Copied to
clipboard

## G = C53⋊7C4order 500 = 22·53

### 7th semidirect product of C53 and C4 acting faithfully

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

 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 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 4A 4B 5A 5B 5C ··· 5AF 10A 10B order 1 2 4 4 5 5 5 ··· 5 10 10 size 1 25 125 125 2 2 4 ··· 4 50 50

38 irreducible representations

 dim 1 1 1 2 2 4 4 type + + + - + image C1 C2 C4 D5 Dic5 F5 D5.D5 kernel C53⋊7C4 C5×C5⋊D5 C53 C5⋊D5 C52 C52 C5 # reps 1 1 2 2 2 6 24

Matrix representation of C537C4 in GL8(𝔽41)

 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 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 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0

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

׿
×
𝔽