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G = C4216D10order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4216D10, C10.182+ 1+4, C4⋊C449D10, (C4×D4)⋊18D5, (D4×C20)⋊20C2, (C22×C4)⋊5D10, (C4×C20)⋊32C22, D10⋊Q88C2, C22⋊C448D10, C4⋊Dic59C22, (C2×D4).217D10, C23⋊D10.5C2, C422D516C2, C42⋊D532C2, D10.31(C4○D4), D10.12D47C2, C20.48D411C2, C23.D59C22, (C2×C10).100C24, (C2×C20).699C23, (C22×C20)⋊37C22, Dic5.5D47C2, (C2×Dic10)⋊6C22, (C4×Dic5)⋊52C22, C2.19(D46D10), C53(C22.45C24), (D4×C10).307C22, C22.12(C4○D20), C10.D442C22, (C23×D5).41C22, (C22×D5).35C23, C23.174(C22×D5), C22.125(C23×D5), D10⋊C4.85C22, C23.11D1029C2, C23.23D1016C2, C23.18D1018C2, (C22×C10).170C23, (C2×Dic5).217C23, (C22×Dic5).98C22, C4⋊C4⋊D57C2, (C4×C5⋊D4)⋊43C2, C2.23(D5×C4○D4), (C5×C4⋊C4)⋊61C22, (D5×C22⋊C4)⋊29C2, C2.49(C2×C4○D20), C10.140(C2×C4○D4), (C2×D10⋊C4)⋊22C2, (C2×C4×D5).252C22, (C2×C10).16(C4○D4), (C5×C22⋊C4)⋊57C22, (C2×C4).284(C22×D5), (C2×C5⋊D4).16C22, SmallGroup(320,1228)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4216D10
C1C5C10C2×C10C22×D5C23×D5D5×C22⋊C4 — C4216D10
C5C2×C10 — C4216D10
C1C22C4×D4

Generators and relations for C4216D10
 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 [×3], C2 [×6], C4 [×11], C22, C22 [×2], C22 [×16], C5, C2×C4 [×5], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], D5 [×3], C10 [×3], C10 [×3], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×7], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic5 [×6], C20 [×5], D10 [×2], D10 [×9], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4, C4×D4, C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C422C2 [×2], Dic10, C4×D5 [×3], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×3], C2×C20 [×5], C2×C20 [×2], C5×D4 [×2], C22×D5 [×2], C22×D5 [×5], C22×C10 [×2], C22.45C24, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5, D10⋊C4 [×8], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×2], C22×Dic5, C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C23×D5, C42⋊D5, C422D5, C23.11D10, D5×C22⋊C4, D10.12D4, Dic5.5D4, D10⋊Q8, C4⋊C4⋊D5, C20.48D4, C2×D10⋊C4, C4×C5⋊D4, C23.23D10, C23.18D10, C23⋊D10, D4×C20, C4216D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.45C24, C4○D20 [×2], C23×D5, C2×C4○D20, D46D10, D5×C4○D4, C4216D10

Smallest permutation representation of C4216D10
On 80 points
Generators in S80
(1 56 20 46)(2 52 16 42)(3 58 17 48)(4 54 18 44)(5 60 19 50)(6 49 11 59)(7 45 12 55)(8 41 13 51)(9 47 14 57)(10 43 15 53)(21 78 36 62)(22 74 37 68)(23 80 38 64)(24 76 39 70)(25 72 40 66)(26 67 31 73)(27 63 32 79)(28 69 33 75)(29 65 34 71)(30 61 35 77)
(1 28 8 23)(2 29 9 24)(3 30 10 25)(4 26 6 21)(5 27 7 22)(11 36 18 31)(12 37 19 32)(13 38 20 33)(14 39 16 34)(15 40 17 35)(41 80 56 69)(42 71 57 70)(43 72 58 61)(44 73 59 62)(45 74 60 63)(46 75 51 64)(47 76 52 65)(48 77 53 66)(49 78 54 67)(50 79 55 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 5)(2 4)(6 9)(7 8)(11 14)(12 13)(16 18)(19 20)(21 39)(22 38)(23 37)(24 36)(25 40)(26 34)(27 33)(28 32)(29 31)(30 35)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 72)(62 71)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)

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

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

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

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+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2D5C4○D4C4○D4D10D10D10D10D10C4○D202+ 1+4D46D10D5×C4○D4
kernelC4216D10C42⋊D5C422D5C23.11D10D5×C22⋊C4D10.12D4Dic5.5D4D10⋊Q8C4⋊C4⋊D5C20.48D4C2×D10⋊C4C4×C5⋊D4C23.23D10C23.18D10C23⋊D10D4×C20C4×D4D10C2×C10C42C22⋊C4C4⋊C4C22×C4C2×D4C22C10C2C2
# reps11111111111111112442424216144

Matrix representation of C4216D10 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] >;

C4216D10 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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