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G = C42:16D10order 320 = 26·5

16th semidirect product of C42 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42:16D10, C10.182+ 1+4, C4:C4:49D10, (C4xD4):18D5, (D4xC20):20C2, (C22xC4):5D10, (C4xC20):32C22, D10:Q8:8C2, C22:C4:48D10, C4:Dic5:9C22, (C2xD4).217D10, C23:D10.5C2, C42:2D5:16C2, C42:D5:32C2, D10.31(C4oD4), D10.12D4:7C2, C20.48D4:11C2, C23.D5:9C22, (C2xC10).100C24, (C2xC20).699C23, (C22xC20):37C22, Dic5.5D4:7C2, (C2xDic10):6C22, (C4xDic5):52C22, C2.19(D4:6D10), C5:3(C22.45C24), (D4xC10).307C22, C22.12(C4oD20), C10.D4:42C22, (C23xD5).41C22, (C22xD5).35C23, C23.174(C22xD5), C22.125(C23xD5), D10:C4.85C22, C23.11D10:29C2, C23.23D10:16C2, C23.18D10:18C2, (C22xC10).170C23, (C2xDic5).217C23, (C22xDic5).98C22, C4:C4:D5:7C2, (C4xC5:D4):43C2, C2.23(D5xC4oD4), (C5xC4:C4):61C22, (D5xC22:C4):29C2, C2.49(C2xC4oD20), C10.140(C2xC4oD4), (C2xD10:C4):22C2, (C2xC4xD5).252C22, (C2xC10).16(C4oD4), (C5xC22:C4):57C22, (C2xC4).284(C22xD5), (C2xC5:D4).16C22, SmallGroup(320,1228)

Series: Derived Chief Lower central Upper central

C1C2xC10 — C42:16D10
C1C5C10C2xC10C22xD5C23xD5D5xC22:C4 — C42:16D10
C5C2xC10 — C42:16D10
C1C22C4xD4

Generators and relations for C42:16D10
 G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 934 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2xC4, C2xC4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C24, Dic5, C20, D10, D10, C2xC10, C2xC10, C2xC10, C2xC22:C4, C42:C2, C4xD4, C4xD4, C22wrC2, C22:Q8, C22.D4, C4.4D4, C42:2C2, Dic10, C4xD5, C2xDic5, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C22xD5, C22xD5, C22xC10, C22.45C24, C4xDic5, C10.D4, C4:Dic5, D10:C4, C23.D5, C4xC20, C5xC22:C4, C5xC4:C4, C2xDic10, C2xC4xD5, C22xDic5, C2xC5:D4, C22xC20, D4xC10, C23xD5, C42:D5, C42:2D5, C23.11D10, D5xC22:C4, D10.12D4, Dic5.5D4, D10:Q8, C4:C4:D5, C20.48D4, C2xD10:C4, C4xC5:D4, C23.23D10, C23.18D10, C23:D10, D4xC20, C42:16D10
Quotients: C1, C2, C22, C23, D5, C4oD4, C24, D10, C2xC4oD4, 2+ 1+4, C22xD5, C22.45C24, C4oD20, C23xD5, C2xC4oD20, D4:6D10, D5xC4oD4, C42:16D10

Smallest permutation representation of C42:16D10
On 80 points
Generators in S80
(1 58 19 66)(2 54 20 62)(3 60 16 68)(4 56 17 64)(5 52 18 70)(6 69 15 51)(7 65 11 57)(8 61 12 53)(9 67 13 59)(10 63 14 55)(21 46 39 79)(22 42 40 75)(23 48 36 71)(24 44 37 77)(25 50 38 73)(26 72 35 49)(27 78 31 45)(28 74 32 41)(29 80 33 47)(30 76 34 43)
(1 28 8 21)(2 29 9 22)(3 30 10 23)(4 26 6 24)(5 27 7 25)(11 38 18 31)(12 39 19 32)(13 40 20 33)(14 36 16 34)(15 37 17 35)(41 53 79 66)(42 54 80 67)(43 55 71 68)(44 56 72 69)(45 57 73 70)(46 58 74 61)(47 59 75 62)(48 60 76 63)(49 51 77 64)(50 52 78 65)
(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 2)(3 5)(7 10)(8 9)(11 14)(12 13)(16 18)(19 20)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 75)(42 74)(43 73)(44 72)(45 71)(46 80)(47 79)(48 78)(49 77)(50 76)(51 56)(52 55)(53 54)(57 60)(58 59)(61 62)(63 70)(64 69)(65 68)(66 67)

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4F4G4H4I4J4K···4O5A5B10A···10F10G···10N20A···20H20I···20X
order12222222224···444444···45510···1010···1020···2020···20
size11112241010202···244101020···20222···24···42···24···4

65 irreducible representations

dim1111111111111111222222222444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2D5C4oD4C4oD4D10D10D10D10D10C4oD202+ 1+4D4:6D10D5xC4oD4
kernelC42:16D10C42:D5C42:2D5C23.11D10D5xC22:C4D10.12D4Dic5.5D4D10:Q8C4:C4:D5C20.48D4C2xD10:C4C4xC5:D4C23.23D10C23.18D10C23:D10D4xC20C4xD4D10C2xC10C42C22:C4C4:C4C22xC4C2xD4C22C10C2C2
# reps11111111111111112442424216144

Matrix representation of C42:16D10 in GL6(F41)

4000000
0400000
009000
000900
00001418
00003727
,
100000
010000
00153700
00362600
000090
000009
,
070000
3560000
001000
000100
000010
0000340
,
3570000
3660000
001000
00284000
000010
0000340

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,14,37,0,0,0,0,18,27],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,36,0,0,0,0,37,26,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[0,35,0,0,0,0,7,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,40],[35,36,0,0,0,0,7,6,0,0,0,0,0,0,1,28,0,0,0,0,0,40,0,0,0,0,0,0,1,3,0,0,0,0,0,40] >;

C42:16D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{16}D_{10}
% in TeX

G:=Group("C4^2:16D10");
// GroupNames label

G:=SmallGroup(320,1228);
// by ID

G=gap.SmallGroup(320,1228);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,1571,136,12550]);
// 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*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

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