C2^3:2Dic10 - GroupNames

G = C₂³⋊₂Dic₁₀ order 320 = 2⁶·5

2^nd semidirect product of C₂³ and Dic₁₀ acting via Dic₁₀/C₁₀=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³⋊₂Dic₁₀, C₂⁴.₂₃D₁₀, C₁₀.₁2₊¹⁺⁴, (C₂²×C₁₀)⋊₅Q₈, C₄⋊Dic₅⋊₃C₂², C₅⋊₁(C₂³⋊₂Q₈), C₂₀.₄₈D₄⋊₃C₂, C₁₀.₆(C₂²×Q₈), (C₂×C₁₀).₂₇C₂⁴, C₂²⋊C₄.₈₆D₁₀, C₂.₆(D₄⋊₆D₁₀), (C₂×C₂₀).₁₂₇C₂³, (C₂×Dic₁₀)⋊₂C₂², (C₂²×C₄).₁₆₉D₁₀, C₁₀.D₄⋊₁C₂², (C₂×Dic₅).₈C₂³, C₂².₅(C₂×Dic₁₀), C₂.₈(C₂²×Dic₁₀), C₂².₆₉(C₂³×D₅), Dic₅.₁₄D₄⋊₁C₂, (C₂³×C₁₀).₅₃C₂², (C₂²×C₂₀).₇₁C₂², C₂³.₁₄₄(C₂²×D₅), C₂³.D₅.₈₅C₂², (C₂²×C₁₀).₁₁₉C₂³, (C₂²×Dic₅).₇₆C₂², (C₂×C₁₀).₄₉(C₂×Q₈), (C₂×C₂²⋊C₄).₁₈D₅, (C₁₀×C₂²⋊C₄).₁₈C₂, (C₂×C₄).₁₃₃(C₂²×D₅), (C₂×C₂³.D₅).₂₂C₂, (C₅×C₂²⋊C₄).₉₇C₂², SmallGroup(320,1155)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₂³⋊₂Dic₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×Dic₅ — C₂²×Dic₅ — C₂×C₂³.D₅ — C₂³⋊₂Dic₁₀

Lower central

C₅ — C₂×C₁₀ — C₂³⋊₂Dic₁₀

Upper central

C₁ — C₂² — C₂×C₂²⋊C₄

Generators and relations for C₂³⋊₂Dic₁₀
G = < a,b,c,d,e | a²=b²=c²=d²⁰=1, e²=d¹⁰, ab=ba, dad^-1=ac=ca, ae=ea, ebe^-1=bc=cb, bd=db, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 782 in 242 conjugacy classes, 111 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂²⋊Q₈, Dic₁₀, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂³⋊₂Q₈, C₁₀.D₄, C₄⋊Dic₅, C₂³.D₅, C₅×C₂²⋊C₄, C₂×Dic₁₀, C₂²×Dic₅, C₂²×C₂₀, C₂³×C₁₀, Dic₅.₁₄D₄, C₂₀.₄₈D₄, C₂×C₂³.D₅, C₁₀×C₂²⋊C₄, C₂³⋊₂Dic₁₀
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₅, C₂×Q₈, C₂⁴, D₁₀, C₂²×Q₈, 2₊¹⁺⁴, Dic₁₀, C₂²×D₅, C₂³⋊₂Q₈, C₂×Dic₁₀, C₂³×D₅, C₂²×Dic₁₀, D₄⋊₆D₁₀, C₂³⋊₂Dic₁₀

Smallest permutation representation of C₂³⋊₂Dic₁₀
►On 80 points

Generators in S₈₀

(2 53)(4 55)(6 57)(8 59)(10 41)(12 43)(14 45)(16 47)(18 49)(20 51)(21 73)(23 75)(25 77)(27 79)(29 61)(31 63)(33 65)(35 67)(37 69)(39 71)

(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)

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

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

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

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

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

62 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	4A	4B	4C	4D	4E	···	4L	5A	5B	10A	···	10N	10O	···	10V	20A	···	20P
order	1	2	2	2	2	···	2	4	4	4	4	4	···	4	5	5	10	···	10	10	···	10	20	···	20
size	1	1	1	1	2	···	2	4	4	4	4	20	···	20	2	2	2	···	2	4	···	4	4	···	4

62 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+	+	-	+	+	+	+	-	+
image	C₁	C₂	C₂	C₂	C₂	Q₈	D₅	D₁₀	D₁₀	D₁₀	Dic₁₀	2₊¹⁺⁴	D₄⋊₆D₁₀
kernel	C₂³⋊₂Dic₁₀	Dic₅.₁₄D₄	C₂₀.₄₈D₄	C₂×C₂³.D₅	C₁₀×C₂²⋊C₄	C₂²×C₁₀	C₂×C₂²⋊C₄	C₂²⋊C₄	C₂²×C₄	C₂⁴	C₂³	C₁₀	C₂
# reps	1	8	4	2	1	4	2	8	4	2	16	2	8

Matrix representation of C₂³⋊₂Dic₁₀ ►in GL₆(𝔽₄₁)

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

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	0	40	0
0	0	0	0	0	40

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

0	40	0	0	0	0
1	0	0	0	0	0
0	0	0	18	0	0
0	0	23	0	0	0
0	0	0	0	0	25
0	0	0	0	16	0

1	30	0	0	0	0
30	40	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	40	0	0	0
0	0	0	40	0	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,23,0,0,0,0,18,0,0,0,0,0,0,0,0,16,0,0,0,0,25,0],[1,30,0,0,0,0,30,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂³⋊₂Dic₁₀ in GAP, Magma, Sage, TeX

C_2^3\rtimes_2{\rm Dic}_{10}

% in TeX

G:=Group("C2^3:2Dic10");

// GroupNames label

G:=SmallGroup(320,1155);

// by ID

G=gap.SmallGroup(320,1155);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,758,675,570,80,12550]);

// Polycyclic

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

// generators/relations

G = C23⋊2Dic10 order 320 = 26·5

2nd semidirect product of C23 and Dic10 acting via Dic10/C10=C22

G = C₂³⋊₂Dic₁₀ order 320 = 2⁶·5

2^nd semidirect product of C₂³ and Dic₁₀ acting via Dic₁₀/C₁₀=C₂²