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## G = A4×Dic5order 240 = 24·3·5

### Direct product of A4 and Dic5

Aliases: A4×Dic5, C52(C4×A4), (C5×A4)⋊4C4, C2.1(D5×A4), (C2×C10)⋊2C12, C23.(C3×D5), C10.3(C2×A4), (C2×A4).2D5, (C22×C10).C6, C22⋊(C3×Dic5), (C22×Dic5)⋊C3, (C10×A4).2C2, SmallGroup(240,110)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — A4×Dic5
 Chief series C1 — C5 — C2×C10 — C22×C10 — C10×A4 — A4×Dic5
 Lower central C2×C10 — 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=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

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

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

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

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

A4×Dic5 is a maximal subgroup of   Dic5.S4  A4⋊Dic10  Dic52S4  Dic5⋊S4  C4×D5×A4
A4×Dic5 is a maximal quotient of   SL2(𝔽3).Dic5

32 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 5A 5B 6A 6B 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 15A 15B 15C 15D 30A 30B 30C 30D order 1 2 2 2 3 3 4 4 4 4 5 5 6 6 10 10 10 10 10 10 12 12 12 12 15 15 15 15 30 30 30 30 size 1 1 3 3 4 4 5 5 15 15 2 2 4 4 2 2 6 6 6 6 20 20 20 20 8 8 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + - + + + - image C1 C2 C3 C4 C6 C12 D5 Dic5 C3×D5 C3×Dic5 A4 C2×A4 C4×A4 D5×A4 A4×Dic5 kernel A4×Dic5 C10×A4 C22×Dic5 C5×A4 C22×C10 C2×C10 C2×A4 A4 C23 C22 Dic5 C10 C5 C2 C1 # reps 1 1 2 2 2 4 2 2 4 4 1 1 2 2 2

Matrix representation of A4×Dic5 in GL5(𝔽61)

 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 23 1 0 0 0 48 36 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 38 60 0 0 0 48 0 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 47 20 0 0 0 56 28 6 0 0 0 0 47
,
 0 60 0 0 0 1 18 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60
,
 8 53 0 0 0 31 53 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,60,23,48,0,0,0,1,36,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,1,38,48,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,47,56,0,0,0,20,28,0,0,0,0,6,47],[0,1,0,0,0,60,18,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[8,31,0,0,0,53,53,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11] >;

A4×Dic5 in GAP, Magma, Sage, TeX

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

G:=Group("A4xDic5");
// GroupNames label

G:=SmallGroup(240,110);
// by ID

G=gap.SmallGroup(240,110);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,2,-5,36,441,190,6917]);
// 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=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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