C5^3:C4 - GroupNames

G = C₅³⋊C₄ order 500 = 2²·5³

5^th semidirect product of C₅³ and C₄ acting faithfully

metabelian, supersoluble, monomial, A-group

Aliases: C₅³⋊₅C₄, C₅²⋊₁₂F₅, C₅²⋊₅Dic₅, D₅.(C₅⋊D₅), C₅⋊(C₅²⋊₆C₄), (C₅×D₅).₄D₅, C₅⋊₃(D₅.D₅), (D₅×C₅²).₃C₂, SmallGroup(500,45)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅³ — C₅³⋊C₄

Chief series

C₁ — C₅ — C₅² — C₅³ — D₅×C₅² — C₅³⋊C₄

Lower central

C₅³ — C₅³⋊C₄

Upper central

C₁

Generators and relations for C₅³⋊C₄
G = < a,b,c,d | a⁵=b⁵=c⁵=d⁴=1, ab=ba, ac=ca, dad^-1=a^-1, bc=cb, dbd^-1=b^-1, dcd^-1=c² >

Subgroups: 448 in 60 conjugacy classes, 25 normal (7 characteristic)
C₁, C₂, C₄, C₅, C₅, C₅, D₅, C₁₀, Dic₅, F₅, C₅², C₅², C₅², C₅×D₅, C₅×C₁₀, C₅²⋊₆C₄, D₅.D₅, C₅³, D₅×C₅², C₅³⋊C₄
Quotients: C₁, C₂, C₄, D₅, Dic₅, F₅, C₅⋊D₅, C₅²⋊₆C₄, D₅.D₅, C₅³⋊C₄

Smallest permutation representation of C₅³⋊C₄
►On 100 points

Generators in S₁₀₀

(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)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)

(1 29 44 39 32)(2 30 45 40 33)(3 26 41 36 34)(4 27 42 37 35)(5 28 43 38 31)(6 25 14 97 17)(7 21 15 98 18)(8 22 11 99 19)(9 23 12 100 20)(10 24 13 96 16)(46 65 54 68 57)(47 61 55 69 58)(48 62 51 70 59)(49 63 52 66 60)(50 64 53 67 56)(71 82 93 79 90)(72 83 94 80 86)(73 84 95 76 87)(74 85 91 77 88)(75 81 92 78 89)

(1 38 27 34 45)(2 39 28 35 41)(3 40 29 31 42)(4 36 30 32 43)(5 37 26 33 44)(6 19 96 15 23)(7 20 97 11 24)(8 16 98 12 25)(9 17 99 13 21)(10 18 100 14 22)(46 55 59 63 67)(47 51 60 64 68)(48 52 56 65 69)(49 53 57 61 70)(50 54 58 62 66)(71 85 94 78 87)(72 81 95 79 88)(73 82 91 80 89)(74 83 92 76 90)(75 84 93 77 86)

(1 22 54 77)(2 21 55 76)(3 25 51 80)(4 24 52 79)(5 23 53 78)(6 70 94 26)(7 69 95 30)(8 68 91 29)(9 67 92 28)(10 66 93 27)(11 65 88 32)(12 64 89 31)(13 63 90 35)(14 62 86 34)(15 61 87 33)(16 60 82 42)(17 59 83 41)(18 58 84 45)(19 57 85 44)(20 56 81 43)(36 97 48 72)(37 96 49 71)(38 100 50 75)(39 99 46 74)(40 98 47 73)

G:=sub<Sym(100)| (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)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100), (1,29,44,39,32)(2,30,45,40,33)(3,26,41,36,34)(4,27,42,37,35)(5,28,43,38,31)(6,25,14,97,17)(7,21,15,98,18)(8,22,11,99,19)(9,23,12,100,20)(10,24,13,96,16)(46,65,54,68,57)(47,61,55,69,58)(48,62,51,70,59)(49,63,52,66,60)(50,64,53,67,56)(71,82,93,79,90)(72,83,94,80,86)(73,84,95,76,87)(74,85,91,77,88)(75,81,92,78,89), (1,38,27,34,45)(2,39,28,35,41)(3,40,29,31,42)(4,36,30,32,43)(5,37,26,33,44)(6,19,96,15,23)(7,20,97,11,24)(8,16,98,12,25)(9,17,99,13,21)(10,18,100,14,22)(46,55,59,63,67)(47,51,60,64,68)(48,52,56,65,69)(49,53,57,61,70)(50,54,58,62,66)(71,85,94,78,87)(72,81,95,79,88)(73,82,91,80,89)(74,83,92,76,90)(75,84,93,77,86), (1,22,54,77)(2,21,55,76)(3,25,51,80)(4,24,52,79)(5,23,53,78)(6,70,94,26)(7,69,95,30)(8,68,91,29)(9,67,92,28)(10,66,93,27)(11,65,88,32)(12,64,89,31)(13,63,90,35)(14,62,86,34)(15,61,87,33)(16,60,82,42)(17,59,83,41)(18,58,84,45)(19,57,85,44)(20,56,81,43)(36,97,48,72)(37,96,49,71)(38,100,50,75)(39,99,46,74)(40,98,47,73)>;

G:=Group( (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)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100), (1,29,44,39,32)(2,30,45,40,33)(3,26,41,36,34)(4,27,42,37,35)(5,28,43,38,31)(6,25,14,97,17)(7,21,15,98,18)(8,22,11,99,19)(9,23,12,100,20)(10,24,13,96,16)(46,65,54,68,57)(47,61,55,69,58)(48,62,51,70,59)(49,63,52,66,60)(50,64,53,67,56)(71,82,93,79,90)(72,83,94,80,86)(73,84,95,76,87)(74,85,91,77,88)(75,81,92,78,89), (1,38,27,34,45)(2,39,28,35,41)(3,40,29,31,42)(4,36,30,32,43)(5,37,26,33,44)(6,19,96,15,23)(7,20,97,11,24)(8,16,98,12,25)(9,17,99,13,21)(10,18,100,14,22)(46,55,59,63,67)(47,51,60,64,68)(48,52,56,65,69)(49,53,57,61,70)(50,54,58,62,66)(71,85,94,78,87)(72,81,95,79,88)(73,82,91,80,89)(74,83,92,76,90)(75,84,93,77,86), (1,22,54,77)(2,21,55,76)(3,25,51,80)(4,24,52,79)(5,23,53,78)(6,70,94,26)(7,69,95,30)(8,68,91,29)(9,67,92,28)(10,66,93,27)(11,65,88,32)(12,64,89,31)(13,63,90,35)(14,62,86,34)(15,61,87,33)(16,60,82,42)(17,59,83,41)(18,58,84,45)(19,57,85,44)(20,56,81,43)(36,97,48,72)(37,96,49,71)(38,100,50,75)(39,99,46,74)(40,98,47,73) );

G=PermutationGroup([[(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),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100)], [(1,29,44,39,32),(2,30,45,40,33),(3,26,41,36,34),(4,27,42,37,35),(5,28,43,38,31),(6,25,14,97,17),(7,21,15,98,18),(8,22,11,99,19),(9,23,12,100,20),(10,24,13,96,16),(46,65,54,68,57),(47,61,55,69,58),(48,62,51,70,59),(49,63,52,66,60),(50,64,53,67,56),(71,82,93,79,90),(72,83,94,80,86),(73,84,95,76,87),(74,85,91,77,88),(75,81,92,78,89)], [(1,38,27,34,45),(2,39,28,35,41),(3,40,29,31,42),(4,36,30,32,43),(5,37,26,33,44),(6,19,96,15,23),(7,20,97,11,24),(8,16,98,12,25),(9,17,99,13,21),(10,18,100,14,22),(46,55,59,63,67),(47,51,60,64,68),(48,52,56,65,69),(49,53,57,61,70),(50,54,58,62,66),(71,85,94,78,87),(72,81,95,79,88),(73,82,91,80,89),(74,83,92,76,90),(75,84,93,77,86)], [(1,22,54,77),(2,21,55,76),(3,25,51,80),(4,24,52,79),(5,23,53,78),(6,70,94,26),(7,69,95,30),(8,68,91,29),(9,67,92,28),(10,66,93,27),(11,65,88,32),(12,64,89,31),(13,63,90,35),(14,62,86,34),(15,61,87,33),(16,60,82,42),(17,59,83,41),(18,58,84,45),(19,57,85,44),(20,56,81,43),(36,97,48,72),(37,96,49,71),(38,100,50,75),(39,99,46,74),(40,98,47,73)]])

53 conjugacy classes

class	1	2	4A	4B	5A	···	5L	5M	···	5AK	10A	···	10L
order	1	2	4	4	5	···	5	5	···	5	10	···	10
size	1	5	125	125	2	···	2	4	···	4	10	···	10

53 irreducible representations

dim	1	1	1	2	2	4	4
type	+	+		+	-	+
image	C₁	C₂	C₄	D₅	Dic₅	F₅	D₅.D₅
kernel	C₅³⋊C₄	D₅×C₅²	C₅³	C₅×D₅	C₅²	C₅²	C₅
# reps	1	1	2	12	12	1	24

Matrix representation of C₅³⋊C₄ ►in GL₆(𝔽₄₁)

40	7	0	0	0	0
34	7	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	40	0	0	0	0
1	34	0	0	0	0
0	0	10	0	7	7
0	0	0	10	0	0
0	0	0	0	37	0
0	0	0	0	0	37

1	0	0	0	0	0
0	1	0	0	0	0
0	0	16	17	10	3
0	0	0	18	0	0
0	0	0	0	37	0
0	0	0	0	0	10

6	2	0	0	0	0
3	35	0	0	0	0
0	0	12	0	35	35
0	0	0	0	0	1
0	0	0	1	0	0
0	0	17	40	29	29

G:=sub<GL(6,GF(41))| [40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,40,34,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,7,0,37,0,0,0,7,0,0,37],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,17,18,0,0,0,0,10,0,37,0,0,0,3,0,0,10],[6,3,0,0,0,0,2,35,0,0,0,0,0,0,12,0,0,17,0,0,0,0,1,40,0,0,35,0,0,29,0,0,35,1,0,29] >;

C₅³⋊C₄ in GAP, Magma, Sage, TeX

C_5^3\rtimes C_4

% in TeX

G:=Group("C5^3:C4");

// GroupNames label

G:=SmallGroup(500,45);

// by ID

G=gap.SmallGroup(500,45);

# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,10,242,1603,7504,5009]);

// Polycyclic

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

// generators/relations

G = C53⋊C4 order 500 = 22·53

5th semidirect product of C53 and C4 acting faithfully

G = C₅³⋊C₄ order 500 = 2²·5³

5^th semidirect product of C₅³ and C₄ acting faithfully