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## G = F5×C23order 460 = 22·5·23

### Direct product of C23 and F5

Aliases: F5×C23, C5⋊C92, C1152C4, D5.C46, (D5×C23).2C2, SmallGroup(460,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — F5×C23
 Chief series C1 — C5 — D5 — D5×C23 — F5×C23
 Lower central C5 — F5×C23
 Upper central C1 — C23

Generators and relations for F5×C23
G = < a,b,c | a23=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of F5×C23
On 115 points
Generators in S115
(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)
(1 106 31 66 86)(2 107 32 67 87)(3 108 33 68 88)(4 109 34 69 89)(5 110 35 47 90)(6 111 36 48 91)(7 112 37 49 92)(8 113 38 50 70)(9 114 39 51 71)(10 115 40 52 72)(11 93 41 53 73)(12 94 42 54 74)(13 95 43 55 75)(14 96 44 56 76)(15 97 45 57 77)(16 98 46 58 78)(17 99 24 59 79)(18 100 25 60 80)(19 101 26 61 81)(20 102 27 62 82)(21 103 28 63 83)(22 104 29 64 84)(23 105 30 65 85)
(24 79 59 99)(25 80 60 100)(26 81 61 101)(27 82 62 102)(28 83 63 103)(29 84 64 104)(30 85 65 105)(31 86 66 106)(32 87 67 107)(33 88 68 108)(34 89 69 109)(35 90 47 110)(36 91 48 111)(37 92 49 112)(38 70 50 113)(39 71 51 114)(40 72 52 115)(41 73 53 93)(42 74 54 94)(43 75 55 95)(44 76 56 96)(45 77 57 97)(46 78 58 98)

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

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

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

115 conjugacy classes

 class 1 2 4A 4B 5 23A ··· 23V 46A ··· 46V 92A ··· 92AR 115A ··· 115V order 1 2 4 4 5 23 ··· 23 46 ··· 46 92 ··· 92 115 ··· 115 size 1 5 5 5 4 1 ··· 1 5 ··· 5 5 ··· 5 4 ··· 4

115 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + image C1 C2 C4 C23 C46 C92 F5 F5×C23 kernel F5×C23 D5×C23 C115 F5 D5 C5 C23 C1 # reps 1 1 2 22 22 44 1 22

Matrix representation of F5×C23 in GL4(𝔽461) generated by

 23 0 0 0 0 23 0 0 0 0 23 0 0 0 0 23
,
 460 460 460 460 1 0 0 0 0 1 0 0 0 0 1 0
,
 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(461))| [23,0,0,0,0,23,0,0,0,0,23,0,0,0,0,23],[460,1,0,0,460,0,1,0,460,0,0,1,460,0,0,0],[0,1,0,0,0,0,0,1,1,0,0,0,0,0,1,0] >;

F5×C23 in GAP, Magma, Sage, TeX

F_5\times C_{23}
% in TeX

G:=Group("F5xC23");
// GroupNames label

G:=SmallGroup(460,5);
// by ID

G=gap.SmallGroup(460,5);
# by ID

G:=PCGroup([4,-2,-23,-2,-5,184,2947,139]);
// Polycyclic

G:=Group<a,b,c|a^23=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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