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G = C5×D23order 230 = 2·5·23

Direct product of C5 and D23

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

Aliases: C5×D23, C23⋊C10, C1152C2, SmallGroup(230,2)

Series: Derived Chief Lower central Upper central

C1C23 — C5×D23
C1C23C115 — C5×D23
C23 — C5×D23
C1C5

Generators and relations for C5×D23
 G = < a,b,c | a5=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >

23C2
23C10

Smallest permutation representation of C5×D23
On 115 points
Generators in S115
(1 114 87 56 31)(2 115 88 57 32)(3 93 89 58 33)(4 94 90 59 34)(5 95 91 60 35)(6 96 92 61 36)(7 97 70 62 37)(8 98 71 63 38)(9 99 72 64 39)(10 100 73 65 40)(11 101 74 66 41)(12 102 75 67 42)(13 103 76 68 43)(14 104 77 69 44)(15 105 78 47 45)(16 106 79 48 46)(17 107 80 49 24)(18 108 81 50 25)(19 109 82 51 26)(20 110 83 52 27)(21 111 84 53 28)(22 112 85 54 29)(23 113 86 55 30)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(38 46)(39 45)(40 44)(41 43)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(65 69)(66 68)(70 80)(71 79)(72 78)(73 77)(74 76)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)(112 115)(113 114)

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

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

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

65 conjugacy classes

class 1  2 5A5B5C5D10A10B10C10D23A···23K115A···115AR
order1255551010101023···23115···115
size1231111232323232···22···2

65 irreducible representations

dim111122
type+++
imageC1C2C5C10D23C5×D23
kernelC5×D23C115D23C23C5C1
# reps11441144

Matrix representation of C5×D23 in GL2(𝔽461) generated by

3680
0368
,
01
460418
,
01
10
G:=sub<GL(2,GF(461))| [368,0,0,368],[0,460,1,418],[0,1,1,0] >;

C5×D23 in GAP, Magma, Sage, TeX

C_5\times D_{23}
% in TeX

G:=Group("C5xD23");
// GroupNames label

G:=SmallGroup(230,2);
// by ID

G=gap.SmallGroup(230,2);
# by ID

G:=PCGroup([3,-2,-5,-23,1982]);
// Polycyclic

G:=Group<a,b,c|a^5=b^23=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D23 in TeX

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