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G = C42:22D10order 320 = 26·5

22nd semidirect product of C42 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42:22D10, C10.1292+ 1+4, (C2xQ8):10D10, (C4xC20):29C22, C22:C4:21D10, C4.4D4:16D5, C23:D10:25C2, C22:D20:26C2, (C2xD4).112D10, C4.D20:30C2, (C2xC20).83C23, (Q8xC10):16C22, C20.23D4:24C2, D10:C4:6C22, (C2xC10).227C24, C5:2(C24:C22), (C4xDic5):37C22, (C2xD20).36C22, (C23xD5):12C22, C2.77(D4:6D10), C2.53(D4:8D10), C23.D5:35C22, C23.49(C22xD5), Dic5.5D4:43C2, (C2xDic10):10C22, (D4xC10).212C22, (C22xC10).57C23, (C22xD5).99C23, C22.248(C23xD5), (C2xDic5).117C23, (C5xC4.4D4):19C2, (C5xC22:C4):32C22, (C2xC4).200(C22xD5), (C2xC5:D4).65C22, SmallGroup(320,1355)

Series: Derived Chief Lower central Upper central

C1C2xC10 — C42:22D10
C1C5C10C2xC10C22xD5C23xD5C23:D10 — C42:22D10
C5C2xC10 — C42:22D10
C1C22C4.4D4

Generators and relations for C42:22D10
 G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1, dad=ab2, cbc-1=a2b-1, dbd=a2b, dcd=c-1 >

Subgroups: 1286 in 260 conjugacy classes, 91 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C42, C22:C4, C22:C4, C2xD4, C2xD4, C2xQ8, C2xQ8, C24, Dic5, C20, D10, C2xC10, C2xC10, C22wrC2, C4.4D4, C4.4D4, Dic10, D20, C2xDic5, C5:D4, C2xC20, C2xC20, C5xD4, C5xQ8, C22xD5, C22xD5, C22xC10, C24:C22, C4xDic5, D10:C4, C23.D5, C4xC20, C5xC22:C4, C2xDic10, C2xD20, C2xC5:D4, D4xC10, Q8xC10, C23xD5, C4.D20, C22:D20, Dic5.5D4, C23:D10, C20.23D4, C5xC4.4D4, C42:22D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22xD5, C24:C22, C23xD5, D4:6D10, D4:8D10, C42:22D10

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4E4F4G4H4I5A5B10A···10F10G10H10I10J20A···20L20M20N20O20P
order12222222224···444445510···101010101020···2020202020
size111144202020204···420202020222···288884···48888

47 irreducible representations

dim111111122222444
type++++++++++++++
imageC1C2C2C2C2C2C2D5D10D10D10D102+ 1+4D4:6D10D4:8D10
kernelC42:22D10C4.D20C22:D20Dic5.5D4C23:D10C20.23D4C5xC4.4D4C4.4D4C42C22:C4C2xD4C2xQ8C10C2C2
# reps124422122822348

Matrix representation of C42:22D10 in GL8(F41)

171000000
4024000000
001710000
0040240000
0000400341
0000040734
000014210
0000141401
,
104000000
010400000
204000000
020400000
000030322730
00009112727
00001322309
000028133211
,
4071340000
3477340000
001340000
007340000
0000343400
00007100
0000214407
000002347
,
740000000
734000000
007400000
007340000
00007700
0000403400
00000010
000000740

G:=sub<GL(8,GF(41))| [17,40,0,0,0,0,0,0,1,24,0,0,0,0,0,0,0,0,17,40,0,0,0,0,0,0,1,24,0,0,0,0,0,0,0,0,40,0,14,14,0,0,0,0,0,40,2,14,0,0,0,0,34,7,1,0,0,0,0,0,1,34,0,1],[1,0,2,0,0,0,0,0,0,1,0,2,0,0,0,0,40,0,40,0,0,0,0,0,0,40,0,40,0,0,0,0,0,0,0,0,30,9,13,28,0,0,0,0,32,11,22,13,0,0,0,0,27,27,30,32,0,0,0,0,30,27,9,11],[40,34,0,0,0,0,0,0,7,7,0,0,0,0,0,0,1,7,1,7,0,0,0,0,34,34,34,34,0,0,0,0,0,0,0,0,34,7,2,0,0,0,0,0,34,1,14,2,0,0,0,0,0,0,40,34,0,0,0,0,0,0,7,7],[7,7,0,0,0,0,0,0,40,34,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,40,34,0,0,0,0,0,0,0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,40] >;

C42:22D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{22}D_{10}
% in TeX

G:=Group("C4^2:22D10");
// GroupNames label

G:=SmallGroup(320,1355);
// by ID

G=gap.SmallGroup(320,1355);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,1571,570,297,192,12550]);
// Polycyclic

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

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