(C2xD20):C4 - GroupNames

G = (C₂×D₂₀)⋊C₄ order 320 = 2⁶·5

1^st semidirect product of C₂×D₂₀ and C₄ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×D₂₀)⋊₁C₄, C₄⋊Dic₅⋊₁C₄, C₁₀.₁₃C₄≀C₂, (C₂×C₁₀).₂₉D₈, (C₂×C₄).₁₀₃D₂₀, C₂₀⋊₇D₄.₇C₂, (C₂×C₂₀).₂₂₁D₄, C₂₀.₅₅D₄⋊₁C₂, C₂².₇(D₄⋊D₅), (C₂×C₁₀).₃₇SD₁₆, C₂.C₄²⋊₆D₅, (C₂²×C₄).₅₂D₁₀, C₅⋊₃(C₂².SD₁₆), C₂².₇(Q₈⋊D₅), C₂.₃(D₂₀⋊₆C₄), C₂.₄(D₂₀⋊₄C₄), C₁₀.₂₂(C₂³⋊C₄), (C₂²×C₁₀).₁₇₄D₄, C₂³.₇₀(C₅⋊D₄), C₁₀.₁₆(D₄⋊C₄), (C₂²×C₂₀).₈₉C₂², C₂.₄(C₂³.₁D₁₀), C₂².₅₄(D₁₀⋊C₄), (C₂×C₄).₈(C₄×D₅), (C₂×C₂₀).₁₉₃(C₂×C₄), (C₂×C₁₀).₉₇(C₂²⋊C₄), (C₅×C₂.C₄²)⋊₁₃C₂, SmallGroup(320,9)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — (C₂×D₂₀)⋊C₄

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — C₂²×C₂₀ — C₂₀⋊₇D₄ — (C₂×D₂₀)⋊C₄

Lower central

C₅ — C₂×C₁₀ — C₂×C₂₀ — (C₂×D₂₀)⋊C₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂.C₄²

Generators and relations for (C₂×D₂₀)⋊C₄
G = < a,b,c,d | a²=b²⁰=c²=d⁴=1, ab=ba, ac=ca, ad=da, cbc=b^-1, dbd^-1=ab¹¹, dcd^-1=b¹⁵c >

Subgroups: 430 in 90 conjugacy classes, 29 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, D₅, C₁₀, C₁₀, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂.C₄², C₂²⋊C₈, C₄⋊D₄, C₅⋊₂C₈, D₂₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₂²×D₅, C₂²×C₁₀, C₂².SD₁₆, C₂×C₅⋊₂C₈, C₄⋊Dic₅, D₁₀⋊C₄, C₂×D₂₀, C₂×C₅⋊D₄, C₂²×C₂₀, C₂²×C₂₀, C₂₀.₅₅D₄, C₅×C₂.C₄², C₂₀⋊₇D₄, (C₂×D₂₀)⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₅, C₂²⋊C₄, D₈, SD₁₆, D₁₀, C₂³⋊C₄, D₄⋊C₄, C₄≀C₂, C₄×D₅, D₂₀, C₅⋊D₄, C₂².SD₁₆, D₁₀⋊C₄, D₄⋊D₅, Q₈⋊D₅, D₂₀⋊₄C₄, C₂³.₁D₁₀, D₂₀⋊₆C₄, (C₂×D₂₀)⋊C₄

Smallest permutation representation of (C₂×D₂₀)⋊C₄
►On 80 points

Generators in S₈₀

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

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

(2 70)(4 72)(6 74)(8 76)(10 78)(12 80)(14 62)(16 64)(18 66)(20 68)(21 45 60 26)(22 37 41 56)(23 47 42 28)(24 39 43 58)(25 49 44 30)(27 51 46 32)(29 53 48 34)(31 55 50 36)(33 57 52 38)(35 59 54 40)

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

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

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

59 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	···	4G	4H	5A	5B	8A	8B	8C	8D	10A	···	10N	20A	···	20X
order	1	2	2	2	2	2	2	4	4	4	···	4	4	5	5	8	8	8	8	10	···	10	20	···	20
size	1	1	1	1	2	2	40	2	2	4	···	4	40	2	2	20	20	20	20	2	···	2	4	···	4

59 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+			+	+	+	+		+			+			+	+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₄	D₅	D₈	SD₁₆	D₁₀	C₄≀C₂	C₄×D₅	D₂₀	C₅⋊D₄	D₂₀⋊₄C₄	C₂³⋊C₄	D₄⋊D₅	Q₈⋊D₅	C₂³.₁D₁₀
kernel	(C₂×D₂₀)⋊C₄	C₂₀.₅₅D₄	C₅×C₂.C₄²	C₂₀⋊₇D₄	C₄⋊Dic₅	C₂×D₂₀	C₂×C₂₀	C₂²×C₁₀	C₂.C₄²	C₂×C₁₀	C₂×C₁₀	C₂²×C₄	C₁₀	C₂×C₄	C₂×C₄	C₂³	C₂	C₁₀	C₂²	C₂²	C₂
# reps	1	1	1	1	2	2	1	1	2	2	2	2	4	4	4	4	16	1	2	2	4

Matrix representation of (C₂×D₂₀)⋊C₄ ►in GL₆(𝔽₄₁)

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	1	0
0	0	0	0	0	1

32	26	0	0	0	0
0	9	0	0	0	0
0	0	9	0	0	0
0	0	38	32	0	0
0	0	0	0	10	0
0	0	0	0	29	37

14	16	0	0	0	0
16	27	0	0	0	0
0	0	38	23	0	0
0	0	5	3	0	0
0	0	0	0	5	40
0	0	0	0	24	36

1	0	0	0	0	0
7	40	0	0	0	0
0	0	1	0	0	0
0	0	15	9	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,0,0,0,0,0,26,9,0,0,0,0,0,0,9,38,0,0,0,0,0,32,0,0,0,0,0,0,10,29,0,0,0,0,0,37],[14,16,0,0,0,0,16,27,0,0,0,0,0,0,38,5,0,0,0,0,23,3,0,0,0,0,0,0,5,24,0,0,0,0,40,36],[1,7,0,0,0,0,0,40,0,0,0,0,0,0,1,15,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

(C₂×D₂₀)⋊C₄ in GAP, Magma, Sage, TeX

(C_2\times D_{20})\rtimes C_4

% in TeX

G:=Group("(C2xD20):C4");

// GroupNames label

G:=SmallGroup(320,9);

// by ID

G=gap.SmallGroup(320,9);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,422,1571,570,192,12550]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=a*b^11,d*c*d^-1=b^15*c>;

// generators/relations

G = (C2×D20)⋊C4 order 320 = 26·5

1st semidirect product of C2×D20 and C4 acting faithfully

G = (C₂×D₂₀)⋊C₄ order 320 = 2⁶·5

1^st semidirect product of C₂×D₂₀ and C₄ acting faithfully