C2xD5:C8 - GroupNames

G = C₂×D₅⋊C₈ order 160 = 2⁵·5

Direct product of C₂ and D₅⋊C₈

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

Aliases: C₂×D₅⋊C₈, D₁₀⋊₃C₈, Dic₅.₉C₂³, D₅⋊(C₂×C₈), C₁₀⋊₁(C₂×C₈), C₅⋊C₈⋊₄C₂², C₅⋊₁(C₂²×C₈), (C₄×D₅).₇C₄, C₄.₂₀(C₂×F₅), (C₂×C₄).₁₁F₅, (C₂×C₂₀).₁₁C₄, C₂₀.₁₉(C₂×C₄), C₂.₁(C₂²×F₅), D₁₀.₁₂(C₂×C₄), C₁₀.₁(C₂²×C₄), (C₂²×D₅).₇C₄, C₂².₁₅(C₂×F₅), Dic₅.₁₄(C₂×C₄), (C₄×D₅).₃₃C₂², (C₂×Dic₅).₅₄C₂², (C₂×C₅⋊C₈)⋊₆C₂, (C₂×C₄×D₅).₁₇C₂, (C₂×C₁₀).₁₃(C₂×C₄), SmallGroup(160,200)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅ — C₂×D₅⋊C₈

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₅⋊C₈ — C₂×C₅⋊C₈ — C₂×D₅⋊C₈

Lower central

C₅ — C₂×D₅⋊C₈

Upper central

C₁ — C₂×C₄

Generators and relations for C₂×D₅⋊C₈
G = < a,b,c,d | a²=b⁵=c²=d⁸=1, ab=ba, ac=ca, ad=da, cbc=b^-1, dbd^-1=b³, dcd^-1=b²c >

Subgroups: 196 in 76 conjugacy classes, 46 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, C₂³, D₅, C₁₀, C₁₀, C₂×C₈, C₂²×C₄, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂²×C₈, C₅⋊C₈, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂²×D₅, D₅⋊C₈, C₂×C₅⋊C₈, C₂×C₄×D₅, C₂×D₅⋊C₈
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, C₂³, C₂×C₈, C₂²×C₄, F₅, C₂²×C₈, C₂×F₅, D₅⋊C₈, C₂²×F₅, C₂×D₅⋊C₈

Smallest permutation representation of C₂×D₅⋊C₈
►On 80 points

Generators in S₈₀

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

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

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

(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)

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

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

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

C₂×D₅⋊C₈ is a maximal subgroup of
D₁₀.₁₈D₈ D₁₀.₃M₄(2) M₄(2).F₅ C₄².₅F₅ C₄².₁₁F₅ C₅⋊C₈⋊₈D₄ D₁₀⋊M₄(2) D₁₀.C₄² D₂₀⋊₂C₈ D₁₀⋊₂M₄(2) C₄⋊C₄.₉F₅ C₂×C₈×F₅ M₄(2)⋊₅F₅ D₁₀.₁₁M₄(2) (C₂×D₄).₈F₅ (C₂×Q₈).₅F₅ C₄○D₂₀⋊C₄ Dic₅.₂₁C₂⁴
C₂×D₅⋊C₈ is a maximal quotient of
C₄².₆F₅ C₄².₁₁F₅ C₄².₁₂F₅ C₅⋊C₈⋊₈D₄ D₂₀⋊₂C₈ Dic₁₀⋊C₈ D₅⋊M₅(2) Dic₁₀.C₈ C₂×C₄×C₅⋊C₈ D₁₀.₁₁M₄(2) C₂₀.₃₄M₄(2)

40 conjugacy classes

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

40 irreducible representations

dim	1	1	1	1	1	1	1	1	4	4	4	4
type	+	+	+	+					+	+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	C₈	F₅	C₂×F₅	C₂×F₅	D₅⋊C₈
kernel	C₂×D₅⋊C₈	D₅⋊C₈	C₂×C₅⋊C₈	C₂×C₄×D₅	C₄×D₅	C₂×C₂₀	C₂²×D₅	D₁₀	C₂×C₄	C₄	C₂²	C₂
# reps	1	4	2	1	4	2	2	16	1	2	1	4

Matrix representation of C₂×D₅⋊C₈ ►in GL₅(𝔽₄₁)

40	0	0	0	0
0	40	0	0	0
0	0	40	0	0
0	0	0	40	0
0	0	0	0	40

1	0	0	0	0
0	0	0	0	40
0	1	0	0	40
0	0	1	0	40
0	0	0	1	40

1	0	0	0	0
0	0	0	40	1
0	0	40	0	1
0	40	0	0	1
0	0	0	0	1

40	0	0	0	0
0	11	30	12	0
0	23	30	0	11
0	11	0	30	23
0	0	12	30	11

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40,40,40],[1,0,0,0,0,0,0,0,40,0,0,0,40,0,0,0,40,0,0,0,0,1,1,1,1],[40,0,0,0,0,0,11,23,11,0,0,30,30,0,12,0,12,0,30,30,0,0,11,23,11] >;

C₂×D₅⋊C₈ in GAP, Magma, Sage, TeX

C_2\times D_5\rtimes C_8

% in TeX

G:=Group("C2xD5:C8");

// GroupNames label

G:=SmallGroup(160,200);

// by ID

G=gap.SmallGroup(160,200);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,86,69,2309,599]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^2*c>;

// generators/relations

G = C2×D5⋊C8 order 160 = 25·5

Direct product of C2 and D5⋊C8

G = C₂×D₅⋊C₈ order 160 = 2⁵·5

Direct product of C₂ and D₅⋊C₈