(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^[×3], C₂^[×3], C₄^[×5], C₂²^[×3], C₂²^[×5], C₅, C₈, C₂×C₄^[×2], C₂×C₄^[×6], D₄^[×3], C₂³, C₂³, D₅, C₁₀^[×3], C₁₀^[×2], C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄^[×2], Dic₅, C₂₀^[×4], D₁₀^[×3], C₂×C₁₀^[×3], C₂×C₁₀^[×2], C₂.C₄², C₂²⋊C₈, C₄⋊D₄, C₅⋊₂C₈, D₂₀, C₂×Dic₅, C₅⋊D₄^[×2], C₂×C₂₀^[×2], C₂×C₂₀^[×5], 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^[×3], C₄^[×2], C₂², C₂×C₄, D₄^[×2], 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 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 61)(20 62)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 41)

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

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

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

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

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

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