄, C₂³xC₄, C₂xC₄oD₄, C₄xD₅, C₂xDic₅, C₂xDic₅, C₅:D₄, C₂xC₂₀, C₂xC₂₀, C₂xC₂₀, C₅xD₄, C₅xQ₈, C₂²xD₅, C₂²xD₅, C₂²xC₁₀, C₂²xC₁₀, C₂².₁₉C₂⁴, C₄xDic₅, C₁₀.D₄, C₄:Dic₅, D₁₀:C₄, C₂³.D₅, C₂xC₄xD₅, C₂xC₄xD₅, C₂²xDic₅, C₂xC₅:D₄, C₂²xC₂₀, C₂²xC₂₀, D₄xC₁₀, D₄xC₁₀, Q₈xC₁₀, C₅xC₄oD₄, C₂³xD₅, C₂³.₂₁D₁₀, C₄xC₅:D₄, C₂³.₁₈D₁₀, C₂³:D₁₀, C₂₀:₂D₄, D₁₀:₃Q₈, D₅xC₂²xC₄, C₁₀xC₄oD₄, (C₂xC₂₀):₁₅D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂xD₄, C₄oD₄, C₂⁴, D₁₀, C₂²xD₄, C₂xC₄oD₄, C₅:D₄, C₂²xD₅, C₂².₁₉C₂⁴, C₂xC₅:D₄, C₂³xD₅, D₅xC₄oD₄, C₂²xC₅:D₄, (C₂xC₂₀):₁₅D₄

Smallest permutation representation of (C₂xC₂₀):₁₅D₄
►On 80 points

Generators in S₈₀

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

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

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

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

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

68 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	4O	4P	5A	5B	10A	···	10F	10G	···	10R	20A	···	20H	20I	···	20T
order	1	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	5	5	10	···	10	10	···	10	20	···	20	20	···	20
size	1	1	1	1	2	2	4	4	10	10	10	10	1	1	1	1	2	2	4	4	10	10	10	10	20	20	20	20	2	2	2	···	2	4	···	4	2	···	2	4	···	4

68 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

C₄oD₄

D₁₀

C₅:D₄

D₅xC₄oD₄

kernel

(C₂xC₂₀):₁₅D₄

C₂³.₂₁D₁₀

C₄xC₅:D₄

C₂³.₁₈D₁₀

C₂³:D₁₀

C₂₀:₂D₄

D₁₀:₃Q₈

D₅xC₂²xC₄

C₁₀xC₄oD₄

C₂xC₂₀

C₂xC₄oD₄

D₁₀

C₂²xC₄

C₂xD₄

C₂xQ₈

C₂xC₄

C₂

# reps

Matrix representation of (C₂xC₂₀):₁₅D₄ ►in GL₄(F₄₁) generated by

1	0	0	0
0	1	0	0
0	0	1	0
0	0	40	40

1	1	0	0
5	6	0	0
0	0	32	0
0	0	0	32

20	3	0	0
3	21	0	0
0	0	40	39
0	0	1	1

6	7	0	0
36	35	0	0
0	0	40	0
0	0	1	1

G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,40,0,0,0,40],[1,5,0,0,1,6,0,0,0,0,32,0,0,0,0,32],[20,3,0,0,3,21,0,0,0,0,40,1,0,0,39,1],[6,36,0,0,7,35,0,0,0,0,40,1,0,0,0,1] >;

(C₂xC₂₀):₁₅D₄ in GAP, Magma, Sage, TeX

(C_2\times C_{20})\rtimes_{15}D_4

% in TeX

G:=Group("(C2xC20):15D4");

// GroupNames label

G:=SmallGroup(320,1500);

// by ID

G=gap.SmallGroup(320,1500);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,570,12550]);

// Polycyclic

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

// generators/relations

G = (C2xC20):15D4 order 320 = 26·5

11st semidirect product of C2xC20 and D4 acting via D4/C2=C22

G = (C₂xC₂₀):₁₅D₄ order 320 = 2⁶·5

11^st semidirect product of C₂xC₂₀ and D₄ acting via D₄/C₂=C₂²