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G = F5×C23order 460 = 22·5·23

Direct product of C23 and F5

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C5 — F5×C23
C1C5D5D5×C23 — F5×C23
C5 — F5×C23
C1C23

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

5C2
5C4
5C46
5C92

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 4A4B 5 23A···23V46A···46V92A···92AR115A···115V
order1244523···2346···4692···92115···115
size155541···15···55···54···4

115 irreducible representations

dim11111144
type+++
imageC1C2C4C23C46C92F5F5×C23
kernelF5×C23D5×C23C115F5D5C5C23C1
# reps112222244122

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

23000
02300
00230
00023
,
460460460460
1000
0100
0010
,
0010
1000
0001
0100
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

Export

Subgroup lattice of F5×C23 in TeX

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