C4^2:5F5 - GroupNames

G = C₄²⋊₅F₅ order 320 = 2⁶·5

2^nd semidirect product of C₄² and F₅ acting via F₅/D₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄²⋊₅F₅, (C₄×C₂₀)⋊₄C₄, C₅⋊(C₄²⋊₅C₄), (C₄×Dic₅)⋊₂₂C₄, (D₅×C₄²).₁₇C₂, D₁₀.₂₄(C₄○D₄), D₁₀.₃Q₈.₅C₂, C₁₀.₈(C₄²⋊C₂), (C₂²×F₅).₃C₂², C₂².₆₈(C₂²×F₅), D₅.₁(C₄²⋊₂C₂), (C₂²×D₅).₂₆₇C₂³, C₂.₁₁(D₁₀.C₂³), (C₂×C₄).₁₀₂(C₂×F₅), (C₂×C₂₀).₁₀₂(C₂×C₄), (C₂×C₄×D₅).₃₆₂C₂², (C₂×C₁₀).₂₈(C₂²×C₄), (C₂×Dic₅).₁₇₆(C₂×C₄), SmallGroup(320,1028)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — C₄²⋊₅F₅

Chief series

C₁ — C₅ — D₅ — D₁₀ — C₂²×D₅ — C₂²×F₅ — D₁₀.₃Q₈ — C₄²⋊₅F₅

Lower central

C₅ — C₂×C₁₀ — C₄²⋊₅F₅

Upper central

C₁ — C₂² — C₄²

Generators and relations for C₄²⋊₅F₅
G = < a,b,c,d | a⁴=b⁴=c⁵=d⁴=1, ab=ba, ac=ca, dad^-1=ab², bc=cb, dbd^-1=a²b^-1, dcd^-1=c³ >

Subgroups: 570 in 138 conjugacy classes, 50 normal (9 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₂×C₄, C₂×C₄, C₂³, D₅, C₁₀, C₄², C₄², C₂²×C₄, Dic₅, C₂₀, F₅, D₁₀, C₂×C₁₀, C₂.C₄², C₂×C₄², C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂×F₅, C₂²×D₅, C₄²⋊₅C₄, C₄×Dic₅, C₄×C₂₀, C₂×C₄×D₅, C₂²×F₅, D₁₀.₃Q₈, D₅×C₄², C₄²⋊₅F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₄○D₄, F₅, C₄²⋊C₂, C₄²⋊₂C₂, C₂×F₅, C₄²⋊₅C₄, C₂²×F₅, D₁₀.C₂³, C₄²⋊₅F₅

Smallest permutation representation of C₄²⋊₅F₅
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	2	4	4	4
type	+	+	+				+	+
image	C₁	C₂	C₂	C₄	C₄	C₄○D₄	F₅	C₂×F₅	D₁₀.C₂³
kernel	C₄²⋊₅F₅	D₁₀.₃Q₈	D₅×C₄²	C₄×Dic₅	C₄×C₂₀	D₁₀	C₄²	C₂×C₄	C₂
# reps	1	6	1	6	2	12	1	3	12

Matrix representation of C₄²⋊₅F₅ ►in GL₆(𝔽₄₁)

40	39	0	0	0	0
1	1	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

32	0	0	0	0	0
0	32	0	0	0	0
0	0	22	0	3	3
0	0	38	19	38	0
0	0	0	38	19	38
0	0	3	3	0	22

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	39	0	0	0	0
0	1	0	0	0	0
0	0	34	0	27	27
0	0	27	27	0	34
0	0	14	7	14	0
0	0	7	34	34	7

G:=sub<GL(6,GF(41))| [40,1,0,0,0,0,39,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,22,38,0,3,0,0,0,19,38,3,0,0,3,38,19,0,0,0,3,0,38,22],[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],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,34,27,14,7,0,0,0,27,7,34,0,0,27,0,14,34,0,0,27,34,0,7] >;

C₄²⋊₅F₅ in GAP, Magma, Sage, TeX

C_4^2\rtimes_5F_5

% in TeX

G:=Group("C4^2:5F5");

// GroupNames label

G:=SmallGroup(320,1028);

// by ID

G=gap.SmallGroup(320,1028);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,120,1094,184,6278,1595]);

// Polycyclic

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

// generators/relations

G = C42⋊5F5 order 320 = 26·5

2nd semidirect product of C42 and F5 acting via F5/D5=C2

G = C₄²⋊₅F₅ order 320 = 2⁶·5

2^nd semidirect product of C₄² and F₅ acting via F₅/D₅=C₂