C4^2:8D10 - GroupNames

G = C₄²:₈D₁₀ order 320 = 2⁶·5

8^th semidirect product of C₄² and D₁₀ acting via D₁₀/C₅=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄²:₈D₁₀, C₄:C₄:₄₂D₁₀, (C₄xD₂₀):₇C₂, (C₂xC₄):₁₁D₂₀, (C₂xC₂₀):₁₀D₄, (C₄xC₂₀):₅C₂², C₄.₇₀(C₂xD₂₀), D₁₀:₁(C₄oD₄), C₄:D₂₀:₄₂C₂, C₂₀.₂₂₃(C₂xD₄), C₄²:C₂:₈D₅, C₂²:D₂₀:₂₉C₂, D₁₀:₂Q₈:₄₇C₂, (C₂xD₂₀):₅₂C₂², (C₂xC₁₀).₆₈C₂⁴, C₄:Dic₅:₅₅C₂², C₂²:C₄.₉₂D₁₀, C₂.₁₄(C₂²xD₂₀), C₂².₁₉(C₂xD₂₀), C₁₀.₁₂(C₂²xD₄), (C₂xC₂₀).₁₄₃C₂³, C₅:₁(C₂².₁₉C₂⁴), (C₂²xC₄).₃₆₅D₁₀, D₁₀:C₄:₅₁C₂², C₂².₉₇(C₂³xD₅), (C₂xDic₁₀):₆₁C₂², C₂².D₂₀:₃₂C₂, (C₂²xD₅).₁₈C₂³, C₂³.₁₅₆(C₂²xD₅), (C₂²xC₂₀).₂₂₈C₂², (C₂²xC₁₀).₁₃₈C₂³, (C₂xDic₅).₂₀₇C₂³, (C₂³xD₅).₁₁₆C₂², (C₂²xDic₅).₂₄₀C₂², C₂.₉(D₅xC₄oD₄), (D₅xC₂²xC₄):₂C₂, (C₂xC₄xD₅):₄₄C₂², (C₂xC₄oD₂₀):₁₇C₂, (C₅xC₄:C₄):₅₂C₂², (C₂xC₁₀).₄₉(C₂xD₄), C₁₀.₁₃₃(C₂xC₄oD₄), (C₅xC₄²:C₂):₁₀C₂, (C₂xC₄).₅₇₅(C₂²xD₅), (C₂xC₅:D₄).₁₀₇C₂², (C₅xC₂²:C₄).₁₀₀C₂², SmallGroup(320,1196)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₀ — C₄²:₈D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂xC₁₀ — C₂²xD₅ — C₂³xD₅ — D₅xC₂²xC₄ — C₄²:₈D₁₀

Lower central

C₅ — C₂xC₁₀ — C₄²:₈D₁₀

Upper central

C₁ — C₂xC₄ — C₄²:C₂

Generators and relations for C₄²:₈D₁₀
G = < a,b,c,d | a⁴=b⁴=c¹⁰=d²=1, ab=ba, cac^-1=ab², dad=a^-1, bc=cb, bd=db, dcd=c^-1 >

Subgroups: 1310 in 330 conjugacy classes, 115 normal (23 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₅, C₂xC₄, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, D₅, C₁₀, C₁₀, C₁₀, C₄², C₂²:C₄, C₂²:C₄, C₄:C₄, C₄:C₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, Dic₅, C₂₀, C₂₀, D₁₀, D₁₀, C₂xC₁₀, C₂xC₁₀, C₂xC₁₀, C₄²:C₂, C₄xD₄, C₂²wrC₂, C₄:D₄, C₂²:Q₈, C₂².D₄, C₂³xC₄, C₂xC₄oD₄, Dic₁₀, C₄xD₅, D₂₀, C₂xDic₅, C₂xDic₅, C₅:D₄, C₂xC₂₀, C₂xC₂₀, C₂²xD₅, C₂²xD₅, C₂²xC₁₀, C₂².₁₉C₂⁴, C₄:Dic₅, D₁₀:C₄, C₄xC₂₀, C₅xC₂²:C₄, C₅xC₄:C₄, C₂xDic₁₀, C₂xC₄xD₅, C₂xC₄xD₅, C₂xD₂₀, C₂xD₂₀, C₄oD₂₀, C₂²xDic₅, C₂xC₅:D₄, C₂²xC₂₀, C₂³xD₅, C₄xD₂₀, C₂²:D₂₀, C₂².D₂₀, C₄:D₂₀, D₁₀:₂Q₈, C₅xC₄²:C₂, D₅xC₂²xC₄, C₂xC₄oD₂₀, C₄²:₈D₁₀
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂xD₄, C₄oD₄, C₂⁴, D₁₀, C₂²xD₄, C₂xC₄oD₄, D₂₀, C₂²xD₅, C₂².₁₉C₂⁴, C₂xD₂₀, C₂³xD₅, C₂²xD₂₀, D₅xC₄oD₄, C₄²:₈D₁₀

Smallest permutation representation of C₄²:₈D₁₀
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

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	10H	10I	10J	20A	···	20H	20I	···	20AB
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	10	10	20	···	20	20	···	20
size	1	1	1	1	2	2	10	10	10	10	20	20	1	1	1	1	2	2	4	4	4	4	10	10	10	10	20	20	2	2	2	···	2	4	4	4	4	2	···	2	4	···	4

68 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

C₄oD₄

D₁₀

D₂₀

D₅xC₄oD₄

kernel

C₄²:₈D₁₀

C₄xD₂₀

C₂²:D₂₀

C₂².D₂₀

C₄:D₂₀

D₁₀:₂Q₈

C₅xC₄²:C₂

D₅xC₂²xC₄

C₂xC₄oD₂₀

C₂xC₂₀

C₄²:C₂

D₁₀

C₄²

C₂²:C₄

C₄:C₄

C₂²xC₄

C₂xC₄

C₂

# reps

Matrix representation of C₄²:₈D₁₀ ►in GL₄(F₄₁) generated by

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

1	0	0	0
0	1	0	0
0	0	9	0
0	0	0	9

34	34	0	0
7	1	0	0
0	0	1	0
0	0	18	40

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

G:=sub<GL(4,GF(41))| [30,9,0,0,32,11,0,0,0,0,40,0,0,0,32,1],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[34,7,0,0,34,1,0,0,0,0,1,18,0,0,0,40],[34,1,0,0,34,7,0,0,0,0,40,0,0,0,0,40] >;

C₄²:₈D₁₀ in GAP, Magma, Sage, TeX

C_4^2\rtimes_8D_{10}

% in TeX

G:=Group("C4^2:8D10");

// GroupNames label

G:=SmallGroup(320,1196);

// by ID

G=gap.SmallGroup(320,1196);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,570,80,12550]);

// Polycyclic

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

// generators/relations

G = C42:8D10 order 320 = 26·5

8th semidirect product of C42 and D10 acting via D10/C5=C22

G = C₄²:₈D₁₀ order 320 = 2⁶·5

8^th semidirect product of C₄² and D₁₀ acting via D₁₀/C₅=C₂²