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G = C53⋊7C4  order 500 = 22·53

7th semidirect product of C53 and C4 acting faithfully

metabelian, supersoluble, monomial, A-group

Aliases: C53⋊7C4, C52⋊8F5, C52⋊7Dic5, C5⋊D5.3D5, C5⋊2(D5.D5), C5⋊3(C5⋊F5), (C5×C5⋊D5).4C2, SmallGroup(500,47)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C53 — C53⋊7C4
Chief series
C1 — C5 — C52 — C53 — C5×C5⋊D5 — C53⋊7C4
Lower central
C53 — C53⋊7C4
Upper central
C1

Generators and relations for C53⋊7C4
 G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b3, dcd-1=c3 >

Subgroups: 512 in 60 conjugacy classes, 19 normal (7 characteristic)
C1, C2, C4, C5, C5, C5, D5, C10, Dic5, F5, C52, C52, C52, C5×D5, C5⋊D5, D5.D5, C5⋊F5, C53, C5×C5⋊D5, C53⋊7C4
Quotients: C1, C2, C4, D5, Dic5, F5, D5.D5, C5⋊F5, C53⋊7C4

Smallest permutation representation of C53⋊7C4
►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 41 28 32 36)(2 42 29 33 37)(3 43 30 34 38)(4 44 26 35 39)(5 45 27 31 40)(6 23 15 96 19)(7 24 11 97 20)(8 25 12 98 16)(9 21 13 99 17)(10 22 14 100 18)(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 88 80 92 84)(72 89 76 93 85)(73 90 77 94 81)(74 86 78 95 82)(75 87 79 91 83)

(1 42 30 35 40)(2 43 26 31 36)(3 44 27 32 37)(4 45 28 33 38)(5 41 29 34 39)(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 54 57 65 68)(47 55 58 61 69)(48 51 59 62 70)(49 52 60 63 66)(50 53 56 64 67)(71 86 76 91 81)(72 87 77 92 82)(73 88 78 93 83)(74 89 79 94 84)(75 90 80 95 85)

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

38 conjugacy classes

class 1  2 4A4B5A5B5C···5AF10A10B
order1244555···51010
size125125125224···45050

38 irreducible representations

dim1112244
type+++-+
imageC1C2C4D5Dic5F5D5.D5
kernelC53⋊7C4C5×C5⋊D5C53C5⋊D5C52C52C5
# reps11222624

Matrix representation of C53⋊7C4 ►in GL8(𝔽41)

100000000
010000000
003700000
000370000
000018000
000001800
000000160
000000016
,
180000000
016000000
003700000
000100000
000018000
000001600
000000370
000000010
,
160000000
018000000
001000000
000370000
000037000
000001000
000000160
000000018
,
00100000
00010000
01000000
10000000
00000010
00000001
00000100
00001000

G:=sub<GL(8,GF(41))| [10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[18,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,10],[16,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,18],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
 

C53⋊7C4 in GAP, Magma, Sage, TeX 

C_5^3\rtimes_7C_4
 
% in TeX
 
G:=Group("C5^3:7C4");
 
// GroupNames label
 
G:=SmallGroup(500,47);
 
// by ID
 
G=gap.SmallGroup(500,47);
 
# by ID
 
G:=PCGroup([5,-2,-2,-5,-5,-5,10,242,803,808,5004,5009]);
 
// 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^3,d*c*d^-1=c^3>;
 
// generators/relations
 

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