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G = C53⋊8C4  order 500 = 22·53

8th semidirect product of C53 and C4 acting faithfully

metabelian, supersoluble, monomial, A-group

Aliases: C53⋊8C4, C52⋊9F5, C5⋊1(C5⋊F5), C53⋊C2.1C2, SmallGroup(500,48)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C53 — C53⋊8C4
Chief series
C1 — C5 — C52 — C53 — C53⋊C2 — C53⋊8C4
Lower central
C53 — C53⋊8C4
Upper central
C1

Generators and relations for C53⋊8C4
 G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, ac=ca, dad-1=a3, bc=cb, dbd-1=b3, dcd-1=c3 >

Subgroups: 2176 in 192 conjugacy classes, 66 normal (4 characteristic)
C1, C2, C4, C5, D5, F5, C52, C5⋊D5, C5⋊F5, C53, C53⋊C2, C53⋊8C4
Quotients: C1, C2, C4, F5, C5⋊F5, C53⋊8C4

Smallest permutation representation of C53⋊8C4
►On 125 points
Generators in S125
(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)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)(121 122 123 124 125)

(1 7 84 59 34)(2 8 85 60 35)(3 9 81 56 31)(4 10 82 57 32)(5 6 83 58 33)(11 110 86 61 36)(12 106 87 62 37)(13 107 88 63 38)(14 108 89 64 39)(15 109 90 65 40)(16 115 91 66 41)(17 111 92 67 42)(18 112 93 68 43)(19 113 94 69 44)(20 114 95 70 45)(21 120 96 71 46)(22 116 97 72 47)(23 117 98 73 48)(24 118 99 74 49)(25 119 100 75 50)(26 125 101 76 51)(27 121 102 77 52)(28 122 103 78 53)(29 123 104 79 54)(30 124 105 80 55)

(1 124 24 19 14)(2 125 25 20 15)(3 121 21 16 11)(4 122 22 17 12)(5 123 23 18 13)(6 104 117 112 107)(7 105 118 113 108)(8 101 119 114 109)(9 102 120 115 110)(10 103 116 111 106)(26 50 45 40 35)(27 46 41 36 31)(28 47 42 37 32)(29 48 43 38 33)(30 49 44 39 34)(51 75 70 65 60)(52 71 66 61 56)(53 72 67 62 57)(54 73 68 63 58)(55 74 69 64 59)(76 100 95 90 85)(77 96 91 86 81)(78 97 92 87 82)(79 98 93 88 83)(80 99 94 89 84)

(2 3 5 4)(6 82 35 56)(7 84 34 59)(8 81 33 57)(9 83 32 60)(10 85 31 58)(11 18 122 25)(12 20 121 23)(13 17 125 21)(14 19 124 24)(15 16 123 22)(26 71 107 92)(27 73 106 95)(28 75 110 93)(29 72 109 91)(30 74 108 94)(36 68 103 100)(37 70 102 98)(38 67 101 96)(39 69 105 99)(40 66 104 97)(41 54 116 90)(42 51 120 88)(43 53 119 86)(44 55 118 89)(45 52 117 87)(46 63 111 76)(47 65 115 79)(48 62 114 77)(49 64 113 80)(50 61 112 78)
 
G:=sub<Sym(125)| (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)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125), (1,7,84,59,34)(2,8,85,60,35)(3,9,81,56,31)(4,10,82,57,32)(5,6,83,58,33)(11,110,86,61,36)(12,106,87,62,37)(13,107,88,63,38)(14,108,89,64,39)(15,109,90,65,40)(16,115,91,66,41)(17,111,92,67,42)(18,112,93,68,43)(19,113,94,69,44)(20,114,95,70,45)(21,120,96,71,46)(22,116,97,72,47)(23,117,98,73,48)(24,118,99,74,49)(25,119,100,75,50)(26,125,101,76,51)(27,121,102,77,52)(28,122,103,78,53)(29,123,104,79,54)(30,124,105,80,55), (1,124,24,19,14)(2,125,25,20,15)(3,121,21,16,11)(4,122,22,17,12)(5,123,23,18,13)(6,104,117,112,107)(7,105,118,113,108)(8,101,119,114,109)(9,102,120,115,110)(10,103,116,111,106)(26,50,45,40,35)(27,46,41,36,31)(28,47,42,37,32)(29,48,43,38,33)(30,49,44,39,34)(51,75,70,65,60)(52,71,66,61,56)(53,72,67,62,57)(54,73,68,63,58)(55,74,69,64,59)(76,100,95,90,85)(77,96,91,86,81)(78,97,92,87,82)(79,98,93,88,83)(80,99,94,89,84), (2,3,5,4)(6,82,35,56)(7,84,34,59)(8,81,33,57)(9,83,32,60)(10,85,31,58)(11,18,122,25)(12,20,121,23)(13,17,125,21)(14,19,124,24)(15,16,123,22)(26,71,107,92)(27,73,106,95)(28,75,110,93)(29,72,109,91)(30,74,108,94)(36,68,103,100)(37,70,102,98)(38,67,101,96)(39,69,105,99)(40,66,104,97)(41,54,116,90)(42,51,120,88)(43,53,119,86)(44,55,118,89)(45,52,117,87)(46,63,111,76)(47,65,115,79)(48,62,114,77)(49,64,113,80)(50,61,112,78)>;
 
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)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125), (1,7,84,59,34)(2,8,85,60,35)(3,9,81,56,31)(4,10,82,57,32)(5,6,83,58,33)(11,110,86,61,36)(12,106,87,62,37)(13,107,88,63,38)(14,108,89,64,39)(15,109,90,65,40)(16,115,91,66,41)(17,111,92,67,42)(18,112,93,68,43)(19,113,94,69,44)(20,114,95,70,45)(21,120,96,71,46)(22,116,97,72,47)(23,117,98,73,48)(24,118,99,74,49)(25,119,100,75,50)(26,125,101,76,51)(27,121,102,77,52)(28,122,103,78,53)(29,123,104,79,54)(30,124,105,80,55), (1,124,24,19,14)(2,125,25,20,15)(3,121,21,16,11)(4,122,22,17,12)(5,123,23,18,13)(6,104,117,112,107)(7,105,118,113,108)(8,101,119,114,109)(9,102,120,115,110)(10,103,116,111,106)(26,50,45,40,35)(27,46,41,36,31)(28,47,42,37,32)(29,48,43,38,33)(30,49,44,39,34)(51,75,70,65,60)(52,71,66,61,56)(53,72,67,62,57)(54,73,68,63,58)(55,74,69,64,59)(76,100,95,90,85)(77,96,91,86,81)(78,97,92,87,82)(79,98,93,88,83)(80,99,94,89,84), (2,3,5,4)(6,82,35,56)(7,84,34,59)(8,81,33,57)(9,83,32,60)(10,85,31,58)(11,18,122,25)(12,20,121,23)(13,17,125,21)(14,19,124,24)(15,16,123,22)(26,71,107,92)(27,73,106,95)(28,75,110,93)(29,72,109,91)(30,74,108,94)(36,68,103,100)(37,70,102,98)(38,67,101,96)(39,69,105,99)(40,66,104,97)(41,54,116,90)(42,51,120,88)(43,53,119,86)(44,55,118,89)(45,52,117,87)(46,63,111,76)(47,65,115,79)(48,62,114,77)(49,64,113,80)(50,61,112,78) );
 
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),(101,102,103,104,105),(106,107,108,109,110),(111,112,113,114,115),(116,117,118,119,120),(121,122,123,124,125)], [(1,7,84,59,34),(2,8,85,60,35),(3,9,81,56,31),(4,10,82,57,32),(5,6,83,58,33),(11,110,86,61,36),(12,106,87,62,37),(13,107,88,63,38),(14,108,89,64,39),(15,109,90,65,40),(16,115,91,66,41),(17,111,92,67,42),(18,112,93,68,43),(19,113,94,69,44),(20,114,95,70,45),(21,120,96,71,46),(22,116,97,72,47),(23,117,98,73,48),(24,118,99,74,49),(25,119,100,75,50),(26,125,101,76,51),(27,121,102,77,52),(28,122,103,78,53),(29,123,104,79,54),(30,124,105,80,55)], [(1,124,24,19,14),(2,125,25,20,15),(3,121,21,16,11),(4,122,22,17,12),(5,123,23,18,13),(6,104,117,112,107),(7,105,118,113,108),(8,101,119,114,109),(9,102,120,115,110),(10,103,116,111,106),(26,50,45,40,35),(27,46,41,36,31),(28,47,42,37,32),(29,48,43,38,33),(30,49,44,39,34),(51,75,70,65,60),(52,71,66,61,56),(53,72,67,62,57),(54,73,68,63,58),(55,74,69,64,59),(76,100,95,90,85),(77,96,91,86,81),(78,97,92,87,82),(79,98,93,88,83),(80,99,94,89,84)], [(2,3,5,4),(6,82,35,56),(7,84,34,59),(8,81,33,57),(9,83,32,60),(10,85,31,58),(11,18,122,25),(12,20,121,23),(13,17,125,21),(14,19,124,24),(15,16,123,22),(26,71,107,92),(27,73,106,95),(28,75,110,93),(29,72,109,91),(30,74,108,94),(36,68,103,100),(37,70,102,98),(38,67,101,96),(39,69,105,99),(40,66,104,97),(41,54,116,90),(42,51,120,88),(43,53,119,86),(44,55,118,89),(45,52,117,87),(46,63,111,76),(47,65,115,79),(48,62,114,77),(49,64,113,80),(50,61,112,78)]])
 

35 conjugacy classes

class 1  2 4A4B5A···5AE
order12445···5
size11251251254···4

35 irreducible representations

dim1114
type+++
imageC1C2C4F5
kernelC53⋊8C4C53⋊C2C53C52
# reps11231

Matrix representation of C53⋊8C4 ►in GL12(ℤ)

010000000000
425000000000
-2-2-2100000000
111-100000000
000010000000
000001000000
000000100000
000000010000
000000000010
000000000001
00000000-1-1-1-1
000000001000
,
010000000000
425000000000
-2-2-2100000000
111-100000000
000000010000
0000-1-1-1-10000
000010000000
000001000000
000000000100
000000000010
000000000001
00000000-1-1-1-1
,
100000000000
010000000000
001000000000
000100000000
000001000000
000000100000
000000010000
0000-1-1-1-10000
000000000100
000000000010
000000000001
00000000-1-1-1-1
,
100000000000
-2-20500000000
010-200000000
-1-1-1100000000
000010000000
000000010000
000001000000
0000-1-1-1-10000
00000000-100-1
000000000-10-1
000000001011
000000001101

G:=sub<GL(12,Integers())| [0,4,-2,1,0,0,0,0,0,0,0,0,1,2,-2,1,0,0,0,0,0,0,0,0,0,5,-2,1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0],[0,4,-2,1,0,0,0,0,0,0,0,0,1,2,-2,1,0,0,0,0,0,0,0,0,0,5,-2,1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[1,-2,0,-1,0,0,0,0,0,0,0,0,0,-2,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,5,-2,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,1,1] >;
 

C53⋊8C4 in GAP, Magma, Sage, TeX 

C_5^3\rtimes_8C_4
 
% in TeX
 
G:=Group("C5^3:8C4");
 
// GroupNames label
 
G:=SmallGroup(500,48);
 
// by ID
 
G=gap.SmallGroup(500,48);
 
# by ID
 
G:=PCGroup([5,-2,-2,-5,-5,-5,10,122,127,803,808,5004,5009]);
 
// 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^3,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^3>;
 
// generators/relations
 

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