C2^2xC4xF5 - GroupNames

G = C₂²×C₄×F₅ order 320 = 2⁶·5

Direct product of C₂²×C₄ and F₅

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

Aliases: C₂²×C₄×F₅, D₁₀⋊₅C₄², D₁₀.₁₁C₂⁴, C₁₀⋊(C₂×C₄²), D₅⋊(C₂×C₄²), C₅⋊(C₂²×C₄²), D₅.(C₂³×C₄), C₂₀⋊₄(C₂²×C₄), (C₂×C₁₀)⋊₃C₄², (C₂²×C₂₀)⋊₁₆C₄, C₂.₂(C₂³×F₅), C₁₀.₄(C₂³×C₄), (C₂³×F₅).₅C₂, (C₂×F₅).₉C₂³, C₂³.₆₆(C₂×F₅), Dic₅⋊₇(C₂²×C₄), (C₄×D₅).₉₂C₂³, (C₂²×Dic₅)⋊₂₀C₄, D₁₀.₃₈(C₂²×C₄), C₂².₅₆(C₂²×F₅), (C₂²×F₅).₂₄C₂², (C₂²×D₅).₂₈₂C₂³, (C₂³×D₅).₁₃₇C₂², (C₂×C₄×D₅)⋊₂₃C₄, (C₂×C₂₀)⋊₁₄(C₂×C₄), (C₄×D₅)⋊₂₁(C₂×C₄), (D₅×C₂²×C₄).₃₇C₂, (C₂×Dic₅)⋊₃₅(C₂×C₄), (C₂×C₄×D₅).₄₁₆C₂², (C₂²×C₁₀).₈₀(C₂×C₄), (C₂×C₁₀).₉₈(C₂²×C₄), (C₂²×D₅).₉₁(C₂×C₄), SmallGroup(320,1590)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅ — C₂²×C₄×F₅

Chief series

C₁ — C₅ — D₅ — D₁₀ — C₂×F₅ — C₂²×F₅ — C₂³×F₅ — C₂²×C₄×F₅

Lower central

C₅ — C₂²×C₄×F₅

Upper central

C₁ — C₂²×C₄

Generators and relations for C₂²×C₄×F₅
G = < a,b,c,d,e | a²=b²=c⁴=d⁵=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d³ >

Subgroups: 1386 in 498 conjugacy classes, 276 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₂×C₄, C₂×C₄, C₂³, C₂³, D₅, C₁₀, C₁₀, C₄², C₂²×C₄, C₂²×C₄, C₂⁴, Dic₅, C₂₀, F₅, D₁₀, D₁₀, C₂×C₁₀, C₂×C₄², C₂³×C₄, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂×F₅, C₂²×D₅, C₂²×C₁₀, C₂²×C₄², C₄×F₅, C₂×C₄×D₅, C₂²×Dic₅, C₂²×C₂₀, C₂²×F₅, C₂³×D₅, C₂×C₄×F₅, D₅×C₂²×C₄, C₂³×F₅, C₂²×C₄×F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₄², C₂²×C₄, C₂⁴, F₅, C₂×C₄², C₂³×C₄, C₂×F₅, C₂²×C₄², C₄×F₅, C₂²×F₅, C₂×C₄×F₅, C₂³×F₅, C₂²×C₄×F₅

Smallest permutation representation of C₂²×C₄×F₅
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

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

80 conjugacy classes

class	1	2A	···	2G	2H	···	2O	4A	···	4H	4I	···	4AV	5	10A	···	10G	20A	···	20H
order	1	2	···	2	2	···	2	4	···	4	4	···	4	5	10	···	10	20	···	20
size	1	1	···	1	5	···	5	1	···	1	5	···	5	4	4	···	4	4	···	4

80 irreducible representations

dim	1	1	1	1	1	1	1	1	4	4	4	4
type	+	+	+	+					+	+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	C₄	F₅	C₂×F₅	C₂×F₅	C₄×F₅
kernel	C₂²×C₄×F₅	C₂×C₄×F₅	D₅×C₂²×C₄	C₂³×F₅	C₂×C₄×D₅	C₂²×Dic₅	C₂²×C₂₀	C₂²×F₅	C₂²×C₄	C₂×C₄	C₂³	C₂²
# reps	1	12	1	2	12	2	2	32	1	6	1	8

Matrix representation of C₂²×C₄×F₅ ►in GL₆(𝔽₄₁)

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

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	0	0	1	0
0	0	0	0	0	1

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

9	0	0	0	0	0
0	9	0	0	0	0
0	0	0	0	32	0
0	0	32	0	0	0
0	0	0	0	0	32
0	0	0	32	0	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,32,0,0,32,0,0,0,0,0,0,0,32,0] >;

C₂²×C₄×F₅ in GAP, Magma, Sage, TeX

C_2^2\times C_4\times F_5

% in TeX

G:=Group("C2^2xC4xF5");

// GroupNames label

G:=SmallGroup(320,1590);

// by ID

G=gap.SmallGroup(320,1590);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,136,6278,818]);

// Polycyclic

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

// generators/relations

G = C22×C4×F5 order 320 = 26·5

Direct product of C22×C4 and F5

G = C₂²×C₄×F₅ order 320 = 2⁶·5

Direct product of C₂²×C₄ and F₅