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G = C538C4order 500 = 22·53

8th semidirect product of C53 and C4 acting faithfully

metabelian, supersoluble, monomial, A-group

Aliases: C538C4, C529F5, C51(C5⋊F5), C53⋊C2.1C2, SmallGroup(500,48)

Series: Derived Chief Lower central Upper central

C1C53 — C538C4
C1C5C52C53C53⋊C2 — C538C4
C53 — C538C4
C1

Generators and relations for C538C4
 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 [×31], D5 [×31], F5 [×31], C52 [×31], C5⋊D5 [×31], C5⋊F5 [×31], C53, C53⋊C2, C538C4
Quotients: C1, C2, C4, F5 [×31], C5⋊F5 [×31], C538C4

Smallest permutation representation of C538C4
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)
(1 108 89 14 64)(2 109 90 15 65)(3 110 86 11 61)(4 106 87 12 62)(5 107 88 13 63)(6 56 31 39 81)(7 57 32 40 82)(8 58 33 36 83)(9 59 34 37 84)(10 60 35 38 85)(16 66 41 115 91)(17 67 42 111 92)(18 68 43 112 93)(19 69 44 113 94)(20 70 45 114 95)(21 71 46 120 96)(22 72 47 116 97)(23 73 48 117 98)(24 74 49 118 99)(25 75 50 119 100)(26 76 51 125 101)(27 77 52 121 102)(28 78 53 122 103)(29 79 54 123 104)(30 80 55 124 105)
(1 124 49 34 44)(2 125 50 35 45)(3 121 46 31 41)(4 122 47 32 42)(5 123 48 33 43)(6 16 11 77 21)(7 17 12 78 22)(8 18 13 79 23)(9 19 14 80 24)(10 20 15 76 25)(26 100 85 95 90)(27 96 81 91 86)(28 97 82 92 87)(29 98 83 93 88)(30 99 84 94 89)(36 112 107 104 117)(37 113 108 105 118)(38 114 109 101 119)(39 115 110 102 120)(40 111 106 103 116)(51 75 60 70 65)(52 71 56 66 61)(53 72 57 67 62)(54 73 58 68 63)(55 74 59 69 64)
(2 3 5 4)(6 104 97 70)(7 101 96 68)(8 103 100 66)(9 105 99 69)(10 102 98 67)(11 107 87 65)(12 109 86 63)(13 106 90 61)(14 108 89 64)(15 110 88 62)(16 36 28 75)(17 38 27 73)(18 40 26 71)(19 37 30 74)(20 39 29 72)(21 112 82 51)(22 114 81 54)(23 111 85 52)(24 113 84 55)(25 115 83 53)(31 123 47 45)(32 125 46 43)(33 122 50 41)(34 124 49 44)(35 121 48 42)(56 79 116 95)(57 76 120 93)(58 78 119 91)(59 80 118 94)(60 77 117 92)

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

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

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

35 conjugacy classes

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

35 irreducible representations

dim1114
type+++
imageC1C2C4F5
kernelC538C4C53⋊C2C53C52
# reps11231

Matrix representation of C538C4 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] >;

C538C4 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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