C2xC8:F5 - GroupNames

G = C₂×C₈⋊F₅ order 320 = 2⁶·5

Direct product of C₂ and C₈⋊F₅

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

Aliases: C₂×C₈⋊F₅, C₂₀.₃₀C₄², D₁₀.₁₂C₄², D₁₀.₄M₄(2), Dic₅.₁₀C₄², C₈⋊₉(C₂×F₅), (C₂×C₈)⋊₈F₅, (C₂×C₄₀)⋊₉C₄, D₅⋊C₈⋊₆C₄, C₄₀⋊₁₀(C₂×C₄), (C₈×D₅)⋊₁₂C₄, (C₄×F₅).₅C₄, C₄.₂₃(C₄×F₅), C₁₀⋊₁(C₈⋊C₄), D₅⋊₂(C₈⋊C₄), (C₂²×F₅).₃C₄, C₄.₄₉(C₂²×F₅), C₂².₁₉(C₄×F₅), (C₂×C₁₀).₁₇C₄², C₁₀.₁₂(C₂×C₄²), C₂₀.₈₉(C₂²×C₄), D₅⋊C₈.₁₇C₂², (C₄×D₅).₈₆C₂³, (C₈×D₅).₆₅C₂², D₅.₁(C₂×M₄(2)), (C₄×F₅).₁₆C₂², D₁₀.₃₂(C₂²×C₄), Dic₅.₃₁(C₂²×C₄), (C₂×C₅⋊C₈)⋊₆C₄, C₅⋊C₈⋊₅(C₂×C₄), C₅⋊₂(C₂×C₈⋊C₄), C₂.₁₃(C₂×C₄×F₅), (D₅×C₂×C₈).₃₃C₂, (C₂×C₅⋊₂C₈)⋊₂₃C₄, C₅⋊₂C₈⋊₃₇(C₂×C₄), (C₂×C₄×F₅).₁₂C₂, (C₂×F₅).₆(C₂×C₄), (C₂×D₅⋊C₈).₁₀C₂, (C₄×D₅).₉₅(C₂×C₄), (C₂×C₄).₁₆₆(C₂×F₅), (C₂×C₂₀).₁₇₅(C₂×C₄), (C₂×C₄×D₅).₄₁₃C₂², (C₂²×D₅).₈₇(C₂×C₄), (C₂×Dic₅).₁₂₅(C₂×C₄), SmallGroup(320,1055)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₂×C₈⋊F₅

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₄×D₅ — C₄×F₅ — C₂×C₄×F₅ — C₂×C₈⋊F₅

Lower central

C₅ — C₁₀ — C₂×C₈⋊F₅

Upper central

C₁ — C₂×C₄ — C₂×C₈

Generators and relations for C₂×C₈⋊F₅
G = < a,b,c,d | a²=b⁸=c⁵=d⁴=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b⁵, dcd^-1=c³ >

Subgroups: 442 in 146 conjugacy classes, 76 normal (30 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂×C₄, C₂×C₄, C₂³, D₅, C₁₀, C₁₀, C₄², C₂×C₈, C₂×C₈, C₂²×C₄, Dic₅, C₂₀, F₅, D₁₀, D₁₀, C₂×C₁₀, C₈⋊C₄, C₂×C₄², C₂²×C₈, C₅⋊₂C₈, C₄₀, C₅⋊C₈, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂×F₅, C₂×F₅, C₂²×D₅, C₂×C₈⋊C₄, C₈×D₅, C₂×C₅⋊₂C₈, C₂×C₄₀, D₅⋊C₈, C₄×F₅, C₂×C₅⋊C₈, C₂×C₄×D₅, C₂²×F₅, C₈⋊F₅, D₅×C₂×C₈, C₂×D₅⋊C₈, C₂×C₄×F₅, C₂×C₈⋊F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₄², M₄(2), C₂²×C₄, F₅, C₈⋊C₄, C₂×C₄², C₂×M₄(2), C₂×F₅, C₂×C₈⋊C₄, C₄×F₅, C₂²×F₅, C₈⋊F₅, C₂×C₄×F₅, C₂×C₈⋊F₅

Smallest permutation representation of C₂×C₈⋊F₅
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	4	4	4	4	4	4
type	+	+	+	+	+									+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	C₄	C₄	C₄	M₄(2)	F₅	C₂×F₅	C₂×F₅	C₄×F₅	C₄×F₅	C₈⋊F₅
kernel	C₂×C₈⋊F₅	C₈⋊F₅	D₅×C₂×C₈	C₂×D₅⋊C₈	C₂×C₄×F₅	C₈×D₅	C₂×C₅⋊₂C₈	C₂×C₄₀	D₅⋊C₈	C₄×F₅	C₂×C₅⋊C₈	C₂²×F₅	D₁₀	C₂×C₈	C₈	C₂×C₄	C₄	C₂²	C₂
# reps	1	4	1	1	1	4	2	2	4	4	4	4	8	1	2	1	2	2	8

Matrix representation of C₂×C₈⋊F₅ ►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	40	0
0	0	0	0	0	40

1	39	0	0	0	0
37	40	0	0	0	0
0	0	32	0	0	0
0	0	0	32	0	0
0	0	0	0	32	0
0	0	0	0	0	32

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

40	20	0	0	0	0
0	1	0	0	0	0
0	0	1	0	40	0
0	0	0	0	40	1
0	0	0	1	40	0
0	0	0	0	40	0

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,37,0,0,0,0,39,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[40,0,0,0,0,0,20,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,40,40,40,40,0,0,0,1,0,0] >;

C₂×C₈⋊F₅ in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes F_5

% in TeX

G:=Group("C2xC8:F5");

// GroupNames label

G:=SmallGroup(320,1055);

// by ID

G=gap.SmallGroup(320,1055);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,100,102,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=c^3>;

// generators/relations

G = C2×C8⋊F5 order 320 = 26·5

Direct product of C2 and C8⋊F5

G = C₂×C₈⋊F₅ order 320 = 2⁶·5

Direct product of C₂ and C₈⋊F₅