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G = D5×C27order 270 = 2·33·5

Direct product of C27 and D5

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

Aliases: D5×C27, C5⋊C54, C45.C6, C1353C2, C15.C18, C9.(C3×D5), C3.(C9×D5), (C9×D5).C3, (C3×D5).C9, SmallGroup(270,2)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C27
C1C5C15C45C135 — D5×C27
C5 — D5×C27
C1C27

Generators and relations for D5×C27
 G = < a,b,c | a27=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C6
5C18
5C54

Smallest permutation representation of D5×C27
On 135 points
Generators in S135
(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)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135)
(1 70 111 106 52)(2 71 112 107 53)(3 72 113 108 54)(4 73 114 82 28)(5 74 115 83 29)(6 75 116 84 30)(7 76 117 85 31)(8 77 118 86 32)(9 78 119 87 33)(10 79 120 88 34)(11 80 121 89 35)(12 81 122 90 36)(13 55 123 91 37)(14 56 124 92 38)(15 57 125 93 39)(16 58 126 94 40)(17 59 127 95 41)(18 60 128 96 42)(19 61 129 97 43)(20 62 130 98 44)(21 63 131 99 45)(22 64 132 100 46)(23 65 133 101 47)(24 66 134 102 48)(25 67 135 103 49)(26 68 109 104 50)(27 69 110 105 51)
(1 52)(2 53)(3 54)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)

G:=sub<Sym(135)| (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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135), (1,70,111,106,52)(2,71,112,107,53)(3,72,113,108,54)(4,73,114,82,28)(5,74,115,83,29)(6,75,116,84,30)(7,76,117,85,31)(8,77,118,86,32)(9,78,119,87,33)(10,79,120,88,34)(11,80,121,89,35)(12,81,122,90,36)(13,55,123,91,37)(14,56,124,92,38)(15,57,125,93,39)(16,58,126,94,40)(17,59,127,95,41)(18,60,128,96,42)(19,61,129,97,43)(20,62,130,98,44)(21,63,131,99,45)(22,64,132,100,46)(23,65,133,101,47)(24,66,134,102,48)(25,67,135,103,49)(26,68,109,104,50)(27,69,110,105,51), (1,52)(2,53)(3,54)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,82)(74,83)(75,84)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90)>;

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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135), (1,70,111,106,52)(2,71,112,107,53)(3,72,113,108,54)(4,73,114,82,28)(5,74,115,83,29)(6,75,116,84,30)(7,76,117,85,31)(8,77,118,86,32)(9,78,119,87,33)(10,79,120,88,34)(11,80,121,89,35)(12,81,122,90,36)(13,55,123,91,37)(14,56,124,92,38)(15,57,125,93,39)(16,58,126,94,40)(17,59,127,95,41)(18,60,128,96,42)(19,61,129,97,43)(20,62,130,98,44)(21,63,131,99,45)(22,64,132,100,46)(23,65,133,101,47)(24,66,134,102,48)(25,67,135,103,49)(26,68,109,104,50)(27,69,110,105,51), (1,52)(2,53)(3,54)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,82)(74,83)(75,84)(76,85)(77,86)(78,87)(79,88)(80,89)(81,90) );

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),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135)], [(1,70,111,106,52),(2,71,112,107,53),(3,72,113,108,54),(4,73,114,82,28),(5,74,115,83,29),(6,75,116,84,30),(7,76,117,85,31),(8,77,118,86,32),(9,78,119,87,33),(10,79,120,88,34),(11,80,121,89,35),(12,81,122,90,36),(13,55,123,91,37),(14,56,124,92,38),(15,57,125,93,39),(16,58,126,94,40),(17,59,127,95,41),(18,60,128,96,42),(19,61,129,97,43),(20,62,130,98,44),(21,63,131,99,45),(22,64,132,100,46),(23,65,133,101,47),(24,66,134,102,48),(25,67,135,103,49),(26,68,109,104,50),(27,69,110,105,51)], [(1,52),(2,53),(3,54),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,51),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(67,103),(68,104),(69,105),(70,106),(71,107),(72,108),(73,82),(74,83),(75,84),(76,85),(77,86),(78,87),(79,88),(80,89),(81,90)])

108 conjugacy classes

class 1  2 3A3B5A5B6A6B9A···9F15A15B15C15D18A···18F27A···27R45A···45L54A···54R135A···135AJ
order123355669···91515151518···1827···2745···4554···54135···135
size151122551···122225···51···12···25···52···2

108 irreducible representations

dim111111112222
type+++
imageC1C2C3C6C9C18C27C54D5C3×D5C9×D5D5×C27
kernelD5×C27C135C9×D5C45C3×D5C15D5C5C27C9C3C1
# reps1122661818241236

Matrix representation of D5×C27 in GL2(𝔽271) generated by

1580
0158
,
01
270254
,
01
10
G:=sub<GL(2,GF(271))| [158,0,0,158],[0,270,1,254],[0,1,1,0] >;

D5×C27 in GAP, Magma, Sage, TeX

D_5\times C_{27}
% in TeX

G:=Group("D5xC27");
// GroupNames label

G:=SmallGroup(270,2);
// by ID

G=gap.SmallGroup(270,2);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-5,36,57,5404]);
// Polycyclic

G:=Group<a,b,c|a^27=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×C27 in TeX

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