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## G = A4⋊Dic5order 240 = 24·3·5

### The semidirect product of A4 and Dic5 acting via Dic5/C10=C2

Aliases: A4⋊Dic5, C10.3S4, C23.D15, C22⋊Dic15, (C2×A4).D5, C52(A4⋊C4), (C5×A4)⋊3C4, C2.1(C5⋊S4), (C10×A4).1C2, (C2×C10)⋊3Dic3, (C22×C10).2S3, SmallGroup(240,107)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×A4 — A4⋊Dic5
 Chief series C1 — C22 — C2×C10 — C5×A4 — C10×A4 — A4⋊Dic5
 Lower central C5×A4 — A4⋊Dic5
 Upper central C1 — C2

Generators and relations for A4⋊Dic5
G = < a,b,c,d,e | a2=b2=c3=d10=1, e2=d5, cac-1=eae-1=ab=ba, ad=da, cbc-1=a, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Character table of A4⋊Dic5

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 6 10A 10B 10C 10D 10E 10F 15A 15B 15C 15D 30A 30B 30C 30D size 1 1 3 3 8 30 30 30 30 2 2 8 2 2 6 6 6 6 8 8 8 8 8 8 8 8 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 -1 1 -i i -i i 1 1 -1 -1 -1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ4 1 -1 1 -1 1 i -i i -i 1 1 -1 -1 -1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ5 2 2 2 2 2 0 0 0 0 -1-√5/2 -1+√5/2 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 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ6 2 2 2 2 -1 0 0 0 0 2 2 -1 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 2 2 -1 0 0 0 0 -1-√5/2 -1+√5/2 -1 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ54+ζ3ζ5-ζ54 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ54+ζ3ζ5-ζ54 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ53+ζ3ζ52-ζ53 orthogonal lifted from D15 ρ8 2 2 2 2 2 0 0 0 0 -1+√5/2 -1-√5/2 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 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ9 2 2 2 2 -1 0 0 0 0 -1+√5/2 -1-√5/2 -1 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -ζ3ζ53+ζ3ζ52-ζ53 -ζ3ζ54+ζ3ζ5-ζ54 -ζ32ζ54+ζ32ζ5-ζ54 ζ3ζ53-ζ3ζ52-ζ52 -ζ32ζ54+ζ32ζ5-ζ54 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 -ζ3ζ54+ζ3ζ5-ζ54 orthogonal lifted from D15 ρ10 2 2 2 2 -1 0 0 0 0 -1-√5/2 -1+√5/2 -1 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -ζ3ζ54+ζ3ζ5-ζ54 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ53+ζ3ζ52-ζ53 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ54+ζ3ζ5-ζ54 ζ3ζ53-ζ3ζ52-ζ52 orthogonal lifted from D15 ρ11 2 2 2 2 -1 0 0 0 0 -1+√5/2 -1-√5/2 -1 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 ζ3ζ53-ζ3ζ52-ζ52 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ54+ζ3ζ5-ζ54 -ζ3ζ53+ζ3ζ52-ζ53 -ζ3ζ54+ζ3ζ5-ζ54 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 -ζ32ζ54+ζ32ζ5-ζ54 orthogonal lifted from D15 ρ12 2 -2 2 -2 -1 0 0 0 0 2 2 1 -2 -2 2 -2 -2 2 -1 -1 -1 -1 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ13 2 -2 2 -2 2 0 0 0 0 -1-√5/2 -1+√5/2 -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 1-√5/2 1-√5/2 1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ14 2 -2 2 -2 -1 0 0 0 0 -1-√5/2 -1+√5/2 1 1-√5/2 1+√5/2 -1+√5/2 1-√5/2 1+√5/2 -1-√5/2 -ζ3ζ54+ζ3ζ5-ζ54 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 -ζ32ζ54+ζ32ζ5-ζ54 ζ3ζ53-ζ3ζ52+ζ53 ζ32ζ54-ζ32ζ5+ζ54 ζ3ζ54-ζ3ζ5+ζ54 -ζ3ζ53+ζ3ζ52+ζ52 symplectic lifted from Dic15, Schur index 2 ρ15 2 -2 2 -2 -1 0 0 0 0 -1-√5/2 -1+√5/2 1 1-√5/2 1+√5/2 -1+√5/2 1-√5/2 1+√5/2 -1-√5/2 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ54+ζ3ζ5-ζ54 -ζ3ζ53+ζ3ζ52+ζ52 ζ3ζ54-ζ3ζ5+ζ54 ζ32ζ54-ζ32ζ5+ζ54 ζ3ζ53-ζ3ζ52+ζ53 symplectic lifted from Dic15, Schur index 2 ρ16 2 -2 2 -2 -1 0 0 0 0 -1+√5/2 -1-√5/2 1 1+√5/2 1-√5/2 -1-√5/2 1+√5/2 1-√5/2 -1+√5/2 ζ3ζ53-ζ3ζ52-ζ52 -ζ32ζ54+ζ32ζ5-ζ54 -ζ3ζ54+ζ3ζ5-ζ54 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5+ζ54 ζ3ζ53-ζ3ζ52+ζ53 -ζ3ζ53+ζ3ζ52+ζ52 ζ32ζ54-ζ32ζ5+ζ54 symplectic lifted from Dic15, Schur index 2 ρ17 2 -2 2 -2 -1 0 0 0 0 -1+√5/2 -1-√5/2 1 1+√5/2 1-√5/2 -1-√5/2 1+√5/2 1-√5/2 -1+√5/2 -ζ3ζ53+ζ3ζ52-ζ53 -ζ3ζ54+ζ3ζ5-ζ54 -ζ32ζ54+ζ32ζ5-ζ54 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5+ζ54 -ζ3ζ53+ζ3ζ52+ζ52 ζ3ζ53-ζ3ζ52+ζ53 ζ3ζ54-ζ3ζ5+ζ54 symplectic lifted from Dic15, Schur index 2 ρ18 2 -2 2 -2 2 0 0 0 0 -1+√5/2 -1-√5/2 -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 1+√5/2 1+√5/2 1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ19 3 3 -1 -1 0 1 1 -1 -1 3 3 0 3 3 -1 -1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ20 3 3 -1 -1 0 -1 -1 1 1 3 3 0 3 3 -1 -1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ21 3 -3 -1 1 0 -i i i -i 3 3 0 -3 -3 -1 1 1 -1 0 0 0 0 0 0 0 0 complex lifted from A4⋊C4 ρ22 3 -3 -1 1 0 i -i -i i 3 3 0 -3 -3 -1 1 1 -1 0 0 0 0 0 0 0 0 complex lifted from A4⋊C4 ρ23 6 6 -2 -2 0 0 0 0 0 -3-3√5/2 -3+3√5/2 0 -3+3√5/2 -3-3√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 0 0 0 0 0 0 0 0 orthogonal lifted from C5⋊S4 ρ24 6 6 -2 -2 0 0 0 0 0 -3+3√5/2 -3-3√5/2 0 -3-3√5/2 -3+3√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 0 0 0 0 0 0 0 0 orthogonal lifted from C5⋊S4 ρ25 6 -6 -2 2 0 0 0 0 0 -3+3√5/2 -3-3√5/2 0 3+3√5/2 3-3√5/2 1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ26 6 -6 -2 2 0 0 0 0 0 -3-3√5/2 -3+3√5/2 0 3-3√5/2 3+3√5/2 1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of A4⋊Dic5
On 60 points
Generators in S60
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 21)(11 59)(12 60)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 51)(19 52)(20 53)(21 26)(22 27)(23 28)(24 29)(25 30)(31 48)(32 49)(33 50)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)
(1 19 44)(2 20 45)(3 11 46)(4 12 47)(5 13 48)(6 14 49)(7 15 50)(8 16 41)(9 17 42)(10 18 43)(21 51 31)(22 52 32)(23 53 33)(24 54 34)(25 55 35)(26 56 36)(27 57 37)(28 58 38)(29 59 39)(30 60 40)
(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 27 6 22)(2 26 7 21)(3 25 8 30)(4 24 9 29)(5 23 10 28)(11 35 16 40)(12 34 17 39)(13 33 18 38)(14 32 19 37)(15 31 20 36)(41 60 46 55)(42 59 47 54)(43 58 48 53)(44 57 49 52)(45 56 50 51)

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

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

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

A4⋊Dic5 is a maximal subgroup of   A4⋊Dic10  Dic5×S4  D5×A4⋊C4  D10⋊S4  C20.1S4  C4×C5⋊S4  C242D15
A4⋊Dic5 is a maximal quotient of   C20.S4  Q8⋊Dic15  C52U2(𝔽3)

Matrix representation of A4⋊Dic5 in GL5(𝔽61)

 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 50 11 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 11 50 60 0 0 56 55 11
,
 44 60 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60
,
 4 57 0 0 0 50 57 0 0 0 0 0 0 1 0 0 0 60 0 0 0 0 6 5 11

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,50,0,0,0,60,11,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,11,56,0,0,0,50,55,0,0,1,60,11],[44,1,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[4,50,0,0,0,57,57,0,0,0,0,0,0,60,6,0,0,1,0,5,0,0,0,0,11] >;

A4⋊Dic5 in GAP, Magma, Sage, TeX

A_4\rtimes {\rm Dic}_5
% in TeX

G:=Group("A4:Dic5");
// GroupNames label

G:=SmallGroup(240,107);
// by ID

G=gap.SmallGroup(240,107);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,2,12,146,1155,3604,916,2165,1637]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^10=1,e^2=d^5,c*a*c^-1=e*a*e^-1=a*b=b*a,a*d=d*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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