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G = C4218D10order 320 = 26·5

18th semidirect product of C42 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4218D10, C10.1242+ 1+4, (C4×D5)⋊4D4, C4.32(D4×D5), (C2×Q8)⋊18D10, C20.61(C2×D4), C4.4D48D5, C20⋊D424C2, C204D414C2, (C4×C20)⋊22C22, C22⋊C420D10, D10.79(C2×D4), (C2×D20)⋊9C22, D10⋊D439C2, C22⋊D2023C2, (C2×D4).171D10, C42⋊D519C2, Dic5.90(C2×D4), (Q8×C10)⋊12C22, C10.88(C22×D4), C20.23D421C2, (C2×C20).186C23, (C2×C10).218C24, C54(C22.29C24), (C4×Dic5)⋊35C22, C2.48(D48D10), D10⋊C423C22, C23.40(C22×D5), (D4×C10).153C22, C10.D455C22, (C22×C10).48C23, (C23×D5).63C22, C22.239(C23×D5), (C2×Dic5).113C23, (C22×D5).223C23, (C2×D4×D5)⋊16C2, C2.61(C2×D4×D5), (C2×C4×D5)⋊25C22, (C2×Q82D5)⋊10C2, (C5×C4.4D4)⋊10C2, (C2×C5⋊D4)⋊22C22, (C5×C22⋊C4)⋊28C22, (C2×C4).193(C22×D5), SmallGroup(320,1346)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4218D10
C1C5C10C2×C10C22×D5C23×D5C2×D4×D5 — C4218D10
C5C2×C10 — C4218D10
C1C22C4.4D4

Generators and relations for C4218D10
 G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=a2b-1, dbd=b-1, dcd=c-1 >

Subgroups: 1598 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×30], C5, C2×C4, C2×C4 [×4], C2×C4 [×11], D4 [×22], Q8 [×2], C23 [×2], C23 [×13], D5 [×6], C10, C10 [×2], C10 [×2], C42, C42, C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×18], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×4], D10 [×2], D10 [×22], C2×C10, C2×C10 [×6], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4, C4.4D4, C41D4 [×2], C22×D4, C2×C4○D4, C4×D5 [×4], C4×D5 [×4], D20 [×12], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×8], C2×C20, C2×C20 [×4], C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×4], C22×D5 [×8], C22×C10 [×2], C22.29C24, C4×Dic5, C10.D4 [×2], D10⋊C4 [×6], C4×C20, C5×C22⋊C4 [×4], C2×C4×D5, C2×C4×D5 [×2], C2×D20 [×2], C2×D20 [×6], D4×D5 [×4], Q82D5 [×4], C2×C5⋊D4 [×6], D4×C10, Q8×C10, C23×D5 [×2], C42⋊D5, C204D4, C22⋊D20 [×4], D10⋊D4 [×4], C20⋊D4, C20.23D4, C5×C4.4D4, C2×D4×D5, C2×Q82D5, C4218D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ 1+4 [×2], C22×D5 [×7], C22.29C24, D4×D5 [×2], C23×D5, C2×D4×D5, D48D10 [×2], C4218D10

Smallest permutation representation of C4218D10
On 80 points
Generators in S80
(1 53 16 58)(2 59 17 54)(3 55 18 60)(4 51 19 56)(5 57 20 52)(6 64 14 69)(7 70 15 65)(8 66 11 61)(9 62 12 67)(10 68 13 63)(21 76 50 36)(22 37 41 77)(23 78 42 38)(24 39 43 79)(25 80 44 40)(26 31 45 71)(27 72 46 32)(28 33 47 73)(29 74 48 34)(30 35 49 75)
(1 74 10 39)(2 80 6 35)(3 76 7 31)(4 72 8 37)(5 78 9 33)(11 77 19 32)(12 73 20 38)(13 79 16 34)(14 75 17 40)(15 71 18 36)(21 65 26 60)(22 56 27 61)(23 67 28 52)(24 58 29 63)(25 69 30 54)(41 51 46 66)(42 62 47 57)(43 53 48 68)(44 64 49 59)(45 55 50 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 19)(2 18)(3 17)(4 16)(5 20)(6 15)(7 14)(8 13)(9 12)(10 11)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 48)(42 47)(43 46)(44 45)(49 50)(51 53)(54 60)(55 59)(56 58)(61 63)(64 70)(65 69)(66 68)(71 80)(72 79)(73 78)(74 77)(75 76)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J5A5B10A···10F10G10H10I10J20A···20L20M20N20O20P
order12222222222244444444445510···101010101020···2020202020
size11114410102020202022444410102020222···288884···48888

50 irreducible representations

dim1111111111222222444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D5D10D10D10D102+ 1+4D4×D5D48D10
kernelC4218D10C42⋊D5C204D4C22⋊D20D10⋊D4C20⋊D4C20.23D4C5×C4.4D4C2×D4×D5C2×Q82D5C4×D5C4.4D4C42C22⋊C4C2×D4C2×Q8C10C4C2
# reps1114411111422822248

Matrix representation of C4218D10 in GL8(𝔽41)

400000000
040000000
00100000
00010000
0000040390
0000400039
00001001
00000110
,
400000000
040000000
00190000
0018400000
00000100
000040000
00001001
0000040400
,
034000000
635000000
004000000
002310000
00001000
00000100
0000040400
0000400040
,
634000000
535000000
00100000
0018400000
00001000
000004000
000000400
00000001

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,40,0,0,1,0,0,0,0,39,0,0,1,0,0,0,0,0,39,1,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,9,40,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0],[0,6,0,0,0,0,0,0,34,35,0,0,0,0,0,0,0,0,40,23,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[6,5,0,0,0,0,0,0,34,35,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1] >;

C4218D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{18}D_{10}
% in TeX

G:=Group("C4^2:18D10");
// GroupNames label

G:=SmallGroup(320,1346);
// by ID

G=gap.SmallGroup(320,1346);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,675,570,297,192,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^2*b^-1,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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