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G = C4221D10order 320 = 26·5

21st semidirect product of C42 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4221D10, C10.732+ 1+4, (C2×Q8)⋊9D10, (C4×C20)⋊33C22, C22⋊C435D10, C4.4D413D5, D103Q831C2, (C2×D4).111D10, C23⋊D10.6C2, C42⋊D536C2, C4⋊Dic542C22, (Q8×C10)⋊15C22, D10.37(C4○D4), Dic54D432C2, (C2×C10).223C24, (C2×C20).632C23, (C4×Dic5)⋊57C22, D10.12D444C2, C23.D534C22, C2.76(D46D10), C23.45(C22×D5), C58(C22.45C24), (D4×C10).211C22, C23.D1040C2, C10.D467C22, (C22×C10).53C23, (C23×D5).66C22, C22.244(C23×D5), C23.18D1025C2, (C2×Dic5).265C23, (C22×Dic5)⋊28C22, (C22×D5).227C23, D10⋊C4.136C22, C2.79(D5×C4○D4), (D5×C22⋊C4)⋊19C2, C10.190(C2×C4○D4), (C5×C4.4D4)⋊15C2, (C2×C4×D5).266C22, (C2×C4).74(C22×D5), (C5×C22⋊C4)⋊31C22, (C2×C5⋊D4).61C22, SmallGroup(320,1351)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4221D10
C1C5C10C2×C10C22×D5C23×D5D5×C22⋊C4 — C4221D10
C5C2×C10 — C4221D10
C1C22C4.4D4

Generators and relations for C4221D10
 G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 950 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×11], C22, C22 [×18], C5, C2×C4, C2×C4 [×4], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], D5 [×4], C10, C10 [×2], C10 [×2], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×8], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic5 [×6], C20 [×5], D10 [×4], D10 [×8], C2×C10, C2×C10 [×6], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C422C2 [×2], C4×D5 [×6], C2×Dic5 [×6], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20 [×4], C5×D4, C5×Q8, C22×D5 [×2], C22×D5 [×5], C22×C10 [×2], C22.45C24, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5 [×2], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×C4×D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], D4×C10, Q8×C10, C23×D5, C42⋊D5 [×2], C23.D10 [×2], D5×C22⋊C4 [×2], Dic54D4 [×2], D10.12D4 [×2], C23.18D10, C23⋊D10, D103Q8 [×2], C5×C4.4D4, C4221D10
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, C23×D5, D46D10, D5×C4○D4 [×2], C4221D10

Smallest permutation representation of C4221D10
On 80 points
Generators in S80
(1 34 11 51)(2 40 12 57)(3 36 13 53)(4 32 14 59)(5 38 15 55)(6 39 16 56)(7 35 17 52)(8 31 18 58)(9 37 19 54)(10 33 20 60)(21 61 26 76)(22 72 27 67)(23 63 28 78)(24 74 29 69)(25 65 30 80)(41 79 46 64)(42 70 47 75)(43 71 48 66)(44 62 49 77)(45 73 50 68)
(1 41 6 29)(2 47 7 25)(3 43 8 21)(4 49 9 27)(5 45 10 23)(11 46 16 24)(12 42 17 30)(13 48 18 26)(14 44 19 22)(15 50 20 28)(31 61 36 71)(32 77 37 67)(33 63 38 73)(34 79 39 69)(35 65 40 75)(51 64 56 74)(52 80 57 70)(53 66 58 76)(54 72 59 62)(55 68 60 78)
(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 14)(2 13)(3 12)(4 11)(5 15)(6 19)(7 18)(8 17)(9 16)(10 20)(21 25)(22 24)(26 30)(27 29)(31 52)(32 51)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 49)(42 48)(43 47)(44 46)(61 65)(62 64)(66 70)(67 69)(71 75)(72 74)(76 80)(77 79)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O5A5B10A···10F10G10H10I10J20A···20L20M20N20O20P
order12222222224444444444444445510···101010101020···2020202020
size1111441010101022224441010101020202020222···288884···48888

53 irreducible representations

dim1111111111222222444
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D5C4○D4D10D10D10D102+ 1+4D46D10D5×C4○D4
kernelC4221D10C42⋊D5C23.D10D5×C22⋊C4Dic54D4D10.12D4C23.18D10C23⋊D10D103Q8C5×C4.4D4C4.4D4D10C42C22⋊C4C2×D4C2×Q8C10C2C2
# reps1222221121282822148

Matrix representation of C4221D10 in GL6(𝔽41)

010000
100000
001000
000100
0000320
0000032
,
900000
090000
001000
000100
000001
000010
,
100000
0400000
006700
0035000
000010
0000040
,
100000
010000
0035100
006600
0000400
000001

G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,6,0,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;

C4221D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{21}D_{10}
% in TeX

G:=Group("C4^2:21D10");
// GroupNames label

G:=SmallGroup(320,1351);
// by ID

G=gap.SmallGroup(320,1351);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,387,100,346,136,12550]);
// Polycyclic

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

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