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## G = Dic5.A4order 240 = 24·3·5

### The non-split extension by Dic5 of A4 acting through Inn(Dic5)

Aliases: Dic5.A4, SL2(𝔽3)⋊2D5, C5⋊(C4.A4), Q82D5⋊C3, (C5×Q8).C6, Q8.(C3×D5), C2.2(D5×A4), C10.1(C2×A4), (C5×SL2(𝔽3))⋊2C2, SmallGroup(240,108)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×Q8 — Dic5.A4
 Chief series C1 — C2 — C10 — C5×Q8 — C5×SL2(𝔽3) — Dic5.A4
 Lower central C5×Q8 — Dic5.A4
 Upper central C1 — C2

Generators and relations for Dic5.A4
G = < a,b,c,d,e | a10=e3=1, b2=c2=d2=a5, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a5c, ece-1=a5cd, ede-1=c >

Character table of Dic5.A4

 class 1 2A 2B 3A 3B 4A 4B 4C 5A 5B 6A 6B 10A 10B 12A 12B 12C 12D 15A 15B 15C 15D 20A 20B 30A 30B 30C 30D size 1 1 30 4 4 5 5 6 2 2 4 4 2 2 20 20 20 20 8 8 8 8 12 12 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 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 1 1 linear of order 2 ρ3 1 1 1 ζ3 ζ32 1 1 1 1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ4 1 1 -1 ζ32 ζ3 -1 -1 1 1 1 ζ3 ζ32 1 1 ζ6 ζ65 ζ6 ζ65 ζ32 ζ32 ζ3 ζ3 1 1 ζ3 ζ32 ζ3 ζ32 linear of order 6 ρ5 1 1 1 ζ32 ζ3 1 1 1 1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ6 1 1 -1 ζ3 ζ32 -1 -1 1 1 1 ζ32 ζ3 1 1 ζ65 ζ6 ζ65 ζ6 ζ3 ζ3 ζ32 ζ32 1 1 ζ32 ζ3 ζ32 ζ3 linear of order 6 ρ7 2 2 0 2 2 0 0 2 -1+√5/2 -1-√5/2 2 2 -1-√5/2 -1+√5/2 0 0 0 0 -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 ρ8 2 2 0 2 2 0 0 2 -1-√5/2 -1+√5/2 2 2 -1+√5/2 -1-√5/2 0 0 0 0 -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 0 -1 -1 2i -2i 0 2 2 1 1 -2 -2 -i -i i i -1 -1 -1 -1 0 0 1 1 1 1 complex lifted from C4.A4 ρ10 2 -2 0 -1 -1 -2i 2i 0 2 2 1 1 -2 -2 i i -i -i -1 -1 -1 -1 0 0 1 1 1 1 complex lifted from C4.A4 ρ11 2 -2 0 ζ65 ζ6 -2i 2i 0 2 2 ζ32 ζ3 -2 -2 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ32 ζ65 ζ65 ζ6 ζ6 0 0 ζ32 ζ3 ζ32 ζ3 complex lifted from C4.A4 ρ12 2 -2 0 ζ6 ζ65 2i -2i 0 2 2 ζ3 ζ32 -2 -2 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ3 ζ6 ζ6 ζ65 ζ65 0 0 ζ3 ζ32 ζ3 ζ32 complex lifted from C4.A4 ρ13 2 -2 0 ζ65 ζ6 2i -2i 0 2 2 ζ32 ζ3 -2 -2 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ32 ζ65 ζ65 ζ6 ζ6 0 0 ζ32 ζ3 ζ32 ζ3 complex lifted from C4.A4 ρ14 2 -2 0 ζ6 ζ65 -2i 2i 0 2 2 ζ3 ζ32 -2 -2 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ3 ζ6 ζ6 ζ65 ζ65 0 0 ζ3 ζ32 ζ3 ζ32 complex lifted from C4.A4 ρ15 2 2 0 -1-√-3 -1+√-3 0 0 2 -1-√5/2 -1+√5/2 -1+√-3 -1-√-3 -1+√5/2 -1-√5/2 0 0 0 0 ζ32ζ54+ζ32ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 ζ3ζ53+ζ3ζ52 -1-√5/2 -1+√5/2 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 complex lifted from C3×D5 ρ16 2 2 0 -1+√-3 -1-√-3 0 0 2 -1+√5/2 -1-√5/2 -1-√-3 -1+√-3 -1-√5/2 -1+√5/2 0 0 0 0 ζ3ζ53+ζ3ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 ζ32ζ54+ζ32ζ5 -1+√5/2 -1-√5/2 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 complex lifted from C3×D5 ρ17 2 2 0 -1+√-3 -1-√-3 0 0 2 -1-√5/2 -1+√5/2 -1-√-3 -1+√-3 -1+√5/2 -1-√5/2 0 0 0 0 ζ3ζ54+ζ3ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 ζ32ζ53+ζ32ζ52 -1-√5/2 -1+√5/2 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 complex lifted from C3×D5 ρ18 2 2 0 -1-√-3 -1+√-3 0 0 2 -1+√5/2 -1-√5/2 -1+√-3 -1-√-3 -1-√5/2 -1+√5/2 0 0 0 0 ζ32ζ53+ζ32ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 ζ3ζ54+ζ3ζ5 -1+√5/2 -1-√5/2 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 complex lifted from C3×D5 ρ19 3 3 1 0 0 -3 -3 -1 3 3 0 0 3 3 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 orthogonal lifted from C2×A4 ρ20 3 3 -1 0 0 3 3 -1 3 3 0 0 3 3 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 orthogonal lifted from A4 ρ21 4 -4 0 -2 -2 0 0 0 -1-√5 -1+√5 2 2 1-√5 1+√5 0 0 0 0 1-√5/2 1+√5/2 1-√5/2 1+√5/2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal faithful ρ22 4 -4 0 -2 -2 0 0 0 -1+√5 -1-√5 2 2 1+√5 1-√5 0 0 0 0 1+√5/2 1-√5/2 1+√5/2 1-√5/2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal faithful ρ23 4 -4 0 1-√-3 1+√-3 0 0 0 -1-√5 -1+√5 -1-√-3 -1+√-3 1-√5 1+√5 0 0 0 0 -ζ3ζ54-ζ3ζ5 -ζ3ζ53-ζ3ζ52 -ζ32ζ54-ζ32ζ5 -ζ32ζ53-ζ32ζ52 0 0 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 complex faithful ρ24 4 -4 0 1-√-3 1+√-3 0 0 0 -1+√5 -1-√5 -1-√-3 -1+√-3 1+√5 1-√5 0 0 0 0 -ζ3ζ53-ζ3ζ52 -ζ3ζ54-ζ3ζ5 -ζ32ζ53-ζ32ζ52 -ζ32ζ54-ζ32ζ5 0 0 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 complex faithful ρ25 4 -4 0 1+√-3 1-√-3 0 0 0 -1-√5 -1+√5 -1+√-3 -1-√-3 1-√5 1+√5 0 0 0 0 -ζ32ζ54-ζ32ζ5 -ζ32ζ53-ζ32ζ52 -ζ3ζ54-ζ3ζ5 -ζ3ζ53-ζ3ζ52 0 0 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 complex faithful ρ26 4 -4 0 1+√-3 1-√-3 0 0 0 -1+√5 -1-√5 -1+√-3 -1-√-3 1+√5 1-√5 0 0 0 0 -ζ32ζ53-ζ32ζ52 -ζ32ζ54-ζ32ζ5 -ζ3ζ53-ζ3ζ52 -ζ3ζ54-ζ3ζ5 0 0 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 complex faithful ρ27 6 6 0 0 0 0 0 -2 -3-3√5/2 -3+3√5/2 0 0 -3+3√5/2 -3-3√5/2 0 0 0 0 0 0 0 0 1+√5/2 1-√5/2 0 0 0 0 orthogonal lifted from D5×A4 ρ28 6 6 0 0 0 0 0 -2 -3+3√5/2 -3-3√5/2 0 0 -3-3√5/2 -3+3√5/2 0 0 0 0 0 0 0 0 1-√5/2 1+√5/2 0 0 0 0 orthogonal lifted from D5×A4

Smallest permutation representation of Dic5.A4
On 80 points
Generators in S80
(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)
(1 46 6 41)(2 45 7 50)(3 44 8 49)(4 43 9 48)(5 42 10 47)(11 54 16 59)(12 53 17 58)(13 52 18 57)(14 51 19 56)(15 60 20 55)(21 64 26 69)(22 63 27 68)(23 62 28 67)(24 61 29 66)(25 70 30 65)(31 74 36 79)(32 73 37 78)(33 72 38 77)(34 71 39 76)(35 80 40 75)
(1 14 6 19)(2 15 7 20)(3 16 8 11)(4 17 9 12)(5 18 10 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(1 24 6 29)(2 25 7 30)(3 26 8 21)(4 27 9 22)(5 28 10 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(11 21 31)(12 22 32)(13 23 33)(14 24 34)(15 25 35)(16 26 36)(17 27 37)(18 28 38)(19 29 39)(20 30 40)(51 61 71)(52 62 72)(53 63 73)(54 64 74)(55 65 75)(56 66 76)(57 67 77)(58 68 78)(59 69 79)(60 70 80)

G:=sub<Sym(80)| (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), (1,46,6,41)(2,45,7,50)(3,44,8,49)(4,43,9,48)(5,42,10,47)(11,54,16,59)(12,53,17,58)(13,52,18,57)(14,51,19,56)(15,60,20,55)(21,64,26,69)(22,63,27,68)(23,62,28,67)(24,61,29,66)(25,70,30,65)(31,74,36,79)(32,73,37,78)(33,72,38,77)(34,71,39,76)(35,80,40,75), (1,14,6,19)(2,15,7,20)(3,16,8,11)(4,17,9,12)(5,18,10,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (1,24,6,29)(2,25,7,30)(3,26,8,21)(4,27,9,22)(5,28,10,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (11,21,31)(12,22,32)(13,23,33)(14,24,34)(15,25,35)(16,26,36)(17,27,37)(18,28,38)(19,29,39)(20,30,40)(51,61,71)(52,62,72)(53,63,73)(54,64,74)(55,65,75)(56,66,76)(57,67,77)(58,68,78)(59,69,79)(60,70,80)>;

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), (1,46,6,41)(2,45,7,50)(3,44,8,49)(4,43,9,48)(5,42,10,47)(11,54,16,59)(12,53,17,58)(13,52,18,57)(14,51,19,56)(15,60,20,55)(21,64,26,69)(22,63,27,68)(23,62,28,67)(24,61,29,66)(25,70,30,65)(31,74,36,79)(32,73,37,78)(33,72,38,77)(34,71,39,76)(35,80,40,75), (1,14,6,19)(2,15,7,20)(3,16,8,11)(4,17,9,12)(5,18,10,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (1,24,6,29)(2,25,7,30)(3,26,8,21)(4,27,9,22)(5,28,10,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (11,21,31)(12,22,32)(13,23,33)(14,24,34)(15,25,35)(16,26,36)(17,27,37)(18,28,38)(19,29,39)(20,30,40)(51,61,71)(52,62,72)(53,63,73)(54,64,74)(55,65,75)(56,66,76)(57,67,77)(58,68,78)(59,69,79)(60,70,80) );

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)], [(1,46,6,41),(2,45,7,50),(3,44,8,49),(4,43,9,48),(5,42,10,47),(11,54,16,59),(12,53,17,58),(13,52,18,57),(14,51,19,56),(15,60,20,55),(21,64,26,69),(22,63,27,68),(23,62,28,67),(24,61,29,66),(25,70,30,65),(31,74,36,79),(32,73,37,78),(33,72,38,77),(34,71,39,76),(35,80,40,75)], [(1,14,6,19),(2,15,7,20),(3,16,8,11),(4,17,9,12),(5,18,10,13),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,71,66,76),(62,72,67,77),(63,73,68,78),(64,74,69,79),(65,75,70,80)], [(1,24,6,29),(2,25,7,30),(3,26,8,21),(4,27,9,22),(5,28,10,23),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,66,46,61),(42,67,47,62),(43,68,48,63),(44,69,49,64),(45,70,50,65),(51,76,56,71),(52,77,57,72),(53,78,58,73),(54,79,59,74),(55,80,60,75)], [(11,21,31),(12,22,32),(13,23,33),(14,24,34),(15,25,35),(16,26,36),(17,27,37),(18,28,38),(19,29,39),(20,30,40),(51,61,71),(52,62,72),(53,63,73),(54,64,74),(55,65,75),(56,66,76),(57,67,77),(58,68,78),(59,69,79),(60,70,80)]])

Dic5.A4 is a maximal subgroup of
C5⋊U2(𝔽3)  SL2(𝔽3).F5  CSU2(𝔽3)⋊D5  Dic5.6S4  Dic5.7S4  GL2(𝔽3)⋊D5  SL2(𝔽3).11D10  Dic10.A4  D5×C4.A4
Dic5.A4 is a maximal quotient of
Dic5×SL2(𝔽3)

Matrix representation of Dic5.A4 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 1 2 0 0 38 16
,
 50 0 0 0 0 50 0 0 0 0 14 12 0 0 60 47
,
 0 1 0 0 60 0 0 0 0 0 1 0 0 0 0 1
,
 48 47 0 0 47 13 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 47 13 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,38,0,0,2,16],[50,0,0,0,0,50,0,0,0,0,14,60,0,0,12,47],[0,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[48,47,0,0,47,13,0,0,0,0,1,0,0,0,0,1],[1,47,0,0,0,13,0,0,0,0,1,0,0,0,0,1] >;

Dic5.A4 in GAP, Magma, Sage, TeX

{\rm Dic}_5.A_4
% in TeX

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

G:=SmallGroup(240,108);
// by ID

G=gap.SmallGroup(240,108);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-5,-2,720,170,374,81,543,261,2884]);
// Polycyclic

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

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