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G = C42:25D10order 320 = 26·5

25th semidirect product of C42 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42:25D10, C10.1422+ 1+4, C4:C4:17D10, C20:4D4:5C2, (C4xC20):2C22, C4:D20:38C2, C42:2C2:8D5, C42:2D5:1C2, D10:D4:46C2, C22:D20:28C2, (C2xD20):10C22, C22:C4.41D10, (C2xC10).255C24, (C2xC20).195C23, C10.D4:5C22, D10.13D4:44C2, C2.67(D4:8D10), D10:C4:24C22, C23.61(C22xD5), C5:4(C22.54C24), (C22xC10).69C23, (C23xD5).70C22, C22.276(C23xD5), (C2xDic5).131C23, (C22xD5).114C23, (C2xC4xD5):28C22, (C5xC4:C4):34C22, (C5xC42:2C2):10C2, (C2xC4).211(C22xD5), (C2xC5:D4).75C22, (C5xC22:C4).80C22, SmallGroup(320,1383)

Series: Derived Chief Lower central Upper central

C1C2xC10 — C42:25D10
C1C5C10C2xC10C22xD5C23xD5C22:D20 — C42:25D10
C5C2xC10 — C42:25D10
C1C22C42:2C2

Generators and relations for C42:25D10
 G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >

Subgroups: 1254 in 252 conjugacy classes, 91 normal (16 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2xC4, C2xC4, D4, C23, C23, D5, C10, C10, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C2xD4, C24, Dic5, C20, D10, C2xC10, C2xC10, C22wrC2, C4:D4, C22.D4, C42:2C2, C42:2C2, C4:1D4, C4xD5, D20, C2xDic5, C5:D4, C2xC20, C22xD5, C22xD5, C22xD5, C22xC10, C22.54C24, C10.D4, D10:C4, C4xC20, C5xC22:C4, C5xC4:C4, C2xC4xD5, C2xD20, C2xC5:D4, C23xD5, C20:4D4, C42:2D5, C22:D20, D10:D4, D10.13D4, C4:D20, C5xC42:2C2, C42:25D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22xD5, C22.54C24, C23xD5, D4:8D10, C42:25D10

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E···2I4A···4F4G4H4I5A5B10A···10F10G10H20A···20L20M···20R
order122222···24···44445510···10101020···2020···20
size1111420···204···4202020222···2884···48···8

47 irreducible representations

dim11111111222244
type++++++++++++++
imageC1C2C2C2C2C2C2C2D5D10D10D102+ 1+4D4:8D10
kernelC42:25D10C20:4D4C42:2D5C22:D20D10:D4D10.13D4C4:D20C5xC42:2C2C42:2C2C42C22:C4C4:C4C10C2
# reps111333312266312

Matrix representation of C42:25D10 in GL8(F41)

22810120000
133910100000
002130000
0028390000
0000113200
000093000
0000001132
000000930
,
1028240000
0113280000
004000000
000400000
00001132390
0000930039
000000309
0000003211
,
4035000000
635000000
3214660000
14323510000
000040700
000034700
00001114134
00002727734
,
10000000
3540000000
92735350000
27304060000
000040000
000034100
00001132400
00002730341

G:=sub<GL(8,GF(41))| [2,13,0,0,0,0,0,0,28,39,0,0,0,0,0,0,10,10,2,28,0,0,0,0,12,10,13,39,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,32,30,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,32,30],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,28,13,40,0,0,0,0,0,24,28,0,40,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,32,30,0,0,0,0,0,0,39,0,30,32,0,0,0,0,0,39,9,11],[40,6,32,14,0,0,0,0,35,35,14,32,0,0,0,0,0,0,6,35,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,40,34,11,27,0,0,0,0,7,7,14,27,0,0,0,0,0,0,1,7,0,0,0,0,0,0,34,34],[1,35,9,27,0,0,0,0,0,40,27,30,0,0,0,0,0,0,35,40,0,0,0,0,0,0,35,6,0,0,0,0,0,0,0,0,40,34,11,27,0,0,0,0,0,1,32,30,0,0,0,0,0,0,40,34,0,0,0,0,0,0,0,1] >;

C42:25D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{25}D_{10}
% in TeX

G:=Group("C4^2:25D10");
// GroupNames label

G:=SmallGroup(320,1383);
// by ID

G=gap.SmallGroup(320,1383);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,184,1571,570,192,12550]);
// 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*b^2,d*a*d=a*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations

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