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G = C4xD4xD5order 320 = 26·5

Direct product of C4, D4 and D5

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

Aliases: C4xD4xD5, C42:32D10, C4:C4:55D10, (D4xC20):8C2, C20:13(C2xD4), D20:22(C2xC4), (C4xD20):23C2, C20:2(C22xC4), (D5xC42):3C2, (C4xC20):15C22, C22:C4:52D10, Dic5:11(C2xD4), (D4xDic5):44C2, D10:5(C22xC4), (C22xC4):36D10, D20:8C4:46C2, (C2xD4).244D10, D10.106(C2xD4), (C2xC10).88C24, C10.42(C23xC4), C4:Dic5:72C22, Dic5:3(C22xC4), C10.46(C22xD4), D10.60(C4oD4), Dic5:4D4:51C2, (C2xC20).586C23, (C22xC20):35C22, (C4xDic5):78C22, C23.D5:47C22, D10:C4:61C22, C22.31(C23xD5), (D4xC10).252C22, (C2xD20).265C22, C10.D4:63C22, C23.167(C22xD5), (C22xC10).158C23, (C2xDic5).375C23, (C22xDic5):43C22, (C22xD5).291C23, (C23xD5).117C22, C5:5(C2xC4xD4), C4:1(C2xC4xD5), C2.5(C2xD4xD5), C22:1(C2xC4xD5), (D5xC4:C4):47C2, C5:D4:6(C2xC4), (C2xD4xD5).19C2, C2.4(D5xC4oD4), (C5xD4):21(C2xC4), (C4xD5):11(C2xC4), (C4xC5:D4):39C2, (C2xC4xD5):69C22, (D5xC22xC4):21C2, (C5xC4:C4):55C22, C2.23(D5xC22xC4), (C2xC10):2(C22xC4), (D5xC22:C4):30C2, C10.138(C2xC4oD4), (C22xD5):17(C2xC4), (C5xC22:C4):62C22, (C2xC4).819(C22xD5), (C2xC5:D4).117C22, SmallGroup(320,1216)

Series: Derived Chief Lower central Upper central

C1C10 — C4xD4xD5
C1C5C10C2xC10C22xD5C23xD5C2xD4xD5 — C4xD4xD5
C5C10 — C4xD4xD5
C1C2xC4C4xD4

Generators and relations for C4xD4xD5
 G = < a,b,c,d,e | a4=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1486 in 426 conjugacy classes, 169 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2xC4, C2xC4, C2xC4, D4, D4, C23, C23, D5, D5, C10, C10, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C24, Dic5, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, C2xC10, C2xC42, C2xC22:C4, C2xC4:C4, C4xD4, C4xD4, C23xC4, C22xD4, C4xD5, C4xD5, D20, C2xDic5, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C2xC20, C5xD4, C22xD5, C22xD5, C22xD5, C22xC10, C2xC4xD4, C4xDic5, C10.D4, C4:Dic5, D10:C4, C23.D5, C4xC20, C5xC22:C4, C5xC4:C4, C2xC4xD5, C2xC4xD5, C2xC4xD5, C2xD20, D4xD5, C22xDic5, C2xC5:D4, C22xC20, D4xC10, C23xD5, D5xC42, C4xD20, D5xC22:C4, Dic5:4D4, D5xC4:C4, D20:8C4, C4xC5:D4, D4xDic5, D4xC20, D5xC22xC4, C2xD4xD5, C4xD4xD5
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D5, C22xC4, C2xD4, C4oD4, C24, D10, C4xD4, C23xC4, C22xD4, C2xC4oD4, C4xD5, C22xD5, C2xC4xD4, C2xC4xD5, D4xD5, C23xD5, D5xC22xC4, C2xD4xD5, D5xC4oD4, C4xD4xD5

Smallest permutation representation of C4xD4xD5
On 80 points
Generators in S80
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)
(1 54 9 59)(2 55 10 60)(3 51 6 56)(4 52 7 57)(5 53 8 58)(11 41 16 46)(12 42 17 47)(13 43 18 48)(14 44 19 49)(15 45 20 50)(21 71 26 76)(22 72 27 77)(23 73 28 78)(24 74 29 79)(25 75 30 80)(31 61 36 66)(32 62 37 67)(33 63 38 68)(34 64 39 69)(35 65 40 70)
(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 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)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D4E···4L4M4N4O4P4Q···4X5A5B10A···10F10G···10N20A···20H20I···20X
order122222222222222244444···444444···45510···1010···1020···2020···20
size1111222255551010101011112···2555510···10222···24···42···24···4

80 irreducible representations

dim111111111111122222222244
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C4D4D5C4oD4D10D10D10D10D10C4xD5D4xD5D5xC4oD4
kernelC4xD4xD5D5xC42C4xD20D5xC22:C4Dic5:4D4D5xC4:C4D20:8C4C4xC5:D4D4xDic5D4xC20D5xC22xC4C2xD4xD5D4xD5C4xD5C4xD4D10C42C22:C4C4:C4C22xC4C2xD4D4C4C2
# reps11122112112116424242421644

Matrix representation of C4xD4xD5 in GL5(F41)

320000
01000
00100
000400
000040
,
10000
040000
004000
000040
00010
,
400000
01000
00100
00010
000040
,
10000
040100
053500
00010
00001
,
10000
040000
05100
00010
00001

G:=sub<GL(5,GF(41))| [32,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,40,0],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40],[1,0,0,0,0,0,40,5,0,0,0,1,35,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,5,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

C4xD4xD5 in GAP, Magma, Sage, TeX

C_4\times D_4\times D_5
% in TeX

G:=Group("C4xD4xD5");
// GroupNames label

G:=SmallGroup(320,1216);
// by ID

G=gap.SmallGroup(320,1216);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,80,12550]);
// Polycyclic

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

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