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G = D5xC4.4D4order 320 = 26·5

Direct product of D5 and C4.4D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5xC4.4D4, C42:33D10, C4.31(D4xD5), (C2xQ8):17D10, (C4xD5).63D4, C20.60(C2xD4), (D5xC42):9C2, (C4xC20):21C22, C22:C4:32D10, D10.108(C2xD4), (C2xD4).170D10, C4.D20:23C2, (C2xC20).79C23, Dic5.21(C2xD4), (Q8xC10):11C22, C10.87(C22xD4), D10.65(C4oD4), C20.17D4:23C2, C20.23D4:20C2, (C2xC10).217C24, (C4xDic5):79C22, C23.D5:32C22, D10:C4:29C22, C23.39(C22xD5), Dic5.5D4:39C2, (C2xDic10):32C22, (D4xC10).152C22, (C2xD20).168C22, (C22xC10).47C23, (C23xD5).62C22, (C22xD5).95C23, C22.238(C23xD5), (C2xDic5).112C23, (C2xQ8xD5):9C2, C2.60(C2xD4xD5), (C2xD4xD5).11C2, C5:3(C2xC4.4D4), C2.75(D5xC4oD4), (D5xC22:C4):16C2, (C5xC4.4D4):9C2, C10.186(C2xC4oD4), (C2xC4xD5).380C22, (C5xC22:C4):27C22, (C2xC4).300(C22xD5), (C2xC5:D4).58C22, SmallGroup(320,1345)

Series: Derived Chief Lower central Upper central

C1C2xC10 — D5xC4.4D4
C1C5C10C2xC10C22xD5C23xD5D5xC22:C4 — D5xC4.4D4
C5C2xC10 — D5xC4.4D4
C1C22C4.4D4

Generators and relations for D5xC4.4D4
 G = < a,b,c,d,e | a5=b2=c4=d4=1, e2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=c2d-1 >

Subgroups: 1358 in 330 conjugacy classes, 111 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, D5, D5, C10, C10, C10, C42, C42, C22:C4, C22:C4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C24, Dic5, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, C2xC42, C2xC22:C4, C4.4D4, C4.4D4, C22xD4, C22xQ8, Dic10, C4xD5, C4xD5, D20, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C5xQ8, C22xD5, C22xD5, C22xD5, C22xC10, C2xC4.4D4, C4xDic5, C4xDic5, D10:C4, C23.D5, C4xC20, C5xC22:C4, C2xDic10, C2xDic10, C2xC4xD5, C2xC4xD5, C2xD20, D4xD5, Q8xD5, C2xC5:D4, D4xC10, Q8xC10, C23xD5, D5xC42, C4.D20, D5xC22:C4, Dic5.5D4, C20.17D4, C20.23D4, C5xC4.4D4, C2xD4xD5, C2xQ8xD5, D5xC4.4D4
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C4oD4, C24, D10, C4.4D4, C22xD4, C2xC4oD4, C22xD5, C2xC4.4D4, D4xD5, C23xD5, C2xD4xD5, D5xC4oD4, D5xC4.4D4

Smallest permutation representation of D5xC4.4D4
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 13)(2 12)(3 11)(4 15)(5 14)(6 16)(7 20)(8 19)(9 18)(10 17)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)(41 51)(42 55)(43 54)(44 53)(45 52)(46 56)(47 60)(48 59)(49 58)(50 57)(61 71)(62 75)(63 74)(64 73)(65 72)(66 76)(67 80)(68 79)(69 78)(70 77)
(1 79 19 64)(2 80 20 65)(3 76 16 61)(4 77 17 62)(5 78 18 63)(6 71 11 66)(7 72 12 67)(8 73 13 68)(9 74 14 69)(10 75 15 70)(21 51 36 46)(22 52 37 47)(23 53 38 48)(24 54 39 49)(25 55 40 50)(26 56 31 41)(27 57 32 42)(28 58 33 43)(29 59 34 44)(30 60 35 45)
(1 44 14 54)(2 45 15 55)(3 41 11 51)(4 42 12 52)(5 43 13 53)(6 46 16 56)(7 47 17 57)(8 48 18 58)(9 49 19 59)(10 50 20 60)(21 61 31 71)(22 62 32 72)(23 63 33 73)(24 64 34 74)(25 65 35 75)(26 66 36 76)(27 67 37 77)(28 68 38 78)(29 69 39 79)(30 70 40 80)
(1 39 19 24)(2 40 20 25)(3 36 16 21)(4 37 17 22)(5 38 18 23)(6 31 11 26)(7 32 12 27)(8 33 13 28)(9 34 14 29)(10 35 15 30)(41 71 56 66)(42 72 57 67)(43 73 58 68)(44 74 59 69)(45 75 60 70)(46 76 51 61)(47 77 52 62)(48 78 53 63)(49 79 54 64)(50 80 55 65)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A···4F4G4H4I···4N4O4P5A5B10A···10F10G10H10I10J20A···20L20M20N20O20P
order1222222222224···4444···4445510···101010101020···2020202020
size111144555520202···24410···102020222···288884···48888

56 irreducible representations

dim1111111111222222244
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D5C4oD4D10D10D10D10D4xD5D5xC4oD4
kernelD5xC4.4D4D5xC42C4.D20D5xC22:C4Dic5.5D4C20.17D4C20.23D4C5xC4.4D4C2xD4xD5C2xQ8xD5C4xD5C4.4D4D10C42C22:C4C2xD4C2xQ8C4C2
# reps1114411111428282248

Matrix representation of D5xC4.4D4 in GL6(F41)

3510000
5400000
001000
000100
000010
000001
,
40400000
010000
001000
000100
0000400
0000040
,
100000
010000
00343600
0010700
000010
000001
,
100000
010000
00193700
0082200
00003218
000009
,
100000
010000
0032000
0017900
00003218
0000329

G:=sub<GL(6,GF(41))| [35,5,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,10,0,0,0,0,36,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,8,0,0,0,0,37,22,0,0,0,0,0,0,32,0,0,0,0,0,18,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,17,0,0,0,0,0,9,0,0,0,0,0,0,32,32,0,0,0,0,18,9] >;

D5xC4.4D4 in GAP, Magma, Sage, TeX

D_5\times C_4._4D_4
% in TeX

G:=Group("D5xC4.4D4");
// GroupNames label

G:=SmallGroup(320,1345);
// by ID

G=gap.SmallGroup(320,1345);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,100,346,136,12550]);
// Polycyclic

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

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