C4^2:4F5 - GroupNames

G = C₄²:₄F₅ order 320 = 2⁶·5

1^st semidirect product of C₄² and F₅ acting via F₅/D₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄²:₄F₅, C₂₀.₂₈C₄², Dic₅.₈C₄², (C₄xF₅):₆C₄, (C₄xC₂₀):₁₁C₄, C₅:(C₄²:₄C₄), C₄.₂₁(C₄xF₅), (C₄xDic₅):₁₉C₄, C₁₀.₅(C₂xC₄²), (D₅xC₄²).₃₂C₂, D₁₀.₂₁(C₄oD₄), D₁₀.₂₇(C₂²xC₄), D₁₀.₃Q₈.₉C₂, C₁₀.₅(C₄²:C₂), C₂².₃₁(C₂²xF₅), D₅.₁(C₄²:C₂), (C₂²xF₅).₁₂C₂², (C₂²xD₅).₂₆₃C₂³, C₂.₃(D₁₀.C₂³), C₂.₇(C₂xC₄xF₅), (C₂xC₄xF₅).₈C₂, (C₂xF₅).₅(C₂xC₄), (C₄xD₅).₇₂(C₂xC₄), (C₂xC₄).₁₆₁(C₂xF₅), (C₂xC₂₀).₁₇₀(C₂xC₄), (C₂xC₄xD₅).₃₆₀C₂², (C₂xC₁₀).₂₄(C₂²xC₄), (C₂xDic₅).₁₇₂(C₂xC₄), SmallGroup(320,1024)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₄²:₄F₅

Chief series

C₁ — C₅ — D₅ — D₁₀ — C₂²xD₅ — C₂²xF₅ — C₂xC₄xF₅ — C₄²:₄F₅

Lower central

C₅ — C₁₀ — C₄²:₄F₅

Upper central

C₁ — C₂xC₄ — C₄²

Generators and relations for C₄²:₄F₅
G = < a,b,c,d | a⁴=b⁴=c⁵=d⁴=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=a²b, dcd^-1=c³ >

Subgroups: 618 in 178 conjugacy classes, 76 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₂xC₄, C₂xC₄, C₂xC₄, C₂³, D₅, C₁₀, C₁₀, C₄², C₄², C₂²xC₄, Dic₅, Dic₅, C₂₀, C₂₀, F₅, D₁₀, C₂xC₁₀, C₂.C₄², C₂xC₄², C₄xD₅, C₄xD₅, C₂xDic₅, C₂xDic₅, C₂xC₂₀, C₂xC₂₀, C₂xF₅, C₂xF₅, C₂²xD₅, C₄²:₄C₄, C₄xDic₅, C₄xDic₅, C₄xC₂₀, C₄xF₅, C₂xC₄xD₅, C₂xC₄xD₅, C₂²xF₅, D₁₀.₃Q₈, D₅xC₄², C₂xC₄xF₅, C₄²:₄F₅
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, C₂³, C₄², C₂²xC₄, C₄oD₄, F₅, C₂xC₄², C₄²:C₂, C₂xF₅, C₄²:₄C₄, C₄xF₅, C₂²xF₅, C₂xC₄xF₅, D₁₀.C₂³, C₄²:₄F₅

Smallest permutation representation of C₄²:₄F₅
►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 31 11 21)(2 32 12 22)(3 33 13 23)(4 34 14 24)(5 35 15 25)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)

(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)

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

dim	1	1	1	1	1	1	1	2	4	4	4	4
type	+	+	+	+					+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	C₄oD₄	F₅	C₂xF₅	C₄xF₅	D₁₀.C₂³
kernel	C₄²:₄F₅	D₁₀.₃Q₈	D₅xC₄²	C₂xC₄xF₅	C₄xDic₅	C₄xC₂₀	C₄xF₅	D₁₀	C₄²	C₂xC₄	C₄	C₂
# reps	1	4	1	2	6	2	16	8	1	3	4	8

Matrix representation of C₄²:₄F₅ ►in GL₆(F₄₁)

9	0	0	0	0	0
0	9	0	0	0	0
0	0	9	0	0	0
0	0	0	9	0	0
0	0	0	0	9	0
0	0	0	0	0	9

1	0	0	0	0	0
2	40	0	0	0	0
0	0	34	0	27	27
0	0	14	7	14	0
0	0	0	14	7	14
0	0	27	27	0	34

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

9	32	0	0	0	0
0	32	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))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,2,0,0,0,0,0,40,0,0,0,0,0,0,34,14,0,27,0,0,0,7,14,27,0,0,27,14,7,0,0,0,27,0,14,34],[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],[9,0,0,0,0,0,32,32,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

C₄²:₄F₅ in GAP, Magma, Sage, TeX

C_4^2\rtimes_4F_5

% in TeX

G:=Group("C4^2:4F5");

// GroupNames label

G:=SmallGroup(320,1024);

// by ID

G=gap.SmallGroup(320,1024);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,100,6278,1595]);

// 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,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;

// generators/relations

G = C42:4F5 order 320 = 26·5

1st semidirect product of C42 and F5 acting via F5/D5=C2

G = C₄²:₄F₅ order 320 = 2⁶·5

1^st semidirect product of C₄² and F₅ acting via F₅/D₅=C₂