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G = C537C4order 500 = 22·53

7th semidirect product of C53 and C4 acting faithfully

metabelian, supersoluble, monomial, A-group

Aliases: C537C4, C528F5, C527Dic5, C5⋊D5.3D5, C52(D5.D5), C53(C5⋊F5), (C5×C5⋊D5).4C2, SmallGroup(500,47)

Series: Derived Chief Lower central Upper central

C1C53 — C537C4
C1C5C52C53C5×C5⋊D5 — C537C4
C53 — C537C4
C1

Generators and relations for C537C4
 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 [×6], C5 [×6], D5 [×6], C10, Dic5, F5 [×6], C52, C52 [×6], C52 [×6], C5×D5 [×6], C5⋊D5, D5.D5 [×6], C5⋊F5, C53, C5×C5⋊D5, C537C4
Quotients: C1, C2, C4, D5, Dic5, F5 [×6], D5.D5 [×6], C5⋊F5, C537C4

Smallest permutation representation of C537C4
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 43 28 36 32)(2 44 29 37 33)(3 45 30 38 34)(4 41 26 39 35)(5 42 27 40 31)(6 23 19 96 15)(7 24 20 97 11)(8 25 16 98 12)(9 21 17 99 13)(10 22 18 100 14)(46 55 63 59 67)(47 51 64 60 68)(48 52 65 56 69)(49 53 61 57 70)(50 54 62 58 66)(71 84 80 92 88)(72 85 76 93 89)(73 81 77 94 90)(74 82 78 95 86)(75 83 79 91 87)
(1 44 30 39 31)(2 45 26 40 32)(3 41 27 36 33)(4 42 28 37 34)(5 43 29 38 35)(6 25 18 97 13)(7 21 19 98 14)(8 22 20 99 15)(9 23 16 100 11)(10 24 17 96 12)(46 54 61 56 68)(47 55 62 57 69)(48 51 63 58 70)(49 52 64 59 66)(50 53 65 60 67)(71 82 76 91 90)(72 83 77 92 86)(73 84 78 93 87)(74 85 79 94 88)(75 81 80 95 89)
(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 98 47 73)(42 97 48 72)(43 96 49 71)(44 100 50 75)(45 99 46 74)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1112244
type+++-+
imageC1C2C4D5Dic5F5D5.D5
kernelC537C4C5×C5⋊D5C53C5⋊D5C52C52C5
# reps11222624

Matrix representation of C537C4 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] >;

C537C4 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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