C2xC20.46D4 - GroupNames

G = C₂×C₂₀.₄₆D₄ order 320 = 2⁶·5

Direct product of C₂ and C₂₀.₄₆D₄

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

Aliases: C₂×C₂₀.₄₆D₄, M₄(2)⋊₂₂D₁₀, C₄.₆₅(C₂×D₂₀), (C₂×C₄).₄₉D₂₀, (C₂×D₂₀).₂₇C₄, C₂₀.₄₁₆(C₂×D₄), (C₂×C₂₀).₁₇₂D₄, (C₂³×D₅).₃C₄, C₂³.₅₅(C₄×D₅), C₁₀⋊₂(C₄.D₄), (C₂×M₄(2))⋊₁₀D₅, (C₁₀×M₄(2))⋊₁₈C₂, (C₂×C₂₀).₄₁₆C₂³, (C₂²×D₂₀).₁₅C₂, (C₂²×C₄).₁₃₈D₁₀, C₄.Dic₅⋊₂₁C₂², C₄.₂₈(D₁₀⋊C₄), C₂₀.₁₀₀(C₂²⋊C₄), (C₂×D₂₀).₂₅₈C₂², (C₅×M₄(2))⋊₃₄C₂², (C₂²×C₂₀).₁₈₇C₂², C₂².₅₀(D₁₀⋊C₄), C₅⋊₄(C₂×C₄.D₄), (C₂×C₄).₅₂(C₄×D₅), C₂².₂₀(C₂×C₄×D₅), C₄.₁₀₉(C₂×C₅⋊D₄), (C₂×C₂₀).₂₈₀(C₂×C₄), C₁₀.₉₈(C₂×C₂²⋊C₄), (C₂×C₄.Dic₅)⋊₁₅C₂, (C₂²×D₅).₅(C₂×C₄), C₂.₂₉(C₂×D₁₀⋊C₄), (C₂×C₄).₂₅₆(C₅⋊D₄), (C₂×C₄).₁₂₀(C₂²×D₅), (C₂²×C₁₀).₁₃₈(C₂×C₄), (C₂×C₁₀).₁₁₅(C₂²×C₄), (C₂×C₁₀).₁₂₉(C₂²⋊C₄), SmallGroup(320,757)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₂×C₂₀.₄₆D₄

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₂×C₂₀ — C₂×D₂₀ — C₂²×D₂₀ — C₂×C₂₀.₄₆D₄

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂₀.₄₆D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂×M₄(2)

Generators and relations for C₂×C₂₀.₄₆D₄
G = < a,b,c,d | a²=b²⁰=d²=1, c⁴=b¹⁰, ab=ba, ac=ca, ad=da, cbc^-1=dbd=b^-1, dcd=b¹⁵c³ >

Subgroups: 958 in 186 conjugacy classes, 63 normal (39 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, D₄, C₂³, C₂³, D₅, C₁₀, C₁₀, C₁₀, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂×D₄, C₂⁴, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₄.D₄, C₂×M₄(2), C₂×M₄(2), C₂²×D₄, C₅⋊₂C₈, C₄₀, D₂₀, C₂×C₂₀, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂×C₄.D₄, C₂×C₅⋊₂C₈, C₄.Dic₅, C₄.Dic₅, C₂×C₄₀, C₅×M₄(2), C₅×M₄(2), C₂×D₂₀, C₂×D₂₀, C₂²×C₂₀, C₂³×D₅, C₂₀.₄₆D₄, C₂×C₄.Dic₅, C₁₀×M₄(2), C₂²×D₂₀, C₂×C₂₀.₄₆D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, D₅, C₂²⋊C₄, C₂²×C₄, C₂×D₄, D₁₀, C₄.D₄, C₂×C₂²⋊C₄, C₄×D₅, D₂₀, C₅⋊D₄, C₂²×D₅, C₂×C₄.D₄, D₁₀⋊C₄, C₂×C₄×D₅, C₂×D₂₀, C₂×C₅⋊D₄, C₂₀.₄₆D₄, C₂×D₁₀⋊C₄, C₂×C₂₀.₄₆D₄

Smallest permutation representation of C₂×C₂₀.₄₆D₄
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

62 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	5A	5B	8A	8B	8C	8D	8E	8F	8G	8H	10A	···	10F	10G	10H	10I	10J	20A	···	20H	20I	20J	20K	20L	40A	···	40P
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	5	5	8	8	8	8	8	8	8	8	10	···	10	10	10	10	10	20	···	20	20	20	20	20	40	···	40
size	1	1	1	1	2	2	20	20	20	20	2	2	2	2	2	2	4	4	4	4	20	20	20	20	2	···	2	4	4	4	4	2	···	2	4	4	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	C₂	C₄	C₄	D₄	D₅	D₁₀	D₁₀	C₄×D₅	D₂₀	C₅⋊D₄	C₄×D₅	C₄.D₄	C₂₀.₄₆D₄
kernel	C₂×C₂₀.₄₆D₄	C₂₀.₄₆D₄	C₂×C₄.Dic₅	C₁₀×M₄(2)	C₂²×D₂₀	C₂×D₂₀	C₂³×D₅	C₂×C₂₀	C₂×M₄(2)	M₄(2)	C₂²×C₄	C₂×C₄	C₂×C₄	C₂×C₄	C₂³	C₁₀	C₂
# reps	1	4	1	1	1	4	4	4	2	4	2	4	8	8	4	2	8

Matrix representation of C₂×C₂₀.₄₆D₄ ►in GL₆(𝔽₄₁)

40	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
0	0	0	0	0	40

6	1	0	0	0	0
40	0	0	0	0	0
0	0	9	11	0	0
0	0	30	14	0	0
0	0	0	0	39	11
0	0	0	0	14	25

6	35	0	0	0	0
40	35	0	0	0	0
0	0	2	5	4	9
0	0	32	39	13	28
0	0	4	21	12	8
0	0	38	17	23	29

6	35	0	0	0	0
40	35	0	0	0	0
0	0	11	9	0	0
0	0	14	30	0	0
0	0	29	39	25	30
0	0	29	10	12	16

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[6,40,0,0,0,0,1,0,0,0,0,0,0,0,9,30,0,0,0,0,11,14,0,0,0,0,0,0,39,14,0,0,0,0,11,25],[6,40,0,0,0,0,35,35,0,0,0,0,0,0,2,32,4,38,0,0,5,39,21,17,0,0,4,13,12,23,0,0,9,28,8,29],[6,40,0,0,0,0,35,35,0,0,0,0,0,0,11,14,29,29,0,0,9,30,39,10,0,0,0,0,25,12,0,0,0,0,30,16] >;

C₂×C₂₀.₄₆D₄ in GAP, Magma, Sage, TeX

C_2\times C_{20}._{46}D_4

% in TeX

G:=Group("C2xC20.46D4");

// GroupNames label

G:=SmallGroup(320,757);

// by ID

G=gap.SmallGroup(320,757);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,58,1123,136,438,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=d^2=1,c^4=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=b^15*c^3>;

// generators/relations

G = C2×C20.46D4 order 320 = 26·5

Direct product of C2 and C20.46D4

G = C₂×C₂₀.₄₆D₄ order 320 = 2⁶·5

Direct product of C₂ and C₂₀.₄₆D₄