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G = C25⋊F5order 500 = 22·53

1st semidirect product of C25 and F5 acting via F5/C5=C4

metabelian, supersoluble, monomial, A-group

Aliases: C251F5, C52.4F5, (C5×C25)⋊5C4, C51(C25⋊C4), C25⋊D5.1C2, C5.(C5⋊F5), SmallGroup(500,22)

Series: Derived Chief Lower central Upper central

C1C5×C25 — C25⋊F5
C1C5C52C5×C25C25⋊D5 — C25⋊F5
C5×C25 — C25⋊F5
C1

Generators and relations for C25⋊F5
 G = < a,b,c | a25=b5=c4=1, ab=ba, cac-1=a18, cbc-1=b3 >

125C2
125C4
25D5
25D5
25D5
25D5
25D5
25D5
25F5
25F5
25F5
25F5
25F5
25F5
5D25
5C5⋊D5
5D25
5D25
5D25
5D25
5C25⋊C4
5C25⋊C4
5C25⋊C4
5C5⋊F5
5C25⋊C4
5C25⋊C4

Smallest permutation representation of C25⋊F5
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 96 124 55 39)(2 97 125 56 40)(3 98 101 57 41)(4 99 102 58 42)(5 100 103 59 43)(6 76 104 60 44)(7 77 105 61 45)(8 78 106 62 46)(9 79 107 63 47)(10 80 108 64 48)(11 81 109 65 49)(12 82 110 66 50)(13 83 111 67 26)(14 84 112 68 27)(15 85 113 69 28)(16 86 114 70 29)(17 87 115 71 30)(18 88 116 72 31)(19 89 117 73 32)(20 90 118 74 33)(21 91 119 75 34)(22 92 120 51 35)(23 93 121 52 36)(24 94 122 53 37)(25 95 123 54 38)
(2 8 25 19)(3 15 24 12)(4 22 23 5)(6 11 21 16)(7 18 20 9)(10 14 17 13)(26 64 84 115)(27 71 83 108)(28 53 82 101)(29 60 81 119)(30 67 80 112)(31 74 79 105)(32 56 78 123)(33 63 77 116)(34 70 76 109)(35 52 100 102)(36 59 99 120)(37 66 98 113)(38 73 97 106)(39 55 96 124)(40 62 95 117)(41 69 94 110)(42 51 93 103)(43 58 92 121)(44 65 91 114)(45 72 90 107)(46 54 89 125)(47 61 88 118)(48 68 87 111)(49 75 86 104)(50 57 85 122)

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

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

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

35 conjugacy classes

class 1  2 4A4B5A···5F25A···25Y
order12445···525···25
size11251251254···44···4

35 irreducible representations

dim111444
type+++++
imageC1C2C4F5F5C25⋊C4
kernelC25⋊F5C25⋊D5C5×C25C25C52C5
# reps1125125

Matrix representation of C25⋊F5 in GL8(𝔽101)

1001001001000000
10000000
01000000
00100000
00007847032
000069467338
00006331835
000066289774
,
1001001001000000
10000000
01000000
00100000
00001000
00000100
00000010
00000001
,
001001000000
100100000000
11100000
010010000000
00001000
00000001
00000100
0000100100100100

G:=sub<GL(8,GF(101))| [100,1,0,0,0,0,0,0,100,0,1,0,0,0,0,0,100,0,0,1,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,0,78,69,63,66,0,0,0,0,4,46,31,28,0,0,0,0,70,73,8,97,0,0,0,0,32,38,35,74],[100,1,0,0,0,0,0,0,100,0,1,0,0,0,0,0,100,0,0,1,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,100,1,0,0,0,0,0,0,100,1,100,0,0,0,0,100,0,1,100,0,0,0,0,100,0,0,0,0,0,0,0,0,0,0,0,1,0,0,100,0,0,0,0,0,0,1,100,0,0,0,0,0,0,0,100,0,0,0,0,0,1,0,100] >;

C25⋊F5 in GAP, Magma, Sage, TeX

C_{25}\rtimes F_5
% in TeX

G:=Group("C25:F5");
// GroupNames label

G:=SmallGroup(500,22);
// by ID

G=gap.SmallGroup(500,22);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,10,1622,3127,387,803,808,5004,5009]);
// Polycyclic

G:=Group<a,b,c|a^25=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^18,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C25⋊F5 in TeX

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