C2^4:8D10 - GroupNames

G = C₂⁴⋊₈D₁₀ order 320 = 2⁶·5

7^th semidirect product of C₂⁴ and D₁₀ acting via D₁₀/C₅=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴⋊₈D₁₀, C₁₀.₈₉2₊¹⁺⁴, (C₂×D₄)⋊₄₀D₁₀, (C₂²×D₄)⋊₁₀D₅, (C₂²×C₁₀)⋊₁₃D₄, (C₂²×C₄)⋊₂₈D₁₀, C₂³⋊₄(C₅⋊D₄), C₅⋊₅(C₂³⋊₃D₄), C₂³⋊D₁₀⋊₃₀C₂, (D₄×C₁₀)⋊₅₈C₂², C₂⁴⋊₂D₅⋊₁₂C₂, Dic₅⋊D₄⋊₄₁C₂, (C₂×C₁₀).₂₉₉C₂⁴, (C₂×C₂₀).₆₄₄C₂³, (C₂³×C₁₀)⋊₁₄C₂², (C₂²×C₂₀)⋊₄₄C₂², C₁₀.₁₄₆(C₂²×D₄), (C₂³×D₅)⋊₁₅C₂², C₂³.D₅⋊₆₄C₂², C₂.₉₂(D₄⋊₆D₁₀), D₁₀⋊C₄⋊₃₆C₂², C₁₀.D₄⋊₃₈C₂², C₂³.₂₀₆(C₂²×D₅), C₂².₃₁₂(C₂³×D₅), C₂³.₂₃D₁₀⋊₂₈C₂, C₂³.₁₈D₁₀⋊₂₉C₂, (C₂²×C₁₀).₂₃₃C₂³, (C₂×Dic₅).₁₅₄C₂³, (C₂²×Dic₅)⋊₃₄C₂², (C₂²×D₅).₁₃₀C₂³, (D₄×C₂×C₁₀)⋊₁₇C₂, (C₂×C₁₀).₅₈₂(C₂×D₄), (C₂×C₅⋊D₄)⋊₄₈C₂², (C₂²×C₅⋊D₄)⋊₁₇C₂, (C₂×C₂³.D₅)⋊₃₀C₂, C₂².₂₀(C₂×C₅⋊D₄), C₂.₁₉(C₂²×C₅⋊D₄), (C₂×C₄).₂₃₈(C₂²×D₅), SmallGroup(320,1476)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₂⁴⋊₈D₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×D₅ — C₂³×D₅ — C₂²×C₅⋊D₄ — C₂⁴⋊₈D₁₀

Lower central

C₅ — C₂×C₁₀ — C₂⁴⋊₈D₁₀

Upper central

C₁ — C₂² — C₂²×D₄

Generators and relations for C₂⁴⋊₈D₁₀
G = < a,b,c,d,e,f | a²=b²=c²=d²=e¹⁰=f²=1, ab=ba, ac=ca, faf=ad=da, ae=ea, bc=cb, ebe^-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 1294 in 346 conjugacy classes, 111 normal (25 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, D₅, C₁₀, C₁₀, C₁₀, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₂²⋊C₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₂²×D₄, C₂²×D₄, C₂×Dic₅, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂³⋊₃D₄, C₁₀.D₄, D₁₀⋊C₄, C₂³.D₅, C₂²×Dic₅, C₂²×Dic₅, C₂×C₅⋊D₄, C₂×C₅⋊D₄, C₂²×C₂₀, D₄×C₁₀, D₄×C₁₀, C₂³×D₅, C₂³×C₁₀, C₂³.₂₃D₁₀, C₂³.₁₈D₁₀, C₂³⋊D₁₀, Dic₅⋊D₄, C₂×C₂³.D₅, C₂⁴⋊₂D₅, C₂²×C₅⋊D₄, D₄×C₂×C₁₀, C₂⁴⋊₈D₁₀
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₂⁴, D₁₀, C₂²×D₄, 2₊¹⁺⁴, C₅⋊D₄, C₂²×D₅, C₂³⋊₃D₄, C₂×C₅⋊D₄, C₂³×D₅, D₄⋊₆D₁₀, C₂²×C₅⋊D₄, C₂⁴⋊₈D₁₀

Smallest permutation representation of C₂⁴⋊₈D₁₀
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

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

62 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	2L	2M	4A	4B	4C	···	4H	5A	5B	10A	···	10N	10O	···	10AD	20A	···	20H
order	1	2	2	2	2	···	2	2	2	2	2	4	4	4	···	4	5	5	10	···	10	10	···	10	20	···	20
size	1	1	1	1	2	···	2	4	4	20	20	4	4	20	···	20	2	2	2	···	2	4	···	4	4	···	4

62 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

D₁₀

C₅⋊D₄

2₊¹⁺⁴

D₄⋊₆D₁₀

kernel

C₂⁴⋊₈D₁₀

C₂³.₂₃D₁₀

C₂³.₁₈D₁₀

C₂³⋊D₁₀

Dic₅⋊D₄

C₂×C₂³.D₅

C₂⁴⋊₂D₅

C₂²×C₅⋊D₄

D₄×C₂×C₁₀

C₂²×C₁₀

C₂²×D₄

C₂²×C₄

C₂×D₄

C₂⁴

C₂³

C₁₀

C₂

# reps

Matrix representation of C₂⁴⋊₈D₁₀ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	24	1	0	0
0	0	40	17	0	0
0	0	0	0	23	1
0	0	0	0	5	18

16	32	0	0	0	0
1	25	0	0	0	0
0	0	24	1	38	23
0	0	40	17	36	23
0	0	0	0	18	40
0	0	0	0	36	23

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

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

40	0	0	0	0	0
0	40	0	0	0	0
0	0	7	7	0	0
0	0	34	40	0	0
0	0	0	14	0	34
0	0	27	39	6	35

1	0	0	0	0	0
40	40	0	0	0	0
0	0	7	7	0	0
0	0	40	34	0	0
0	0	0	0	6	34
0	0	0	0	5	35

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,0,0,23,5,0,0,0,0,1,18],[16,1,0,0,0,0,32,25,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,38,36,18,36,0,0,23,23,40,23],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,34,0,27,0,0,7,40,14,39,0,0,0,0,0,6,0,0,0,0,34,35],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,6,5,0,0,0,0,34,35] >;

C₂⁴⋊₈D₁₀ in GAP, Magma, Sage, TeX

C_2^4\rtimes_8D_{10}

% in TeX

G:=Group("C2^4:8D10");

// GroupNames label

G:=SmallGroup(320,1476);

// by ID

G=gap.SmallGroup(320,1476);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,675,570,12550]);

// 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,a*c=c*a,f*a*f=a*d=d*a,a*e=e*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f=b*c*d,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⋊8D10 order 320 = 26·5

7th semidirect product of C24 and D10 acting via D10/C5=C22

G = C₂⁴⋊₈D₁₀ order 320 = 2⁶·5

7^th semidirect product of C₂⁴ and D₁₀ acting via D₁₀/C₅=C₂²