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G = C2×D42D5order 160 = 25·5

Direct product of C2 and D42D5

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

Aliases: C2×D42D5, D45D10, C10.6C24, C20.20C23, D10.2C23, C23.19D10, Dic107C22, Dic5.3C23, (C2×D4)⋊8D5, (D4×C10)⋊6C2, C102(C4○D4), (C2×C4).60D10, (C5×D4)⋊6C22, (C4×D5)⋊4C22, C5⋊D42C22, C2.7(C23×D5), (C2×C10).1C23, C4.20(C22×D5), (C2×Dic10)⋊12C2, (C2×C20).45C22, (C22×Dic5)⋊8C2, (C2×Dic5)⋊9C22, C22.1(C22×D5), (C22×C10).23C22, (C22×D5).32C22, (C2×C4×D5)⋊4C2, C52(C2×C4○D4), (C2×C5⋊D4)⋊10C2, SmallGroup(160,218)

Series: Derived Chief Lower central Upper central

C1C10 — C2×D42D5
C1C5C10D10C22×D5C2×C4×D5 — C2×D42D5
C5C10 — C2×D42D5
C1C22C2×D4

Generators and relations for C2×D42D5
 G = < a,b,c,d,e | a2=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: 424 in 164 conjugacy classes, 89 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×4], C22 [×8], C5, C2×C4, C2×C4 [×15], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, D5 [×2], C10, C10 [×2], C10 [×4], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], Dic5 [×6], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×10], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×C10 [×2], C2×Dic10, C2×C4×D5, D42D5 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C2×D42D5
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, C22×D5 [×7], D42D5 [×2], C23×D5, C2×D42D5

Smallest permutation representation of C2×D42D5
On 80 points
Generators in S80
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)
(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 5)(2 4)(6 10)(7 9)(11 20)(12 19)(13 18)(14 17)(15 16)(21 25)(22 24)(26 30)(27 29)(31 40)(32 39)(33 38)(34 37)(35 36)(41 45)(42 44)(46 50)(47 49)(51 60)(52 59)(53 58)(54 57)(55 56)(61 65)(62 64)(66 70)(67 69)(71 80)(72 79)(73 78)(74 77)(75 76)

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

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

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

C2×D42D5 is a maximal subgroup of
C23⋊C45D5  M4(2).19D10  D4⋊(C4×D5)  D42D5⋊C4  D43D20  D4.D20  Dic10⋊D4  Dic10.16D4  (C2×D4)⋊6F5  (C2×D4)⋊8F5  (C2×D4).7F5  (C2×D4).8F5  (C2×D4).9F5  C42.108D10  D45D20  D46D20  C24.56D10  C24.33D10  C24.34D10  C20⋊(C4○D4)  C10.682- 1+4  Dic1019D4  Dic1020D4  C4⋊C421D10  C10.392+ 1+4  C10.402+ 1+4  C10.732- 1+4  C10.792- 1+4  C10.822- 1+4  C4⋊C428D10  C42.233D10  C42.141D10  Dic1010D4  C4226D10  C42.238D10  Dic1011D4  SD16⋊D10  C24.42D10  C10.1042- 1+4  Dic5.C24  C2×D5×C4○D4  D20.37C23
C2×D42D5 is a maximal quotient of
C24.31D10  C10.52- 1+4  C42.102D10  C42.105D10  C42.106D10  D46Dic10  D46D20  C42.229D10  C42.117D10  C42.119D10  C24.56D10  C24.32D10  C24.33D10  C24.35D10  C20⋊(C4○D4)  Dic1019D4  C4⋊C4.178D10  C10.342+ 1+4  C10.352+ 1+4  C10.362+ 1+4  C4⋊C421D10  C10.732- 1+4  C10.432+ 1+4  C10.452+ 1+4  C10.462+ 1+4  C10.1152+ 1+4  C10.472+ 1+4  (Q8×Dic5)⋊C2  C22⋊Q825D5  C10.152- 1+4  C10.1182+ 1+4  C10.212- 1+4  C10.232- 1+4  C10.772- 1+4  C10.242- 1+4  C4⋊C4.197D10  C10.802- 1+4  C10.1222+ 1+4  C10.852- 1+4  C42.139D10  C42.234D10  C42.143D10  C42.144D10  C42.166D10  C42.238D10  Dic1011D4  C42.168D10  Dic108Q8  C42.241D10  C42.176D10  C42.177D10  C2×D4×Dic5  C24.42D10

40 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J5A5B10A···10F10G···10N20A20B20C20D
order122222222244444444445510···1010···1020202020
size11112222101022555510101010222···24···44444

40 irreducible representations

dim1111111222224
type+++++++++++-
imageC1C2C2C2C2C2C2D5C4○D4D10D10D10D42D5
kernelC2×D42D5C2×Dic10C2×C4×D5D42D5C22×Dic5C2×C5⋊D4D4×C10C2×D4C10C2×C4D4C23C2
# reps1118221242844

Matrix representation of C2×D42D5 in GL4(𝔽41) generated by

40000
04000
00400
00040
,
40000
04000
00320
0099
,
1000
0100
003223
0099
,
344000
1000
0010
0001
,
344000
7700
0010
004040
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,32,9,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,32,9,0,0,23,9],[34,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[34,7,0,0,40,7,0,0,0,0,1,40,0,0,0,40] >;

C2×D42D5 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_2D_5
% in TeX

G:=Group("C2xD4:2D5");
// GroupNames label

G:=SmallGroup(160,218);
// by ID

G=gap.SmallGroup(160,218);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,86,579,159,4613]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=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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