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G = C52⋊2Dic3order 300 = 22·3·52

The semidirect product of C52 and Dic3 acting via Dic3/C2=S3

Aliases: C522Dic3, (C5×C10).S3, C52⋊C33C4, C2.(C52⋊S3), (C2×C52⋊C3).1C2, SmallGroup(300,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊C3 — C52⋊2Dic3
 Chief series C1 — C52 — C52⋊C3 — C2×C52⋊C3 — C52⋊2Dic3
 Lower central C52⋊C3 — C52⋊2Dic3
 Upper central C1 — C2

Generators and relations for C522Dic3
G = < a,b,c,d | a5=b5=c6=1, d2=c3, cbc-1=ab=ba, cac-1=a3b2, ad=da, dbd-1=a-1b-1, dcd-1=c-1 >

Character table of C522Dic3

 class 1 2 3 4A 4B 5A 5B 5C 5D 5E 5F 6 10A 10B 10C 10D 10E 10F 20A 20B 20C 20D 20E 20F 20G 20H size 1 1 50 15 15 3 3 3 3 6 6 50 3 3 3 3 6 6 15 15 15 15 15 15 15 15 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 1 i -i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -i -i -i -i i i i i linear of order 4 ρ4 1 -1 1 -i i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 i i i i -i -i -i -i linear of order 4 ρ5 2 2 -1 0 0 2 2 2 2 2 2 -1 2 2 2 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3 ρ6 2 -2 -1 0 0 2 2 2 2 2 2 1 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ7 3 3 0 1 1 2ζ54+ζ52 ζ53+2ζ5 2ζ52+ζ5 ζ54+2ζ53 1+√5/2 1-√5/2 0 2ζ54+ζ52 ζ53+2ζ5 ζ54+2ζ53 2ζ52+ζ5 1+√5/2 1-√5/2 ζ52 ζ53 ζ54 ζ5 ζ5 ζ52 ζ53 ζ54 complex lifted from C52⋊S3 ρ8 3 3 0 -1 -1 2ζ52+ζ5 ζ54+2ζ53 ζ53+2ζ5 2ζ54+ζ52 1-√5/2 1+√5/2 0 2ζ52+ζ5 ζ54+2ζ53 2ζ54+ζ52 ζ53+2ζ5 1-√5/2 1+√5/2 -ζ5 -ζ54 -ζ52 -ζ53 -ζ53 -ζ5 -ζ54 -ζ52 complex lifted from C52⋊S3 ρ9 3 3 0 1 1 ζ53+2ζ5 2ζ54+ζ52 ζ54+2ζ53 2ζ52+ζ5 1+√5/2 1-√5/2 0 ζ53+2ζ5 2ζ54+ζ52 2ζ52+ζ5 ζ54+2ζ53 1+√5/2 1-√5/2 ζ53 ζ52 ζ5 ζ54 ζ54 ζ53 ζ52 ζ5 complex lifted from C52⋊S3 ρ10 3 3 0 -1 -1 ζ53+2ζ5 2ζ54+ζ52 ζ54+2ζ53 2ζ52+ζ5 1+√5/2 1-√5/2 0 ζ53+2ζ5 2ζ54+ζ52 2ζ52+ζ5 ζ54+2ζ53 1+√5/2 1-√5/2 -ζ53 -ζ52 -ζ5 -ζ54 -ζ54 -ζ53 -ζ52 -ζ5 complex lifted from C52⋊S3 ρ11 3 3 0 -1 -1 2ζ54+ζ52 ζ53+2ζ5 2ζ52+ζ5 ζ54+2ζ53 1+√5/2 1-√5/2 0 2ζ54+ζ52 ζ53+2ζ5 ζ54+2ζ53 2ζ52+ζ5 1+√5/2 1-√5/2 -ζ52 -ζ53 -ζ54 -ζ5 -ζ5 -ζ52 -ζ53 -ζ54 complex lifted from C52⋊S3 ρ12 3 3 0 1 1 ζ54+2ζ53 2ζ52+ζ5 2ζ54+ζ52 ζ53+2ζ5 1-√5/2 1+√5/2 0 ζ54+2ζ53 2ζ52+ζ5 ζ53+2ζ5 2ζ54+ζ52 1-√5/2 1+√5/2 ζ54 ζ5 ζ53 ζ52 ζ52 ζ54 ζ5 ζ53 complex lifted from C52⋊S3 ρ13 3 3 0 1 1 2ζ52+ζ5 ζ54+2ζ53 ζ53+2ζ5 2ζ54+ζ52 1-√5/2 1+√5/2 0 2ζ52+ζ5 ζ54+2ζ53 2ζ54+ζ52 ζ53+2ζ5 1-√5/2 1+√5/2 ζ5 ζ54 ζ52 ζ53 ζ53 ζ5 ζ54 ζ52 complex lifted from C52⋊S3 ρ14 3 3 0 -1 -1 ζ54+2ζ53 2ζ52+ζ5 2ζ54+ζ52 ζ53+2ζ5 1-√5/2 1+√5/2 0 ζ54+2ζ53 2ζ52+ζ5 ζ53+2ζ5 2ζ54+ζ52 1-√5/2 1+√5/2 -ζ54 -ζ5 -ζ53 -ζ52 -ζ52 -ζ54 -ζ5 -ζ53 complex lifted from C52⋊S3 ρ15 3 -3 0 i -i 2ζ52+ζ5 ζ54+2ζ53 ζ53+2ζ5 2ζ54+ζ52 1-√5/2 1+√5/2 0 -2ζ52-ζ5 -ζ54-2ζ53 -2ζ54-ζ52 -ζ53-2ζ5 -1+√5/2 -1-√5/2 ζ43ζ5 ζ43ζ54 ζ43ζ52 ζ43ζ53 ζ4ζ53 ζ4ζ5 ζ4ζ54 ζ4ζ52 complex faithful ρ16 3 -3 0 -i i ζ54+2ζ53 2ζ52+ζ5 2ζ54+ζ52 ζ53+2ζ5 1-√5/2 1+√5/2 0 -ζ54-2ζ53 -2ζ52-ζ5 -ζ53-2ζ5 -2ζ54-ζ52 -1+√5/2 -1-√5/2 ζ4ζ54 ζ4ζ5 ζ4ζ53 ζ4ζ52 ζ43ζ52 ζ43ζ54 ζ43ζ5 ζ43ζ53 complex faithful ρ17 3 -3 0 -i i 2ζ54+ζ52 ζ53+2ζ5 2ζ52+ζ5 ζ54+2ζ53 1+√5/2 1-√5/2 0 -2ζ54-ζ52 -ζ53-2ζ5 -ζ54-2ζ53 -2ζ52-ζ5 -1-√5/2 -1+√5/2 ζ4ζ52 ζ4ζ53 ζ4ζ54 ζ4ζ5 ζ43ζ5 ζ43ζ52 ζ43ζ53 ζ43ζ54 complex faithful ρ18 3 -3 0 -i i ζ53+2ζ5 2ζ54+ζ52 ζ54+2ζ53 2ζ52+ζ5 1+√5/2 1-√5/2 0 -ζ53-2ζ5 -2ζ54-ζ52 -2ζ52-ζ5 -ζ54-2ζ53 -1-√5/2 -1+√5/2 ζ4ζ53 ζ4ζ52 ζ4ζ5 ζ4ζ54 ζ43ζ54 ζ43ζ53 ζ43ζ52 ζ43ζ5 complex faithful ρ19 3 -3 0 i -i 2ζ54+ζ52 ζ53+2ζ5 2ζ52+ζ5 ζ54+2ζ53 1+√5/2 1-√5/2 0 -2ζ54-ζ52 -ζ53-2ζ5 -ζ54-2ζ53 -2ζ52-ζ5 -1-√5/2 -1+√5/2 ζ43ζ52 ζ43ζ53 ζ43ζ54 ζ43ζ5 ζ4ζ5 ζ4ζ52 ζ4ζ53 ζ4ζ54 complex faithful ρ20 3 -3 0 -i i 2ζ52+ζ5 ζ54+2ζ53 ζ53+2ζ5 2ζ54+ζ52 1-√5/2 1+√5/2 0 -2ζ52-ζ5 -ζ54-2ζ53 -2ζ54-ζ52 -ζ53-2ζ5 -1+√5/2 -1-√5/2 ζ4ζ5 ζ4ζ54 ζ4ζ52 ζ4ζ53 ζ43ζ53 ζ43ζ5 ζ43ζ54 ζ43ζ52 complex faithful ρ21 3 -3 0 i -i ζ54+2ζ53 2ζ52+ζ5 2ζ54+ζ52 ζ53+2ζ5 1-√5/2 1+√5/2 0 -ζ54-2ζ53 -2ζ52-ζ5 -ζ53-2ζ5 -2ζ54-ζ52 -1+√5/2 -1-√5/2 ζ43ζ54 ζ43ζ5 ζ43ζ53 ζ43ζ52 ζ4ζ52 ζ4ζ54 ζ4ζ5 ζ4ζ53 complex faithful ρ22 3 -3 0 i -i ζ53+2ζ5 2ζ54+ζ52 ζ54+2ζ53 2ζ52+ζ5 1+√5/2 1-√5/2 0 -ζ53-2ζ5 -2ζ54-ζ52 -2ζ52-ζ5 -ζ54-2ζ53 -1-√5/2 -1+√5/2 ζ43ζ53 ζ43ζ52 ζ43ζ5 ζ43ζ54 ζ4ζ54 ζ4ζ53 ζ4ζ52 ζ4ζ5 complex faithful ρ23 6 6 0 0 0 1+√5 1+√5 1-√5 1-√5 -3-√5/2 -3+√5/2 0 1+√5 1+√5 1-√5 1-√5 -3-√5/2 -3+√5/2 0 0 0 0 0 0 0 0 orthogonal lifted from C52⋊S3 ρ24 6 6 0 0 0 1-√5 1-√5 1+√5 1+√5 -3+√5/2 -3-√5/2 0 1-√5 1-√5 1+√5 1+√5 -3+√5/2 -3-√5/2 0 0 0 0 0 0 0 0 orthogonal lifted from C52⋊S3 ρ25 6 -6 0 0 0 1-√5 1-√5 1+√5 1+√5 -3+√5/2 -3-√5/2 0 -1+√5 -1+√5 -1-√5 -1-√5 3-√5/2 3+√5/2 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ26 6 -6 0 0 0 1+√5 1+√5 1-√5 1-√5 -3-√5/2 -3+√5/2 0 -1-√5 -1-√5 -1+√5 -1+√5 3+√5/2 3-√5/2 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C522Dic3
On 60 points
Generators in S60
(2 42 48 35 54)(3 49 36 43 37)(5 39 45 32 51)(6 52 33 46 40)(7 55 19 13 26)(9 28 15 21 57)(10 58 22 16 29)(12 25 18 24 60)
(1 34 41 53 47)(2 35 42 54 48)(3 49 36 43 37)(4 31 38 50 44)(5 32 39 51 45)(6 52 33 46 40)(7 55 19 13 26)(8 20 27 56 14)(9 21 28 57 15)(10 58 22 16 29)(11 23 30 59 17)(12 24 25 60 18)
(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)
(1 59 4 56)(2 58 5 55)(3 57 6 60)(7 54 10 51)(8 53 11 50)(9 52 12 49)(13 48 16 45)(14 47 17 44)(15 46 18 43)(19 42 22 39)(20 41 23 38)(21 40 24 37)(25 36 28 33)(26 35 29 32)(27 34 30 31)

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

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

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

Matrix representation of C522Dic3 in GL5(𝔽61)

 1 0 0 0 0 0 1 0 0 0 0 0 34 0 0 0 0 0 34 0 0 0 0 0 20
,
 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 1 0 0 0 0 0 34
,
 1 60 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 60 0 0 0 0 0 60 0
,
 49 53 0 0 0 41 12 0 0 0 0 0 0 60 0 0 0 60 0 0 0 0 0 0 1

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,34,0,0,0,0,0,34,0,0,0,0,0,20],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,34],[1,1,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0,0],[49,41,0,0,0,53,12,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,1] >;

C522Dic3 in GAP, Magma, Sage, TeX

C_5^2\rtimes_2{\rm Dic}_3
% in TeX

G:=Group("C5^2:2Dic3");
// GroupNames label

G:=SmallGroup(300,13);
// by ID

G=gap.SmallGroup(300,13);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,5,10,122,973,7204,1439]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^6=1,d^2=c^3,c*b*c^-1=a*b=b*a,c*a*c^-1=a^3*b^2,a*d=d*a,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations

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