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G = C23⋊F5order 460 = 22·5·23

The semidirect product of C23 and F5 acting via F5/D5=C2

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

Aliases: C23⋊F5, C5⋊Dic23, C1151C4, D5.D23, (D5×C23).1C2, SmallGroup(460,6)

Series: Derived Chief Lower central Upper central

C1C115 — C23⋊F5
C1C23C115D5×C23 — C23⋊F5
C115 — C23⋊F5
C1

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

5C2
115C4
5C46
23F5
5Dic23

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

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

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

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

49 conjugacy classes

class 1  2 4A4B 5 23A···23K46A···46K115A···115V
order1244523···2346···46115···115
size1511511542···210···104···4

49 irreducible representations

dim1112244
type+++-+
imageC1C2C4D23Dic23F5C23⋊F5
kernelC23⋊F5D5×C23C115D5C5C23C1
# reps1121111122

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

0100
46036300
0001
00460363
,
963834600
783640460
1000
0100
,
1000
36346000
3657836578
2669626696
G:=sub<GL(4,GF(461))| [0,460,0,0,1,363,0,0,0,0,0,460,0,0,1,363],[96,78,1,0,383,364,0,1,460,0,0,0,0,460,0,0],[1,363,365,266,0,460,78,96,0,0,365,266,0,0,78,96] >;

C23⋊F5 in GAP, Magma, Sage, TeX

C_{23}\rtimes F_5
% in TeX

G:=Group("C23:F5");
// GroupNames label

G:=SmallGroup(460,6);
// by ID

G=gap.SmallGroup(460,6);
# by ID

G:=PCGroup([4,-2,-2,-5,-23,8,146,102,7043]);
// Polycyclic

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

Export

Subgroup lattice of C23⋊F5 in TeX

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