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G = C5xD4:2D5order 400 = 24·52

Direct product of C5 and D4:2D5

direct product, metabelian, supersoluble, monomial

Aliases: C5xD4:2D5, C20.37D10, Dic10:3C10, C102.13C22, (C5xD4):5D5, D4:2(C5xD5), (C5xD4):3C10, (C4xD5):2C10, (D5xC20):6C2, C5:D4:2C10, C4.5(D5xC10), C20.5(C2xC10), (C2xC10).6D10, (D4xC52):4C2, (C2xDic5):3C10, (C10xDic5):9C2, (C5xDic10):8C2, D10.2(C2xC10), C52:10(C4oD4), C22.1(D5xC10), (C5xC10).24C23, C10.6(C22xC10), (C5xC20).21C22, Dic5.3(C2xC10), C10.45(C22xD5), (D5xC10).15C22, (C5xDic5).25C22, C5:2(C5xC4oD4), C2.7(D5xC2xC10), (C2xC10).(C2xC10), (C5xC5:D4):6C2, SmallGroup(400,186)

Series: Derived Chief Lower central Upper central

C1C10 — C5xD4:2D5
C1C5C10C5xC10D5xC10D5xC20 — C5xD4:2D5
C5C10 — C5xD4:2D5
C1C10C5xD4

Generators and relations for C5xD4:2D5
 G = < a,b,c,d,e | a5=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 236 in 96 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C5, C2xC4, D4, D4, Q8, D5, C10, C10, C4oD4, Dic5, Dic5, C20, C20, D10, C2xC10, C2xC10, C52, Dic10, C4xD5, C2xDic5, C5:D4, C2xC20, C5xD4, C5xD4, C5xQ8, C5xD5, C5xC10, C5xC10, D4:2D5, C5xC4oD4, C5xDic5, C5xDic5, C5xC20, D5xC10, C102, C5xDic10, D5xC20, C10xDic5, C5xC5:D4, D4xC52, C5xD4:2D5
Quotients: C1, C2, C22, C5, C23, D5, C10, C4oD4, D10, C2xC10, C22xD5, C22xC10, C5xD5, D4:2D5, C5xC4oD4, D5xC10, D5xC2xC10, C5xD4:2D5

Smallest permutation representation of C5xD4:2D5
On 40 points
Generators in S40
(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)
(1 18 8 13)(2 19 9 14)(3 20 10 15)(4 16 6 11)(5 17 7 12)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 16)(7 17)(8 18)(9 19)(10 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 4 2 5 3)(6 9 7 10 8)(11 14 12 15 13)(16 19 17 20 18)(21 23 25 22 24)(26 28 30 27 29)(31 33 35 32 34)(36 38 40 37 39)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 29)(7 30)(8 26)(9 27)(10 28)(11 34)(12 35)(13 31)(14 32)(15 33)(16 39)(17 40)(18 36)(19 37)(20 38)

G:=sub<Sym(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), (1,18,8,13)(2,19,9,14)(3,20,10,15)(4,16,6,11)(5,17,7,12)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (1,13)(2,14)(3,15)(4,11)(5,12)(6,16)(7,17)(8,18)(9,19)(10,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,4,2,5,3)(6,9,7,10,8)(11,14,12,15,13)(16,19,17,20,18)(21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)(36,38,40,37,39), (1,21)(2,22)(3,23)(4,24)(5,25)(6,29)(7,30)(8,26)(9,27)(10,28)(11,34)(12,35)(13,31)(14,32)(15,33)(16,39)(17,40)(18,36)(19,37)(20,38)>;

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), (1,18,8,13)(2,19,9,14)(3,20,10,15)(4,16,6,11)(5,17,7,12)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (1,13)(2,14)(3,15)(4,11)(5,12)(6,16)(7,17)(8,18)(9,19)(10,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,4,2,5,3)(6,9,7,10,8)(11,14,12,15,13)(16,19,17,20,18)(21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)(36,38,40,37,39), (1,21)(2,22)(3,23)(4,24)(5,25)(6,29)(7,30)(8,26)(9,27)(10,28)(11,34)(12,35)(13,31)(14,32)(15,33)(16,39)(17,40)(18,36)(19,37)(20,38) );

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)], [(1,18,8,13),(2,19,9,14),(3,20,10,15),(4,16,6,11),(5,17,7,12),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,16),(7,17),(8,18),(9,19),(10,20),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35)], [(1,4,2,5,3),(6,9,7,10,8),(11,14,12,15,13),(16,19,17,20,18),(21,23,25,22,24),(26,28,30,27,29),(31,33,35,32,34),(36,38,40,37,39)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,29),(7,30),(8,26),(9,27),(10,28),(11,34),(12,35),(13,31),(14,32),(15,33),(16,39),(17,40),(18,36),(19,37),(20,38)]])

100 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B5C5D5E···5N10A10B10C10D10E···10V10W···10AP10AQ10AR10AS10AT20A20B20C20D20E···20N20O···20V20W···20AD
order122224444455555···51010101010···1010···10101010102020202020···2020···2020···20
size112210255101011112···211112···24···41010101022224···45···510···10

100 irreducible representations

dim1111111111112222222244
type+++++++++-
imageC1C2C2C2C2C2C5C10C10C10C10C10D5C4oD4D10D10C5xD5C5xC4oD4D5xC10D5xC10D4:2D5C5xD4:2D5
kernelC5xD4:2D5C5xDic10D5xC20C10xDic5C5xC5:D4D4xC52D4:2D5Dic10C4xD5C2xDic5C5:D4C5xD4C5xD4C52C20C2xC10D4C5C4C22C5C1
# reps11122144488422248881628

Matrix representation of C5xD4:2D5 in GL4(F41) generated by

16000
01600
00370
00037
,
0100
40000
0010
0001
,
0100
1000
0010
0001
,
1000
0100
00180
001916
,
03200
9000
001635
002225
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,37,0,0,0,0,37],[0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,18,19,0,0,0,16],[0,9,0,0,32,0,0,0,0,0,16,22,0,0,35,25] >;

C5xD4:2D5 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes_2D_5
% in TeX

G:=Group("C5xD4:2D5");
// GroupNames label

G:=SmallGroup(400,186);
// by ID

G=gap.SmallGroup(400,186);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,247,794,404,11525]);
// Polycyclic

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

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