C2xC2^3:F5 - GroupNames

G = C₂×C₂³⋊F₅ order 320 = 2⁶·5

Direct product of C₂ and C₂³⋊F₅

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

Aliases: C₂×C₂³⋊F₅, C₂⁴⋊₃F₅, C₂³⋊₄(C₂×F₅), (C₂³×C₁₀)⋊₆C₄, C₁₀⋊₂(C₂³⋊C₄), D₁₀.₁₂(C₂×D₄), C₂²⋊F₅⋊₃C₂², (C₂²×Dic₅)⋊₉C₄, (C₂²×D₅).₇₅D₄, D₁₀.₁₉(C₂²⋊C₄), C₂².₁₇(C₂²×F₅), (C₂³×D₅).₉₀C₂², C₂².₅₂(C₂²⋊F₅), (C₂²×D₅).₁₅₀C₂³, C₅⋊₃(C₂×C₂³⋊C₄), (C₂×C₅⋊D₄)⋊₁₆C₄, (C₂×C₂²⋊F₅)⋊₅C₂, (C₂²×C₁₀)⋊₈(C₂×C₄), (C₂×Dic₅)⋊₄(C₂×C₄), C₂.₃₈(C₂×C₂²⋊F₅), C₁₀.₃₇(C₂×C₂²⋊C₄), (C₂×C₁₀).₉₀(C₂²×C₄), (C₂²×D₅).₅₉(C₂×C₄), (C₂²×C₅⋊D₄).₁₄C₂, (C₂×C₁₀).₆₁(C₂²⋊C₄), (C₂×C₅⋊D₄).₁₅₅C₂², SmallGroup(320,1134)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₂×C₂³⋊F₅

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₂²×D₅ — C₂²⋊F₅ — C₂×C₂²⋊F₅ — C₂×C₂³⋊F₅

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂³⋊F₅

Upper central

C₁ — C₂² — C₂³ — C₂⁴

Generators and relations for C₂×C₂³⋊F₅
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁵=f⁴=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf^-1=bcd, fcf^-1=cd=dc, ce=ec, de=ed, df=fd, fef^-1=e³ >

Subgroups: 1034 in 210 conjugacy classes, 52 normal (20 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₅, C₂×C₄, D₄, C₂³, C₂³, C₂³, D₅, C₁₀, C₁₀, C₁₀, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, Dic₅, F₅, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂²×D₄, C₂×Dic₅, C₂×Dic₅, C₅⋊D₄, C₂×F₅, C₂²×D₅, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂×C₂³⋊C₄, C₂²⋊F₅, C₂²⋊F₅, C₂²×Dic₅, C₂×C₅⋊D₄, C₂×C₅⋊D₄, C₂²×F₅, C₂³×D₅, C₂³×C₁₀, C₂³⋊F₅, C₂×C₂²⋊F₅, C₂²×C₅⋊D₄, C₂×C₂³⋊F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, F₅, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂×F₅, C₂×C₂³⋊C₄, C₂²⋊F₅, C₂²×F₅, C₂³⋊F₅, C₂×C₂²⋊F₅, C₂×C₂³⋊F₅

Smallest permutation representation of C₂×C₂³⋊F₅
►On 80 points

Generators in S₈₀

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

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

(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)

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

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

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

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

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

38 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	4A	···	4J	5	10A	···	10O
order	1	2	2	2	2	2	2	2	2	2	2	2	4	···	4	5	10	···	10
size	1	1	1	1	2	2	4	4	10	10	10	10	20	···	20	4	4	···	4

38 irreducible representations

dim

type

image

C₁

C₂

C₄

D₄

F₅

C₂³⋊C₄

C₂×F₅

C₂²⋊F₅

C₂³⋊F₅

kernel

C₂×C₂³⋊F₅

C₂³⋊F₅

C₂×C₂²⋊F₅

C₂²×C₅⋊D₄

C₂²×Dic₅

C₂×C₅⋊D₄

C₂³×C₁₀

C₂²×D₅

C₂⁴

C₁₀

C₂³

C₂²

C₂

# reps

Matrix representation of C₂×C₂³⋊F₅ ►in GL₆(𝔽₄₁)

40	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	0
0	0	0	0	0	40

0	1	0	0	0	0
1	0	0	0	0	0
0	0	24	35	0	0
0	0	7	17	0	0
0	0	35	0	18	35
0	0	40	6	6	23

40	0	0	0	0	0
0	40	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	38	21	40	0
0	0	38	21	0	40

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	0	0	0	0	0
0	1	0	0	0	0
0	0	40	1	0	0
0	0	33	7	0	0
0	0	7	40	40	6
0	0	2	34	35	35

0	40	0	0	0	0
1	0	0	0	0	0
0	0	0	0	40	1
0	0	6	40	39	6
0	0	3	20	1	0
0	0	10	19	1	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,24,7,35,40,0,0,35,17,0,6,0,0,0,0,18,6,0,0,0,0,35,23],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,38,38,0,0,0,1,21,21,0,0,0,0,40,0,0,0,0,0,0,40],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,7,2,0,0,1,7,40,34,0,0,0,0,40,35,0,0,0,0,6,35],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,6,3,10,0,0,0,40,20,19,0,0,40,39,1,1,0,0,1,6,0,0] >;

C₂×C₂³⋊F₅ in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes F_5

% in TeX

G:=Group("C2xC2^3:F5");

// GroupNames label

G:=SmallGroup(320,1134);

// by ID

G=gap.SmallGroup(320,1134);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,297,1684,6278,1595]);

// Polycyclic

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

// generators/relations

G = C2×C23⋊F5 order 320 = 26·5

Direct product of C2 and C23⋊F5

G = C₂×C₂³⋊F₅ order 320 = 2⁶·5

Direct product of C₂ and C₂³⋊F₅