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G = D5×C23order 230 = 2·5·23

Direct product of C23 and D5

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

Aliases: D5×C23, C5⋊C46, C1153C2, SmallGroup(230,1)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C23
C1C5C115 — D5×C23
C5 — D5×C23
C1C23

Generators and relations for D5×C23
 G = < a,b,c | a23=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C46

Smallest permutation representation of D5×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 83 42 106 57)(2 84 43 107 58)(3 85 44 108 59)(4 86 45 109 60)(5 87 46 110 61)(6 88 24 111 62)(7 89 25 112 63)(8 90 26 113 64)(9 91 27 114 65)(10 92 28 115 66)(11 70 29 93 67)(12 71 30 94 68)(13 72 31 95 69)(14 73 32 96 47)(15 74 33 97 48)(16 75 34 98 49)(17 76 35 99 50)(18 77 36 100 51)(19 78 37 101 52)(20 79 38 102 53)(21 80 39 103 54)(22 81 40 104 55)(23 82 41 105 56)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(70 93)(71 94)(72 95)(73 96)(74 97)(75 98)(76 99)(77 100)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)

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

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

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

D5×C23 is a maximal subgroup of   C23⋊F5

92 conjugacy classes

class 1  2 5A5B23A···23V46A···46V115A···115AR
order125523···2346···46115···115
size15221···15···52···2

92 irreducible representations

dim111122
type+++
imageC1C2C23C46D5D5×C23
kernelD5×C23C115D5C5C23C1
# reps112222244

Matrix representation of D5×C23 in GL2(𝔽461) generated by

4410
0441
,
211
4600
,
01
10
G:=sub<GL(2,GF(461))| [441,0,0,441],[21,460,1,0],[0,1,1,0] >;

D5×C23 in GAP, Magma, Sage, TeX

D_5\times C_{23}
% in TeX

G:=Group("D5xC23");
// GroupNames label

G:=SmallGroup(230,1);
// by ID

G=gap.SmallGroup(230,1);
# by ID

G:=PCGroup([3,-2,-23,-5,1658]);
// Polycyclic

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

Export

Subgroup lattice of D5×C23 in TeX

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