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G = D52⋊C3order 312 = 23·3·13

The semidirect product of D52 and C3 acting faithfully

metacyclic, supersoluble, monomial

Aliases: D52⋊C3, C521C6, D261C6, C4⋊(C13⋊C6), C13⋊C31D4, C131(C3×D4), C26.3(C2×C6), (C2×C13⋊C6)⋊1C2, (C4×C13⋊C3)⋊1C2, C2.4(C2×C13⋊C6), (C2×C13⋊C3).3C22, SmallGroup(312,10)

Series: Derived Chief Lower central Upper central

C1C26 — D52⋊C3
C1C13C26C2×C13⋊C3C2×C13⋊C6 — D52⋊C3
C13C26 — D52⋊C3
C1C2C4

Generators and relations for D52⋊C3
 G = < a,b,c | a52=b2=c3=1, bab=a-1, cac-1=a9, cbc-1=a8b >

26C2
26C2
13C3
13C22
13C22
13C6
26C6
26C6
2D13
2D13
13D4
13C2×C6
13C2×C6
13C12
2C13⋊C6
2C13⋊C6
13C3×D4

Character table of D52⋊C3

 class 12A2B2C3A3B46A6B6C6D6E6F12A12B13A13B26A26B52A52B52C52D
 size 11262613132131326262626262666666666
ρ111111111111111111111111    trivial
ρ211-1111-111-11-11-1-11111-1-1-1-1    linear of order 2
ρ311-1-111111-1-1-1-11111111111    linear of order 2
ρ4111-111-1111-11-1-1-11111-1-1-1-1    linear of order 2
ρ5111-1ζ32ζ3-1ζ32ζ3ζ32ζ65ζ3ζ6ζ6ζ651111-1-1-1-1    linear of order 6
ρ61111ζ3ζ321ζ3ζ32ζ3ζ32ζ32ζ3ζ3ζ3211111111    linear of order 3
ρ711-11ζ32ζ3-1ζ32ζ3ζ6ζ3ζ65ζ32ζ6ζ651111-1-1-1-1    linear of order 6
ρ811-11ζ3ζ32-1ζ3ζ32ζ65ζ32ζ6ζ3ζ65ζ61111-1-1-1-1    linear of order 6
ρ9111-1ζ3ζ32-1ζ3ζ32ζ3ζ6ζ32ζ65ζ65ζ61111-1-1-1-1    linear of order 6
ρ101111ζ32ζ31ζ32ζ3ζ32ζ3ζ3ζ32ζ32ζ311111111    linear of order 3
ρ1111-1-1ζ3ζ321ζ3ζ32ζ65ζ6ζ6ζ65ζ3ζ3211111111    linear of order 6
ρ1211-1-1ζ32ζ31ζ32ζ3ζ6ζ65ζ65ζ6ζ32ζ311111111    linear of order 6
ρ132-200220-2-200000022-2-20000    orthogonal lifted from D4
ρ142-200-1+-3-1--301--31+-300000022-2-20000    complex lifted from C3×D4
ρ152-200-1--3-1+-301+-31--300000022-2-20000    complex lifted from C3×D4
ρ16660000-600000000-1-13/2-1+13/2-1+13/2-1-13/21-13/21+13/21-13/21+13/2    orthogonal lifted from C2×C13⋊C6
ρ17660000600000000-1+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
ρ18660000-600000000-1+13/2-1-13/2-1-13/2-1+13/21+13/21-13/21+13/21-13/2    orthogonal lifted from C2×C13⋊C6
ρ19660000600000000-1-13/2-1+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
ρ206-60000000000000-1-13/2-1+13/21-13/21+13/243ζ131243ζ131043ζ13943ζ13443ζ13343ζ13ζ4ζ13114ζ1384ζ1374ζ1364ζ1354ζ1324ζ13124ζ13104ζ1394ζ1344ζ1334ζ13ζ43ζ131143ζ13843ζ13743ζ13643ζ13543ζ132    orthogonal faithful
ρ216-60000000000000-1-13/2-1+13/21-13/21+13/24ζ13124ζ13104ζ1394ζ1344ζ1334ζ13ζ43ζ131143ζ13843ζ13743ζ13643ζ13543ζ13243ζ131243ζ131043ζ13943ζ13443ζ13343ζ13ζ4ζ13114ζ1384ζ1374ζ1364ζ1354ζ132    orthogonal faithful
ρ226-60000000000000-1+13/2-1-13/21+13/21-13/2ζ4ζ13114ζ1384ζ1374ζ1364ζ1354ζ1324ζ13124ζ13104ζ1394ζ1344ζ1334ζ13ζ43ζ131143ζ13843ζ13743ζ13643ζ13543ζ13243ζ131243ζ131043ζ13943ζ13443ζ13343ζ13    orthogonal faithful
ρ236-60000000000000-1+13/2-1-13/21+13/21-13/2ζ43ζ131143ζ13843ζ13743ζ13643ζ13543ζ13243ζ131243ζ131043ζ13943ζ13443ζ13343ζ13ζ4ζ13114ζ1384ζ1374ζ1364ζ1354ζ1324ζ13124ζ13104ζ1394ζ1344ζ1334ζ13    orthogonal faithful

Smallest permutation representation of D52⋊C3
On 52 points
Generators in S52
(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)
(1 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)
(2 30 10)(3 7 19)(4 36 28)(5 13 37)(6 42 46)(8 48 12)(9 25 21)(11 31 39)(15 43 23)(16 20 32)(17 49 41)(18 26 50)(22 38 34)(24 44 52)(29 33 45)(35 51 47)

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

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

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

Matrix representation of D52⋊C3 in GL6(𝔽157)

4111413444
153461035012646
11181114115138
191474612842151
61132134140136
212115515319155
,
4111413444
1531532315346153
11131107541114
19421534614946
61152911110138
21172315544151
,
100000
70138696913870
000001
010000
8720155898688
156891559015589

G:=sub<GL(6,GF(157))| [4,153,111,19,6,21,111,46,8,147,113,21,4,103,111,46,2,155,134,50,4,128,134,153,4,126,115,42,140,19,4,46,138,151,136,155],[4,153,111,19,6,21,111,153,31,42,115,17,4,23,107,153,29,23,134,153,54,46,111,155,4,46,111,149,10,44,4,153,4,46,138,151],[1,70,0,0,87,156,0,138,0,1,20,89,0,69,0,0,155,155,0,69,0,0,89,90,0,138,0,0,86,155,0,70,1,0,88,89] >;

D52⋊C3 in GAP, Magma, Sage, TeX

D_{52}\rtimes C_3
% in TeX

G:=Group("D52:C3");
// GroupNames label

G:=SmallGroup(312,10);
// by ID

G=gap.SmallGroup(312,10);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-13,141,66,7204,464]);
// Polycyclic

G:=Group<a,b,c|a^52=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^8*b>;
// generators/relations

Export

Subgroup lattice of D52⋊C3 in TeX
Character table of D52⋊C3 in TeX

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