C4^2xF5 - GroupNames

G = C₄²×F₅ order 320 = 2⁶·5

Direct product of C₄² and F₅

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

Aliases: C₄²×F₅, C₅⋊C₄³, C₂₀⋊₃C₄², Dic₅⋊₄C₄², (C₄×C₂₀)⋊₁₀C₄, D₅.(C₂×C₄²), (C₄×Dic₅)⋊₂₅C₄, C₁₀.₄(C₂×C₄²), (D₅×C₄²).₃₁C₂, D₁₀.₂₆(C₂²×C₄), C₂².₃₀(C₂²×F₅), (C₂²×F₅).₂₁C₂², (C₂²×D₅).₂₆₂C₂³, C₂.₂(C₂×C₄×F₅), (C₂×C₄×F₅).₁₃C₂, (C₂×F₅).₈(C₂×C₄), (C₄×D₅).₇₁(C₂×C₄), (C₂×C₄).₁₆₀(C₂×F₅), (C₂×C₂₀).₁₆₉(C₂×C₄), (C₂×C₄×D₅).₄₁₁C₂², (C₂×C₁₀).₂₃(C₂²×C₄), (C₂×Dic₅).₁₇₁(C₂×C₄), SmallGroup(320,1023)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅ — C₄²×F₅

Chief series

C₁ — C₅ — D₅ — D₁₀ — C₂²×D₅ — C₂²×F₅ — C₂×C₄×F₅ — C₄²×F₅

Lower central

C₅ — C₄²×F₅

Upper central

C₁ — C₄²

Generators and relations for C₄²×F₅
G = < a,b,c,d | a⁴=b⁴=c⁵=d⁴=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c³ >

Subgroups: 714 in 258 conjugacy classes, 144 normal (9 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₂×C₄, C₂×C₄, C₂³, D₅, C₁₀, C₄², C₄², C₂²×C₄, Dic₅, C₂₀, F₅, D₁₀, C₂×C₁₀, C₂×C₄², C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂×F₅, C₂²×D₅, C₄³, C₄×Dic₅, C₄×C₂₀, C₄×F₅, C₂×C₄×D₅, C₂²×F₅, D₅×C₄², C₂×C₄×F₅, C₄²×F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₄², C₂²×C₄, F₅, C₂×C₄², C₂×F₅, C₄³, C₄×F₅, C₂²×F₅, C₂×C₄×F₅, C₄²×F₅

Smallest permutation representation of C₄²×F₅
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

80 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	···	4L	4M	···	4BD	5	10A	10B	10C	20A	···	20L
order	1	2	2	2	2	2	2	2	4	···	4	4	···	4	5	10	10	10	20	···	20
size	1	1	1	1	5	5	5	5	1	···	1	5	···	5	4	4	4	4	4	···	4

80 irreducible representations

dim	1	1	1	1	1	1	4	4	4
type	+	+	+				+	+
image	C₁	C₂	C₂	C₄	C₄	C₄	F₅	C₂×F₅	C₄×F₅
kernel	C₄²×F₅	D₅×C₄²	C₂×C₄×F₅	C₄×Dic₅	C₄×C₂₀	C₄×F₅	C₄²	C₂×C₄	C₄
# reps	1	1	6	6	2	48	1	3	12

Matrix representation of C₄²×F₅ ►in GL₆(𝔽₄₁)

9	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	32	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	40	40	40
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0

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

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

C₄²×F₅ in GAP, Magma, Sage, TeX

C_4^2\times F_5

% in TeX

G:=Group("C4^2xF5");

// GroupNames label

G:=SmallGroup(320,1023);

// by ID

G=gap.SmallGroup(320,1023);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,100,6278,1595]);

// Polycyclic

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

// generators/relations

G = C42×F5 order 320 = 26·5

Direct product of C42 and F5

G = C₄²×F₅ order 320 = 2⁶·5

Direct product of C₄² and F₅