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G = C4xD5xA4order 480 = 25·3·5

Direct product of C4, D5 and A4

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

Aliases: C4xD5xA4, C20:2(C2xA4), C22:(D5xC12), (A4xC20):5C2, (C22xC20):2C6, Dic5:2(C2xA4), (A4xDic5):5C2, D10.5(C2xA4), (C2xA4).14D10, (C22xD5):4C12, C10.2(C22xA4), (C23xD5).2C6, C23.11(C6xD5), (C22xDic5):2C6, (C10xA4).14C22, C5:2(C2xC4xA4), (D5xC22xC4):C3, C2.1(C2xD5xA4), (C2xD5xA4).4C2, (C5xA4):8(C2xC4), (C2xC10):3(C2xC12), (C22xC4):2(C3xD5), (C22xC10).2(C2xC6), SmallGroup(480,1036)

Series: Derived Chief Lower central Upper central

C1C2xC10 — C4xD5xA4
C1C5C2xC10C22xC10C10xA4C2xD5xA4 — C4xD5xA4
C2xC10 — C4xD5xA4
C1C4

Generators and relations for C4xD5xA4
 G = < a,b,c,d,e,f | a4=b5=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 780 in 132 conjugacy classes, 33 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C2xC4, C23, C23, D5, D5, C10, C10, C12, A4, C2xC6, C15, C22xC4, C22xC4, C24, Dic5, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, C2xC12, C2xA4, C2xA4, C3xD5, C30, C23xC4, C4xD5, C4xD5, C2xDic5, C2xC20, C22xD5, C22xD5, C22xC10, C4xA4, C4xA4, C22xA4, C3xDic5, C60, C5xA4, C6xD5, C2xC4xD5, C22xDic5, C22xC20, C23xD5, C2xC4xA4, D5xC12, D5xA4, C10xA4, D5xC22xC4, A4xDic5, A4xC20, C2xD5xA4, C4xD5xA4
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, D5, C12, A4, C2xC6, D10, C2xC12, C2xA4, C3xD5, C4xD5, C4xA4, C22xA4, C6xD5, C2xC4xA4, D5xC12, D5xA4, C2xD5xA4, C4xD5xA4

Smallest permutation representation of C4xD5xA4
On 60 points
Generators in S60
(1 39 9 34)(2 40 10 35)(3 36 6 31)(4 37 7 32)(5 38 8 33)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 56 26 51)(22 57 27 52)(23 58 28 53)(24 59 29 54)(25 60 30 55)
(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 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)
(1 9)(2 10)(3 6)(4 7)(5 8)(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)
(1 24 14)(2 25 15)(3 21 11)(4 22 12)(5 23 13)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(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)

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

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

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

64 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E6F10A10B10C10D10E10F12A12B12C12D12E12F12G12H15A15B15C15D20A20B20C20D20E20F20G20H30A30B30C30D60A···60H
order1222222233444444445566666610101010101012121212121212121515151520202020202020203030303060···60
size113355151544113355151522442020202022666644442020202088882222666688888···8

64 irreducible representations

dim111111111122222233333666
type++++++++++++
imageC1C2C2C2C3C4C6C6C6C12D5D10C3xD5C4xD5C6xD5D5xC12A4C2xA4C2xA4C2xA4C4xA4D5xA4C2xD5xA4C4xD5xA4
kernelC4xD5xA4A4xDic5A4xC20C2xD5xA4D5xC22xC4D5xA4C22xDic5C22xC20C23xD5C22xD5C4xA4C2xA4C22xC4A4C23C22C4xD5Dic5C20D10D5C4C2C1
# reps111124222822444811114224

Matrix representation of C4xD5xA4 in GL5(F61)

600000
060000
005000
000500
000050
,
181000
4260000
00100
00010
00001
,
6060000
01000
006000
000600
000060
,
10000
01000
006000
000600
003601
,
10000
01000
006000
004510
000060
,
10000
01000
001360
000481
000140

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,50,0,0,0,0,0,50,0,0,0,0,0,50],[18,42,0,0,0,1,60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,60,1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,36,0,0,0,60,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,45,0,0,0,0,1,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,13,0,0,0,0,6,48,14,0,0,0,1,0] >;

C4xD5xA4 in GAP, Magma, Sage, TeX

C_4\times D_5\times A_4
% in TeX

G:=Group("C4xD5xA4");
// GroupNames label

G:=SmallGroup(480,1036);
// by ID

G=gap.SmallGroup(480,1036);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,92,648,271,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^5=c^2=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

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