C2^4:4D10 - GroupNames

G = C₂⁴⋊₄D₁₀ order 320 = 2⁶·5

3^rd semidirect product of C₂⁴ and D₁₀ acting via D₁₀/C₅=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴⋊₄D₁₀, C₁₀.₂₆2₊¹⁺⁴, C₅⋊₂D₄², C₅⋊D₄⋊₄D₄, D₁₀⋊₆(C₂×D₄), C₂²⋊₃(D₄×D₅), (C₂×D₄)⋊₁₈D₁₀, C₂²≀C₂⋊₃D₅, Dic₅⋊₃(C₂×D₄), C₂³⋊D₁₀⋊₄C₂, C₂²⋊D₂₀⋊₉C₂, C₂₀⋊D₄⋊₁₁C₂, C₂²⋊C₄⋊₂₄D₁₀, (D₄×C₁₀)⋊₇C₂², D₁₀⋊D₄⋊₁₃C₂, Dic₅⋊₄D₄⋊₂C₂, Dic₅⋊D₄⋊₂C₂, (C₂×D₂₀)⋊₁₉C₂², (C₂×C₂₀).₂₈C₂³, C₁₀.₅₆(C₂²×D₄), (C₂³×D₅)⋊₇C₂², (C₂×C₁₀).₁₃₄C₂⁴, (C₂³×C₁₀)⋊₁₀C₂², (C₄×Dic₅)⋊₁₄C₂², C₁₀.D₄⋊₉C₂², C₂³.D₅⋊₁₅C₂², C₂.₂₈(D₄⋊₆D₁₀), D₁₀⋊C₄⋊₁₁C₂², (C₂²×D₅).₅₃C₂³, C₂³.₁₀₈(C₂²×D₅), C₂².₁₅₅(C₂³×D₅), (C₂²×C₁₀).₁₈₁C₂³, (C₂×Dic₅).₂₃₁C₂³, (C₂²×Dic₅)⋊₁₃C₂², (C₂×D₄×D₅)⋊₇C₂, C₂.₂₉(C₂×D₄×D₅), (C₂×C₁₀)⋊₆(C₂×D₄), (C₂×C₄×D₅)⋊₇C₂², (C₅×C₂²≀C₂)⋊₅C₂, (C₂²×C₅⋊D₄)⋊₈C₂, (C₂×C₅⋊D₄)⋊₃₉C₂², (C₅×C₂²⋊C₄)⋊₅C₂², (C₂×C₄).₂₈(C₂²×D₅), SmallGroup(320,1262)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₂⁴⋊₄D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×D₅ — C₂³×D₅ — C₂×D₄×D₅ — C₂⁴⋊₄D₁₀

Lower central

C₅ — C₂×C₁₀ — C₂⁴⋊₄D₁₀

Upper central

C₁ — C₂² — C₂²≀C₂

Generators and relations for C₂⁴⋊₄D₁₀
G = < a,b,c,d,e,f | a²=b²=c²=d²=e¹⁰=f²=1, ab=ba, eae^-1=faf=ac=ca, ad=da, fbf=bc=cb, ebe^-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 1910 in 428 conjugacy classes, 115 normal (27 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, D₅, C₁₀, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, Dic₅, Dic₅, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₄×D₄, C₂²≀C₂, C₂²≀C₂, C₄⋊D₄, C₄⋊₁D₄, C₂²×D₄, C₄×D₅, D₂₀, C₂×Dic₅, C₂×Dic₅, C₅⋊D₄, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, D₄², C₄×Dic₅, C₁₀.D₄, D₁₀⋊C₄, C₂³.D₅, C₅×C₂²⋊C₄, C₅×C₂²⋊C₄, C₂×C₄×D₅, C₂×D₂₀, C₂×D₂₀, D₄×D₅, C₂²×Dic₅, C₂×C₅⋊D₄, C₂×C₅⋊D₄, D₄×C₁₀, D₄×C₁₀, C₂³×D₅, C₂³×D₅, C₂³×C₁₀, Dic₅⋊₄D₄, C₂²⋊D₂₀, D₁₀⋊D₄, C₂³⋊D₁₀, Dic₅⋊D₄, C₂₀⋊D₄, C₅×C₂²≀C₂, C₂×D₄×D₅, C₂²×C₅⋊D₄, C₂⁴⋊₄D₁₀
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₂⁴, D₁₀, C₂²×D₄, 2₊¹⁺⁴, C₂²×D₅, D₄², D₄×D₅, C₂³×D₅, C₂×D₄×D₅, D₄⋊₆D₁₀, C₂⁴⋊₄D₁₀

Smallest permutation representation of C₂⁴⋊₄D₁₀
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

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

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

53 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	2L	2M	2N	2O	4A	4B	4C	4D	4E	4F	4G	4H	4I	5A	5B	10A	···	10F	10G	···	10R	10S	10T	20A	···	20F
order	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	5	5	10	···	10	10	···	10	10	10	20	···	20
size	1	1	1	1	2	2	2	2	4	4	10	10	10	10	20	20	4	4	4	10	10	10	10	20	20	2	2	2	···	2	4	···	4	8	8	8	···	8

53 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

D₁₀

2₊¹⁺⁴

D₄×D₅

D₄⋊₆D₁₀

kernel

C₂⁴⋊₄D₁₀

Dic₅⋊₄D₄

C₂²⋊D₂₀

D₁₀⋊D₄

C₂³⋊D₁₀

Dic₅⋊D₄

C₂₀⋊D₄

C₅×C₂²≀C₂

C₂×D₄×D₅

C₂²×C₅⋊D₄

C₅⋊D₄

C₂²≀C₂

C₂²⋊C₄

C₂×D₄

C₂⁴

C₁₀

C₂²

C₂

# reps

Matrix representation of C₂⁴⋊₄D₁₀ ►in GL₆(𝔽₄₁)

1	23	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

40	18	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	9
0	0	0	0	0	40

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

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	40	0
0	0	0	0	0	40

40	0	0	0	0	0
9	1	0	0	0	0
0	0	7	7	0	0
0	0	34	40	0	0
0	0	0	0	1	0
0	0	0	0	18	40

1	0	0	0	0	0
32	40	0	0	0	0
0	0	7	7	0	0
0	0	40	34	0	0
0	0	0	0	40	0
0	0	0	0	0	40

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,23,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,18,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,9,40],[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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,9,0,0,0,0,0,1,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,1,18,0,0,0,0,0,40],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

C₂⁴⋊₄D₁₀ in GAP, Magma, Sage, TeX

C_2^4\rtimes_4D_{10}

% in TeX

G:=Group("C2^4:4D10");

// GroupNames label

G:=SmallGroup(320,1262);

// by ID

G=gap.SmallGroup(320,1262);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,1571,297,12550]);

// Polycyclic

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

// generators/relations

G = C24⋊4D10 order 320 = 26·5

3rd semidirect product of C24 and D10 acting via D10/C5=C22

G = C₂⁴⋊₄D₁₀ order 320 = 2⁶·5

3^rd semidirect product of C₂⁴ and D₁₀ acting via D₁₀/C₅=C₂²