(C2xC40):C4 - GroupNames

G = (C₂×C₄₀)⋊C₄ order 320 = 2⁶·5

11^st semidirect product of C₂×C₄₀ and C₄ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×C₄₀)⋊₁₁C₄, (C₂×C₈)⋊₁Dic₅, (C₂×C₄).₁₄D₂₀, C₂₀.₄₂(C₄⋊C₄), (C₂×C₂₀).₁₁Q₈, C₂³.₉(C₄×D₅), (C₂×C₂₀).₁₀₉D₄, C₄.Dic₅⋊₁₂C₄, (C₂×C₄).₅Dic₁₀, C₂³.D₅.₈C₄, (C₂×C₁₀).₄₂C₄², C₄.₁₃(C₄⋊Dic₅), (C₂²×C₄).₆₂D₁₀, (C₂×M₄(2)).₇D₅, C₂₀.₅₈(C₂²⋊C₄), C₅⋊₅(M₄(2)⋊₄C₄), C₄.₂₈(C₂³.D₅), C₄.₃₅(D₁₀⋊C₄), C₂².₁₀(C₄×Dic₅), C₄.₁₁(C₁₀.D₄), (C₁₀×M₄(2)).₁₁C₂, (C₂²×C₂₀).₁₂₅C₂², C₂³.₂₁D₁₀.₉C₂, C₂².₆(C₁₀.D₄), C₂².₁₉(D₁₀⋊C₄), C₁₀.₃₄(C₂.C₄²), C₂.₁₅(C₁₀.₁₀C₄²), (C₂×C₄).₂₁(C₄×D₅), (C₂×C₁₀).₃₆(C₄⋊C₄), (C₂×C₂₀).₄₇₇(C₂×C₄), (C₂×C₄).₇₆(C₂×Dic₅), (C₂×C₄).₁₈₀(C₅⋊D₄), (C₂×C₄.Dic₅).₁₁C₂, (C₂×C₁₀).₇₄(C₂²⋊C₄), (C₂²×C₁₀).₁₀₀(C₂×C₄), SmallGroup(320,114)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — (C₂×C₄₀)⋊C₄

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂₀ — C₂²×C₂₀ — C₂³.₂₁D₁₀ — (C₂×C₄₀)⋊C₄

Lower central

C₅ — C₁₀ — C₂×C₁₀ — (C₂×C₄₀)⋊C₄

Upper central

C₁ — C₄ — C₂²×C₄ — C₂×M₄(2)

Generators and relations for (C₂×C₄₀)⋊C₄
G = < a,b,c | a²=b⁴⁰=c⁴=1, ab=ba, cac^-1=ab²⁰, cbc^-1=ab²⁹ >

Subgroups: 262 in 90 conjugacy classes, 47 normal (39 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂³, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₄²⋊C₂, C₂×M₄(2), C₂×M₄(2), C₅⋊₂C₈, C₄₀, C₂×Dic₅, C₂×C₂₀, C₂²×C₁₀, M₄(2)⋊₄C₄, C₂×C₅⋊₂C₈, C₄.Dic₅, C₄.Dic₅, C₄×Dic₅, C₄⋊Dic₅, C₂³.D₅, C₂×C₄₀, C₅×M₄(2), C₂²×C₂₀, C₂×C₄.Dic₅, C₂³.₂₁D₁₀, C₁₀×M₄(2), (C₂×C₄₀)⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, D₅, C₄², C₂²⋊C₄, C₄⋊C₄, Dic₅, D₁₀, C₂.C₄², Dic₁₀, C₄×D₅, D₂₀, C₂×Dic₅, C₅⋊D₄, M₄(2)⋊₄C₄, C₄×Dic₅, C₁₀.D₄, C₄⋊Dic₅, D₁₀⋊C₄, C₂³.D₅, C₁₀.₁₀C₄², (C₂×C₄₀)⋊C₄

Smallest permutation representation of (C₂×C₄₀)⋊C₄
►On 80 points

Generators in S₈₀

(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(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)

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

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

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

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

62 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	4I	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	4	4	4	4	4	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	2	2	2	1	1	2	2	2	20	20	20	20	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	2	2	4	4
type	+	+	+	+				+	-	+	-	+	-		+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	D₄	Q₈	D₅	Dic₅	D₁₀	Dic₁₀	C₄×D₅	D₂₀	C₅⋊D₄	C₄×D₅	M₄(2)⋊₄C₄	(C₂×C₄₀)⋊C₄
kernel	(C₂×C₄₀)⋊C₄	C₂×C₄.Dic₅	C₂³.₂₁D₁₀	C₁₀×M₄(2)	C₄.Dic₅	C₂³.D₅	C₂×C₄₀	C₂×C₂₀	C₂×C₂₀	C₂×M₄(2)	C₂×C₈	C₂²×C₄	C₂×C₄	C₂×C₄	C₂×C₄	C₂×C₄	C₂³	C₅	C₁
# reps	1	1	1	1	4	4	4	3	1	2	4	2	4	4	4	8	4	2	8

Matrix representation of (C₂×C₄₀)⋊C₄ ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	40	0	0	0
0	0	37	1	12	29
0	0	0	0	0	1
0	0	0	0	1	0

18	21	0	0	0	0
20	21	0	0	0	0
0	0	15	0	6	35
0	0	0	0	0	1
0	0	5	9	26	15
0	0	0	9	0	0

32	0	0	0	0	0
28	9	0	0	0	0
0	0	1	20	35	0
0	0	0	40	0	0
0	0	0	0	0	40
0	0	0	0	1	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,37,0,0,0,0,0,1,0,0,0,0,0,12,0,1,0,0,0,29,1,0],[18,20,0,0,0,0,21,21,0,0,0,0,0,0,15,0,5,0,0,0,0,0,9,9,0,0,6,0,26,0,0,0,35,1,15,0],[32,28,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,20,40,0,0,0,0,35,0,0,1,0,0,0,0,40,0] >;

(C₂×C₄₀)⋊C₄ in GAP, Magma, Sage, TeX

(C_2\times C_{40})\rtimes C_4

% in TeX

G:=Group("(C2xC40):C4");

// GroupNames label

G:=SmallGroup(320,114);

// by ID

G=gap.SmallGroup(320,114);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,136,1684,12550]);

// Polycyclic

G:=Group<a,b,c|a^2=b^40=c^4=1,a*b=b*a,c*a*c^-1=a*b^20,c*b*c^-1=a*b^29>;

// generators/relations

G = (C2×C40)⋊C4 order 320 = 26·5

11st semidirect product of C2×C40 and C4 acting faithfully

G = (C₂×C₄₀)⋊C₄ order 320 = 2⁶·5

11^st semidirect product of C₂×C₄₀ and C₄ acting faithfully