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G = C4226D10order 320 = 26·5

26th semidirect product of C42 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4226D10, C10.762+ 1+4, (C4×D5)⋊5D4, C41D45D5, C4.34(D4×D5), (C2×D4)⋊12D10, C20.65(C2×D4), C20⋊D426C2, (C4×C20)⋊26C22, D10.81(C2×D4), C23⋊D1026C2, C4.D2025C2, (D4×C10)⋊32C22, C42⋊D523C2, Dic5.92(C2×D4), C10.93(C22×D4), Dic5⋊D436C2, C20.17D426C2, (C2×C20).635C23, (C2×C10).259C24, C55(C22.29C24), (C4×Dic5)⋊39C22, C23.D536C22, C2.80(D46D10), D10⋊C434C22, C23.65(C22×D5), (C2×Dic10)⋊34C22, (C2×D20).176C22, C10.D471C22, (C22×C10).73C23, (C23×D5).72C22, C22.280(C23×D5), (C2×Dic5).134C23, (C22×Dic5)⋊29C22, (C22×D5).237C23, (C2×D4×D5)⋊19C2, C2.66(C2×D4×D5), (C5×C41D4)⋊6C2, (C2×D42D5)⋊20C2, (C2×C5⋊D4)⋊26C22, (C2×C4×D5).147C22, (C2×C4).213(C22×D5), SmallGroup(320,1387)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4226D10
C1C5C10C2×C10C22×D5C23×D5C2×D4×D5 — C4226D10
C5C2×C10 — C4226D10
C1C22C41D4

Generators and relations for C4226D10
 G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=b-1, dbd=a2b-1, dcd=c-1 >

Subgroups: 1470 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 [×2], C2×C4 [×13], D4 [×22], Q8 [×2], C23 [×4], C23 [×11], D5 [×4], C10, C10 [×2], C10 [×4], C42, C42, C22⋊C4 [×10], C4⋊C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×4], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×2], D10 [×2], D10 [×16], C2×C10, C2×C10 [×12], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C41D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5, C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20, C2×C20 [×2], C5×D4 [×8], C22×D5, C22×D5 [×2], C22×D5 [×8], C22×C10 [×4], C22.29C24, C4×Dic5, C10.D4 [×2], D10⋊C4 [×6], C23.D5 [×4], C4×C20, C2×Dic10, C2×C4×D5, C2×D20, D4×D5 [×4], D42D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×8], D4×C10 [×2], D4×C10 [×4], C23×D5 [×2], C42⋊D5, C4.D20, C20.17D4, C23⋊D10 [×4], Dic5⋊D4 [×4], C20⋊D4, C5×C41D4, C2×D4×D5, C2×D42D5, C4226D10
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, D46D10 [×2], C4226D10

Smallest permutation representation of C4226D10
On 80 points
Generators in S80
(1 73 16 78)(2 79 17 74)(3 75 18 80)(4 71 19 76)(5 77 20 72)(6 50 14 45)(7 46 15 41)(8 42 11 47)(9 48 12 43)(10 44 13 49)(21 69 38 51)(22 52 39 70)(23 61 40 53)(24 54 31 62)(25 63 32 55)(26 56 33 64)(27 65 34 57)(28 58 35 66)(29 67 36 59)(30 60 37 68)
(1 24 13 29)(2 30 14 25)(3 26 15 21)(4 22 11 27)(5 28 12 23)(6 32 17 37)(7 38 18 33)(8 34 19 39)(9 40 20 35)(10 36 16 31)(41 69 75 56)(42 57 76 70)(43 61 77 58)(44 59 78 62)(45 63 79 60)(46 51 80 64)(47 65 71 52)(48 53 72 66)(49 67 73 54)(50 55 74 68)
(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 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 25)(22 24)(26 30)(27 29)(31 39)(32 38)(33 37)(34 36)(41 79)(42 78)(43 77)(44 76)(45 75)(46 74)(47 73)(48 72)(49 71)(50 80)(51 63)(52 62)(53 61)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J5A5B10A···10F10G···10N20A···20L
order12222222222244444444445510···1010···1020···20
size11114444101020202244101020202020222···28···84···4

50 irreducible representations

dim11111111112222444
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D5D10D102+ 1+4D4×D5D46D10
kernelC4226D10C42⋊D5C4.D20C20.17D4C23⋊D10Dic5⋊D4C20⋊D4C5×C41D4C2×D4×D5C2×D42D5C4×D5C41D4C42C2×D4C10C4C2
# reps111144111142212248

Matrix representation of C4226D10 in GL8(𝔽41)

400000000
040000000
004000000
000400000
000040020
0000213939
000040010
000011040
,
001400000
3512350000
38214000000
39214000000
00000100
00001000
0000114040
00000001
,
06000000
347000000
11635350000
16126400000
00001000
00000100
000010400
000011040
,
735000000
834000000
3035660000
25291350000
00001000
000004000
000010400
0000404021

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,2,40,1,0,0,0,0,0,1,0,1,0,0,0,0,2,39,1,0,0,0,0,0,0,39,0,40],[0,35,38,39,0,0,0,0,0,1,21,21,0,0,0,0,1,2,40,40,0,0,0,0,40,35,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,1],[0,34,11,16,0,0,0,0,6,7,6,12,0,0,0,0,0,0,35,6,0,0,0,0,0,0,35,40,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[7,8,30,25,0,0,0,0,35,34,35,29,0,0,0,0,0,0,6,1,0,0,0,0,0,0,6,35,0,0,0,0,0,0,0,0,1,0,1,40,0,0,0,0,0,40,0,40,0,0,0,0,0,0,40,2,0,0,0,0,0,0,0,1] >;

C4226D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{26}D_{10}
% in TeX

G:=Group("C4^2:26D10");
// GroupNames label

G:=SmallGroup(320,1387);
// by ID

G=gap.SmallGroup(320,1387);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,675,570,297,136,12550]);
// 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,c*b*c^-1=b^-1,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations

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