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G = D65⋊C3order 390 = 2·3·5·13

The semidirect product of D65 and C3 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: D65⋊C3, C651C6, C13⋊C3⋊D5, C5⋊(C13⋊C6), C13⋊(C3×D5), (C5×C13⋊C3)⋊1C2, SmallGroup(390,3)

Series: Derived Chief Lower central Upper central

C1C65 — D65⋊C3
C1C13C65C5×C13⋊C3 — D65⋊C3
C65 — D65⋊C3
C1

Generators and relations for D65⋊C3
 G = < a,b,c | a65=b2=c3=1, bab=a-1, cac-1=a61, cbc-1=a60b >

65C2
13C3
65C6
13D5
13C15
5D13
13C3×D5
5C13⋊C6

Character table of D65⋊C3

 class 123A3B5A5B6A6B13A13B15A15B15C15D65A65B65C65D65E65F65G65H
 size 1651313226565662626262666666666
ρ11111111111111111111111    trivial
ρ21-11111-1-111111111111111    linear of order 2
ρ311ζ3ζ3211ζ32ζ311ζ3ζ3ζ32ζ3211111111    linear of order 3
ρ41-1ζ32ζ311ζ65ζ611ζ32ζ32ζ3ζ311111111    linear of order 6
ρ511ζ32ζ311ζ3ζ3211ζ32ζ32ζ3ζ311111111    linear of order 3
ρ61-1ζ3ζ3211ζ6ζ6511ζ3ζ3ζ32ζ3211111111    linear of order 6
ρ72022-1-5/2-1+5/20022-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ82022-1+5/2-1-5/20022-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1-5/2    orthogonal lifted from D5
ρ920-1--3-1+-3-1+5/2-1-5/20022ζ32ζ5432ζ5ζ32ζ5332ζ52ζ3ζ533ζ52ζ3ζ543ζ5-1+5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1-5/2    complex lifted from C3×D5
ρ1020-1--3-1+-3-1-5/2-1+5/20022ζ32ζ5332ζ52ζ32ζ5432ζ5ζ3ζ543ζ5ζ3ζ533ζ52-1-5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1+5/2    complex lifted from C3×D5
ρ1120-1+-3-1--3-1+5/2-1-5/20022ζ3ζ543ζ5ζ3ζ533ζ52ζ32ζ5332ζ52ζ32ζ5432ζ5-1+5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1-5/2    complex lifted from C3×D5
ρ1220-1+-3-1--3-1-5/2-1+5/20022ζ3ζ533ζ52ζ3ζ543ζ5ζ32ζ5432ζ5ζ32ζ5332ζ52-1-5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1+5/2    complex lifted from C3×D5
ρ1360006600-1+13/2-1-13/20000-1+13/2-1-13/2-1-13/2-1+13/2-1-13/2-1+13/2-1-13/2-1+13/2    orthogonal lifted from C13⋊C6
ρ1460006600-1-13/2-1+13/20000-1-13/2-1+13/2-1+13/2-1-13/2-1+13/2-1-13/2-1+13/2-1-13/2    orthogonal lifted from C13⋊C6
ρ156000-3-35/2-3+35/200-1-13/2-1+13/20000ζ53ζ13653ζ13553ζ13252ζ131152ζ13852ζ137ζ53ζ131253ζ131053ζ13452ζ13952ζ13352ζ13ζ54ζ13954ζ13354ζ135ζ13125ζ13105ζ134ζ53ζ131153ζ13853ζ13752ζ13652ζ13552ζ132ζ53ζ13953ζ13353ζ1352ζ131252ζ131052ζ134ζ54ζ13654ζ13554ζ1325ζ13115ζ1385ζ137ζ54ζ131254ζ131054ζ1345ζ1395ζ1335ζ13ζ54ζ131154ζ13854ζ1375ζ1365ζ1355ζ132    orthogonal faithful
ρ166000-3+35/2-3-35/200-1+13/2-1-13/20000ζ54ζ131254ζ131054ζ1345ζ1395ζ1335ζ13ζ54ζ131154ζ13854ζ1375ζ1365ζ1355ζ132ζ53ζ131153ζ13853ζ13752ζ13652ζ13552ζ132ζ54ζ13954ζ13354ζ135ζ13125ζ13105ζ134ζ54ζ13654ζ13554ζ1325ζ13115ζ1385ζ137ζ53ζ13953ζ13353ζ1352ζ131252ζ131052ζ134ζ53ζ13653ζ13553ζ13252ζ131152ζ13852ζ137ζ53ζ131253ζ131053ζ13452ζ13952ζ13352ζ13    orthogonal faithful
ρ176000-3-35/2-3+35/200-1+13/2-1-13/20000ζ53ζ131253ζ131053ζ13452ζ13952ζ13352ζ13ζ53ζ131153ζ13853ζ13752ζ13652ζ13552ζ132ζ54ζ13654ζ13554ζ1325ζ13115ζ1385ζ137ζ53ζ13953ζ13353ζ1352ζ131252ζ131052ζ134ζ53ζ13653ζ13553ζ13252ζ131152ζ13852ζ137ζ54ζ131254ζ131054ζ1345ζ1395ζ1335ζ13ζ54ζ131154ζ13854ζ1375ζ1365ζ1355ζ132ζ54ζ13954ζ13354ζ135ζ13125ζ13105ζ134    orthogonal faithful
ρ186000-3+35/2-3-35/200-1-13/2-1+13/20000ζ54ζ13654ζ13554ζ1325ζ13115ζ1385ζ137ζ54ζ131254ζ131054ζ1345ζ1395ζ1335ζ13ζ53ζ131253ζ131053ζ13452ζ13952ζ13352ζ13ζ54ζ131154ζ13854ζ1375ζ1365ζ1355ζ132ζ54ζ13954ζ13354ζ135ζ13125ζ13105ζ134ζ53ζ131153ζ13853ζ13752ζ13652ζ13552ζ132ζ53ζ13953ζ13353ζ1352ζ131252ζ131052ζ134ζ53ζ13653ζ13553ζ13252ζ131152ζ13852ζ137    orthogonal faithful
ρ196000-3+35/2-3-35/200-1-13/2-1+13/20000ζ54ζ131154ζ13854ζ1375ζ1365ζ1355ζ132ζ54ζ13954ζ13354ζ135ζ13125ζ13105ζ134ζ53ζ13953ζ13353ζ1352ζ131252ζ131052ζ134ζ54ζ13654ζ13554ζ1325ζ13115ζ1385ζ137ζ54ζ131254ζ131054ζ1345ζ1395ζ1335ζ13ζ53ζ13653ζ13553ζ13252ζ131152ζ13852ζ137ζ53ζ131253ζ131053ζ13452ζ13952ζ13352ζ13ζ53ζ131153ζ13853ζ13752ζ13652ζ13552ζ132    orthogonal faithful
ρ206000-3+35/2-3-35/200-1+13/2-1-13/20000ζ54ζ13954ζ13354ζ135ζ13125ζ13105ζ134ζ54ζ13654ζ13554ζ1325ζ13115ζ1385ζ137ζ53ζ13653ζ13553ζ13252ζ131152ζ13852ζ137ζ54ζ131254ζ131054ζ1345ζ1395ζ1335ζ13ζ54ζ131154ζ13854ζ1375ζ1365ζ1355ζ132ζ53ζ131253ζ131053ζ13452ζ13952ζ13352ζ13ζ53ζ131153ζ13853ζ13752ζ13652ζ13552ζ132ζ53ζ13953ζ13353ζ1352ζ131252ζ131052ζ134    orthogonal faithful
ρ216000-3-35/2-3+35/200-1-13/2-1+13/20000ζ53ζ131153ζ13853ζ13752ζ13652ζ13552ζ132ζ53ζ13953ζ13353ζ1352ζ131252ζ131052ζ134ζ54ζ131254ζ131054ζ1345ζ1395ζ1335ζ13ζ53ζ13653ζ13553ζ13252ζ131152ζ13852ζ137ζ53ζ131253ζ131053ζ13452ζ13952ζ13352ζ13ζ54ζ131154ζ13854ζ1375ζ1365ζ1355ζ132ζ54ζ13954ζ13354ζ135ζ13125ζ13105ζ134ζ54ζ13654ζ13554ζ1325ζ13115ζ1385ζ137    orthogonal faithful
ρ226000-3-35/2-3+35/200-1+13/2-1-13/20000ζ53ζ13953ζ13353ζ1352ζ131252ζ131052ζ134ζ53ζ13653ζ13553ζ13252ζ131152ζ13852ζ137ζ54ζ131154ζ13854ζ1375ζ1365ζ1355ζ132ζ53ζ131253ζ131053ζ13452ζ13952ζ13352ζ13ζ53ζ131153ζ13853ζ13752ζ13652ζ13552ζ132ζ54ζ13954ζ13354ζ135ζ13125ζ13105ζ134ζ54ζ13654ζ13554ζ1325ζ13115ζ1385ζ137ζ54ζ131254ζ131054ζ1345ζ1395ζ1335ζ13    orthogonal faithful

Smallest permutation representation of D65⋊C3
On 65 points
Generators in S65
(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)
(1 65)(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)
(2 17 62)(3 33 58)(4 49 54)(5 65 50)(6 16 46)(7 32 42)(8 48 38)(9 64 34)(10 15 30)(11 31 26)(12 47 22)(13 63 18)(19 29 59)(20 45 55)(21 61 51)(23 28 43)(24 44 39)(25 60 35)(36 41 56)(37 57 52)

G:=sub<Sym(65)| (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), (1,65)(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34), (2,17,62)(3,33,58)(4,49,54)(5,65,50)(6,16,46)(7,32,42)(8,48,38)(9,64,34)(10,15,30)(11,31,26)(12,47,22)(13,63,18)(19,29,59)(20,45,55)(21,61,51)(23,28,43)(24,44,39)(25,60,35)(36,41,56)(37,57,52)>;

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), (1,65)(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34), (2,17,62)(3,33,58)(4,49,54)(5,65,50)(6,16,46)(7,32,42)(8,48,38)(9,64,34)(10,15,30)(11,31,26)(12,47,22)(13,63,18)(19,29,59)(20,45,55)(21,61,51)(23,28,43)(24,44,39)(25,60,35)(36,41,56)(37,57,52) );

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)], [(1,65),(2,64),(3,63),(4,62),(5,61),(6,60),(7,59),(8,58),(9,57),(10,56),(11,55),(12,54),(13,53),(14,52),(15,51),(16,50),(17,49),(18,48),(19,47),(20,46),(21,45),(22,44),(23,43),(24,42),(25,41),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34)], [(2,17,62),(3,33,58),(4,49,54),(5,65,50),(6,16,46),(7,32,42),(8,48,38),(9,64,34),(10,15,30),(11,31,26),(12,47,22),(13,63,18),(19,29,59),(20,45,55),(21,61,51),(23,28,43),(24,44,39),(25,60,35),(36,41,56),(37,57,52)])

Matrix representation of D65⋊C3 in GL6(𝔽1171)

15239239844391126
1045592958805592831
340880101453314427
74453026718770504
6671626931119881188
98389695711107431110
,
15239239844391126
1146110693287621019
87098299711278325
6142861214275301
95711108041538301110
2146121487214214
,
100000
742311741741311742
000001
010000
4298611169431428430
117043111694321169431

G:=sub<GL(6,GF(1171))| [152,1045,340,744,667,983,39,592,880,530,162,896,239,958,101,26,693,957,844,805,453,718,1119,1110,391,592,314,770,881,743,126,831,427,504,188,1110],[152,1146,870,61,957,214,39,1106,982,428,1110,61,239,932,997,61,804,214,844,87,112,214,153,87,391,62,783,275,830,214,126,1019,25,301,1110,214],[1,742,0,0,429,1170,0,311,0,1,861,431,0,741,0,0,1169,1169,0,741,0,0,431,432,0,311,0,0,428,1169,0,742,1,0,430,431] >;

D65⋊C3 in GAP, Magma, Sage, TeX

D_{65}\rtimes C_3
% in TeX

G:=Group("D65:C3");
// GroupNames label

G:=SmallGroup(390,3);
// by ID

G=gap.SmallGroup(390,3);
# by ID

G:=PCGroup([4,-2,-3,-5,-13,290,5763,727]);
// Polycyclic

G:=Group<a,b,c|a^65=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^61,c*b*c^-1=a^60*b>;
// generators/relations

Export

Subgroup lattice of D65⋊C3 in TeX
Character table of D65⋊C3 in TeX

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