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G = A4xDic5order 240 = 24·3·5

Direct product of A4 and Dic5

direct product, metabelian, soluble, monomial, A-group

Aliases: A4xDic5, C5:2(C4xA4), (C5xA4):4C4, C2.1(D5xA4), (C2xC10):2C12, C23.(C3xD5), C10.3(C2xA4), (C2xA4).2D5, (C22xC10).C6, C22:(C3xDic5), (C22xDic5):C3, (C10xA4).2C2, SmallGroup(240,110)

Series: Derived Chief Lower central Upper central

C1C2xC10 — A4xDic5
C1C5C2xC10C22xC10C10xA4 — A4xDic5
C2xC10 — A4xDic5
C1C2

Generators and relations for A4xDic5
 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 >

Subgroups: 148 in 38 conjugacy classes, 15 normal (all characteristic)
Quotients: C1, C2, C3, C4, C6, D5, C12, A4, Dic5, C2xA4, C3xD5, C4xA4, C3xDic5, D5xA4, A4xDic5
3C2
3C2
4C3
3C22
3C22
5C4
15C4
4C6
3C10
3C10
4C15
15C2xC4
15C2xC4
20C12
3C2xC10
3Dic5
3C2xC10
4C30
5C22xC4
3C2xDic5
3C2xDic5
4C3xDic5
5C4xA4

Smallest permutation representation of A4xDic5
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)]])

A4xDic5 is a maximal subgroup of   Dic5.S4  A4:Dic10  Dic5:2S4  Dic5:S4  C4xD5xA4
A4xDic5 is a maximal quotient of   SL2(F3).Dic5

32 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D5A5B6A6B10A10B10C10D10E10F12A12B12C12D15A15B15C15D30A30B30C30D
order12223344445566101010101010121212121515151530303030
size11334455151522442266662020202088888888

32 irreducible representations

dim111111222233366
type+++-+++-
imageC1C2C3C4C6C12D5Dic5C3xD5C3xDic5A4C2xA4C4xA4D5xA4A4xDic5
kernelA4xDic5C10xA4C22xDic5C5xA4C22xC10C2xC10C2xA4A4C23C22Dic5C10C5C2C1
# reps112224224411222

Matrix representation of A4xDic5 in GL5(F61)

10000
01000
006000
002310
00483660
,
10000
01000
00100
0038600
0048060
,
10000
01000
0047200
0056286
000047
,
060000
118000
006000
000600
000060
,
853000
3153000
001100
000110
000011

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] >;

A4xDic5 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

Export

Subgroup lattice of A4xDic5 in TeX

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