C2xC2^3:Dic5 - GroupNames

G = C₂×C₂³⋊Dic₅ order 320 = 2⁶·5

Direct product of C₂ and C₂³⋊Dic₅

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂×C₂³⋊Dic₅, C₂⁴⋊₃Dic₅, C₂⁴.₁₇D₁₀, (D₄×C₁₀)⋊₂₅C₄, (C₂³×C₁₀)⋊₄C₄, (C₂×D₄)⋊₅Dic₅, C₁₀⋊₄(C₂³⋊C₄), (C₂²×C₂₀)⋊₁₇C₄, (C₂²×D₄).₅D₅, C₂³⋊₁(C₂×Dic₅), (C₂²×C₄)⋊₃Dic₅, (C₂×D₄).₂₀₀D₁₀, (C₂²×C₁₀).₁₀₈D₄, C₂³.₆₄(C₅⋊D₄), C₂³.D₅⋊₄₃C₂², C₂³.₇₆(C₂²×D₅), (D₄×C₁₀).₂₈₀C₂², (C₂³×C₁₀).₄₅C₂², C₂².₁(C₂³.D₅), C₂².₆(C₂²×Dic₅), (C₂²×C₁₀).₁₁₅C₂³, C₅⋊₆(C₂×C₂³⋊C₄), (C₂×C₂₀)⋊₁₅(C₂×C₄), (D₄×C₂×C₁₀).₁₀C₂, (C₂×C₄)⋊₁(C₂×Dic₅), (C₂²×C₁₀)⋊₇(C₂×C₄), (C₂×C₁₀).₃₈(C₂×D₄), (C₂×C₂³.D₅)⋊₈C₂, C₂².₁₀(C₂×C₅⋊D₄), C₂.₁₀(C₂×C₂³.D₅), C₁₀.₁₁₅(C₂×C₂²⋊C₄), (C₂×C₁₀).₂₉₆(C₂²×C₄), (C₂×C₁₀).₁₇₇(C₂²⋊C₄), SmallGroup(320,846)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₂×C₂³⋊Dic₅

Chief series

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

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂³⋊Dic₅

Upper central

C₁ — C₂² — C₂⁴ — C₂²×D₄

Generators and relations for C₂×C₂³⋊Dic₅
G = < a,b,c,d,e,f | a²=b²=c²=d²=e¹⁰=1, f²=e⁵, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe^-1=bd=db, fbf^-1=bcd, fcf^-1=cd=dc, ce=ec, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 670 in 210 conjugacy classes, 71 normal (25 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₁₀, C₁₀, C₁₀, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂²×D₄, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂×C₂³⋊C₄, C₂³.D₅, C₂³.D₅, C₂²×Dic₅, C₂²×C₂₀, D₄×C₁₀, D₄×C₁₀, C₂³×C₁₀, C₂³⋊Dic₅, C₂×C₂³.D₅, D₄×C₂×C₁₀, C₂×C₂³⋊Dic₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, D₅, C₂²⋊C₄, C₂²×C₄, C₂×D₄, Dic₅, D₁₀, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂×Dic₅, C₅⋊D₄, C₂²×D₅, C₂×C₂³⋊C₄, C₂³.D₅, C₂²×Dic₅, C₂×C₅⋊D₄, C₂³⋊Dic₅, C₂×C₂³.D₅, C₂×C₂³⋊Dic₅

Smallest permutation representation of C₂×C₂³⋊Dic₅
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+				+	+	-	-	+	-	+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₅	Dic₅	Dic₅	D₁₀	Dic₅	D₁₀	C₅⋊D₄	C₂³⋊C₄	C₂³⋊Dic₅
kernel	C₂×C₂³⋊Dic₅	C₂³⋊Dic₅	C₂×C₂³.D₅	D₄×C₂×C₁₀	C₂²×C₂₀	D₄×C₁₀	C₂³×C₁₀	C₂²×C₁₀	C₂²×D₄	C₂²×C₄	C₂×D₄	C₂×D₄	C₂⁴	C₂⁴	C₂³	C₁₀	C₂
# reps	1	4	2	1	2	4	2	4	2	2	4	4	2	2	16	2	8

Matrix representation of C₂×C₂³⋊Dic₅ ►in GL₆(𝔽₄₁)

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	2	0	0	0	0
0	40	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	39	0	40	0
0	0	0	39	0	40

40	0	0	0	0	0
0	40	0	0	0	0
0	0	18	6	0	0
0	0	35	23	0	0
0	0	0	0	18	6
0	0	0	0	35	23

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	6	18	6	18
0	0	23	21	23	21
0	0	0	0	35	23
0	0	0	0	18	20

9	18	0	0	0	0
32	32	0	0	0	0
0	0	1	6	12	34
0	0	0	40	3	29
0	0	39	29	40	35
0	0	0	2	0	1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,2,40,0,0,0,0,0,0,1,0,39,0,0,0,0,1,0,39,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,18,35,0,0,0,0,6,23,0,0,0,0,0,0,18,35,0,0,0,0,6,23],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,23,0,0,0,0,18,21,0,0,0,0,6,23,35,18,0,0,18,21,23,20],[9,32,0,0,0,0,18,32,0,0,0,0,0,0,1,0,39,0,0,0,6,40,29,2,0,0,12,3,40,0,0,0,34,29,35,1] >;

C₂×C₂³⋊Dic₅ in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes {\rm Dic}_5

% in TeX

G:=Group("C2xC2^3:Dic5");

// GroupNames label

G:=SmallGroup(320,846);

// by ID

G=gap.SmallGroup(320,846);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,297,1684,12550]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^10=1,f^2=e^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;

// generators/relations

G = C2×C23⋊Dic5 order 320 = 26·5

Direct product of C2 and C23⋊Dic5

G = C₂×C₂³⋊Dic₅ order 320 = 2⁶·5

Direct product of C₂ and C₂³⋊Dic₅