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G = C4225D10order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4225D10, C10.1422+ 1+4, C4⋊C417D10, C204D45C2, (C4×C20)⋊2C22, C4⋊D2038C2, C422C28D5, C422D51C2, D10⋊D446C2, C22⋊D2028C2, (C2×D20)⋊10C22, C22⋊C4.41D10, (C2×C10).255C24, (C2×C20).195C23, C10.D45C22, D10.13D444C2, C2.67(D48D10), D10⋊C424C22, C23.61(C22×D5), C54(C22.54C24), (C22×C10).69C23, (C23×D5).70C22, C22.276(C23×D5), (C2×Dic5).131C23, (C22×D5).114C23, (C2×C4×D5)⋊28C22, (C5×C4⋊C4)⋊34C22, (C5×C422C2)⋊10C2, (C2×C4).211(C22×D5), (C2×C5⋊D4).75C22, (C5×C22⋊C4).80C22, SmallGroup(320,1383)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4225D10
C1C5C10C2×C10C22×D5C23×D5C22⋊D20 — C4225D10
C5C2×C10 — C4225D10
C1C22C422C2

Generators and relations for C4225D10
 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 [×3], C2 [×6], C4 [×9], C22, C22 [×22], C5, C2×C4 [×6], C2×C4 [×6], D4 [×12], C23, C23 [×8], D5 [×5], C10 [×3], C10, C42, C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×3], C2×D4 [×12], C24, Dic5 [×3], C20 [×6], D10 [×19], C2×C10, C2×C10 [×3], C22≀C2 [×3], C4⋊D4 [×6], C22.D4 [×3], C422C2, C422C2, C41D4, C4×D5 [×3], D20 [×9], C2×Dic5 [×3], C5⋊D4 [×3], C2×C20 [×6], C22×D5 [×2], C22×D5 [×3], C22×D5 [×3], C22×C10, C22.54C24, C10.D4 [×3], D10⋊C4 [×9], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×C4×D5 [×3], C2×D20 [×9], C2×C5⋊D4 [×3], C23×D5, C204D4, C422D5, C22⋊D20 [×3], D10⋊D4 [×3], D10.13D4 [×3], C4⋊D20 [×3], C5×C422C2, C4225D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4 [×3], C22×D5 [×7], C22.54C24, C23×D5, D48D10 [×3], C4225D10

Smallest permutation representation of C4225D10
On 80 points
Generators in S80
(1 52 6 28)(2 58 7 24)(3 54 8 30)(4 60 9 26)(5 56 10 22)(11 59 18 25)(12 55 19 21)(13 51 20 27)(14 57 16 23)(15 53 17 29)(31 42 36 71)(32 77 37 48)(33 44 38 73)(34 79 39 50)(35 46 40 75)(41 64 80 69)(43 66 72 61)(45 68 74 63)(47 70 76 65)(49 62 78 67)
(1 40 16 64)(2 36 17 70)(3 32 18 66)(4 38 19 62)(5 34 20 68)(6 35 14 69)(7 31 15 65)(8 37 11 61)(9 33 12 67)(10 39 13 63)(21 78 60 73)(22 50 51 45)(23 80 52 75)(24 42 53 47)(25 72 54 77)(26 44 55 49)(27 74 56 79)(28 46 57 41)(29 76 58 71)(30 48 59 43)
(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 19)(12 18)(13 17)(14 16)(15 20)(21 30)(22 29)(23 28)(24 27)(25 26)(31 63)(32 62)(33 61)(34 70)(35 69)(36 68)(37 67)(38 66)(39 65)(40 64)(42 50)(43 49)(44 48)(45 47)(51 58)(52 57)(53 56)(54 55)(59 60)(71 79)(72 78)(73 77)(74 76)

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

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

G=PermutationGroup([(1,52,6,28),(2,58,7,24),(3,54,8,30),(4,60,9,26),(5,56,10,22),(11,59,18,25),(12,55,19,21),(13,51,20,27),(14,57,16,23),(15,53,17,29),(31,42,36,71),(32,77,37,48),(33,44,38,73),(34,79,39,50),(35,46,40,75),(41,64,80,69),(43,66,72,61),(45,68,74,63),(47,70,76,65),(49,62,78,67)], [(1,40,16,64),(2,36,17,70),(3,32,18,66),(4,38,19,62),(5,34,20,68),(6,35,14,69),(7,31,15,65),(8,37,11,61),(9,33,12,67),(10,39,13,63),(21,78,60,73),(22,50,51,45),(23,80,52,75),(24,42,53,47),(25,72,54,77),(26,44,55,49),(27,74,56,79),(28,46,57,41),(29,76,58,71),(30,48,59,43)], [(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),(11,19),(12,18),(13,17),(14,16),(15,20),(21,30),(22,29),(23,28),(24,27),(25,26),(31,63),(32,62),(33,61),(34,70),(35,69),(36,68),(37,67),(38,66),(39,65),(40,64),(42,50),(43,49),(44,48),(45,47),(51,58),(52,57),(53,56),(54,55),(59,60),(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+4D48D10
kernelC4225D10C204D4C422D5C22⋊D20D10⋊D4D10.13D4C4⋊D20C5×C422C2C422C2C42C22⋊C4C4⋊C4C10C2
# reps111333312266312

Matrix representation of C4225D10 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] >;

C4225D10 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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