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G = D40⋊C22order 320 = 26·5

3rd semidirect product of D40 and C22 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q164D10, SD166D10, D403C22, C40.5C23, C20.24C24, M4(2)⋊12D10, D20.17C23, Dic10.17C23, D40⋊C23C2, C8⋊D103C2, D4⋊D57C22, (C2×Q8)⋊23D10, C4.192(D4×D5), (C8×D5)⋊5C22, Q8⋊D56C22, C8.C226D5, Q8.D101C2, C8.5(C22×D5), Q16⋊D52C2, C4○D4.30D10, (C4×D5).101D4, D10.89(C2×D4), C20.245(C2×D4), (D4×D5)⋊10C22, C40⋊C26C22, C8⋊D56C22, (D5×M4(2))⋊4C2, D4⋊D1010C2, (C5×Q16)⋊2C22, (Q8×D5)⋊12C22, C5⋊Q165C22, C4.24(C23×D5), C22.49(D4×D5), SD163D53C2, (C2×D20)⋊37C22, C54(D8⋊C22), C52C8.12C23, (Q8×C10)⋊21C22, (C5×SD16)⋊6C22, (C5×D4).17C23, (C22×D5).52D4, D4.17(C22×D5), (C4×D5).67C23, Q8.17(C22×D5), (C5×Q8).17C23, C20.C2310C2, D42D511C22, (C2×C20).115C23, (C2×Dic5).252D4, Dic5.102(C2×D4), Q82D511C22, C4○D20.31C22, C10.125(C22×D4), (C5×M4(2))⋊6C22, C4.Dic515C22, C2.98(C2×D4×D5), (D5×C4○D4)⋊5C2, (C2×C10).70(C2×D4), (C5×C8.C22)⋊2C2, (C2×Q82D5)⋊17C2, (C2×C4×D5).172C22, (C2×C4).99(C22×D5), (C5×C4○D4).26C22, SmallGroup(320,1449)

Series: Derived Chief Lower central Upper central

C1C20 — D40⋊C22
C1C5C10C20C4×D5C2×C4×D5D5×C4○D4 — D40⋊C22
C5C10C20 — D40⋊C22
C1C2C2×C4C8.C22

Generators and relations for D40⋊C22
 G = < a,b,c,d | a40=b2=c2=d2=1, bab=a-1, cac=a21, dad=a29, cbc=a20b, dbd=a8b, cd=dc >

Subgroups: 1038 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×11], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4, D4 [×13], Q8, Q8 [×2], Q8 [×3], C23 [×3], D5 [×5], C10, C10 [×2], C2×C8 [×2], M4(2), M4(2) [×3], D8 [×4], SD16 [×2], SD16 [×6], Q16 [×2], Q16 [×2], C22×C4 [×3], C2×D4 [×4], C2×Q8, C2×Q8, C4○D4, C4○D4 [×11], Dic5 [×2], Dic5, C20 [×2], C20 [×3], D10 [×2], D10 [×8], C2×C10, C2×C10, C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22, C8.C22 [×3], C2×C4○D4 [×2], C52C8 [×2], C40 [×2], Dic10, Dic10, C4×D5 [×4], C4×D5 [×7], D20, D20 [×2], D20 [×6], C2×Dic5, C2×Dic5, C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×2], C5×Q8, C22×D5, C22×D5 [×2], D8⋊C22, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40 [×2], C4.Dic5, D4⋊D5 [×2], Q8⋊D5 [×4], C5⋊Q16 [×2], C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20, C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q8×D5, Q82D5, Q82D5 [×4], Q82D5 [×2], Q8×C10, C5×C4○D4, D5×M4(2), C8⋊D10, D40⋊C2 [×2], SD163D5 [×2], Q16⋊D5 [×2], Q8.D10 [×2], C20.C23, D4⋊D10, C5×C8.C22, C2×Q82D5, D5×C4○D4, D40⋊C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D8⋊C22, D4×D5 [×2], C23×D5, C2×D4×D5, D40⋊C22

Smallest permutation representation of D40⋊C22
On 80 points
Generators in S80
(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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(77 80)(78 79)
(1 21)(3 23)(5 25)(7 27)(9 29)(11 31)(13 33)(15 35)(17 37)(19 39)(41 61)(43 63)(45 65)(47 67)(49 69)(51 71)(53 73)(55 75)(57 77)(59 79)
(1 49)(2 78)(3 67)(4 56)(5 45)(6 74)(7 63)(8 52)(9 41)(10 70)(11 59)(12 48)(13 77)(14 66)(15 55)(16 44)(17 73)(18 62)(19 51)(20 80)(21 69)(22 58)(23 47)(24 76)(25 65)(26 54)(27 43)(28 72)(29 61)(30 50)(31 79)(32 68)(33 57)(34 46)(35 75)(36 64)(37 53)(38 42)(39 71)(40 60)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A10B10C10D10E10F20A20B20C20D20E···20J40A40B40C40D
order1222222224444444445588881010101010102020202020···2040404040
size11241010202020224445510202244202022448844448···88888

44 irreducible representations

dim1111111111112222222224448
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10D10D10D8⋊C22D4×D5D4×D5D40⋊C22
kernelD40⋊C22D5×M4(2)C8⋊D10D40⋊C2SD163D5Q16⋊D5Q8.D10C20.C23D4⋊D10C5×C8.C22C2×Q82D5D5×C4○D4C4×D5C2×Dic5C22×D5C8.C22M4(2)SD16Q16C2×Q8C4○D4C5C4C22C1
# reps1112222111112112244222222

Matrix representation of D40⋊C22 in GL8(𝔽41)

28130270000
28191400000
321922280000
03213130000
000033250
000022339
000011360
000009230
,
28130270000
221327160000
22919180000
9028220000
00003838169
00003939832
0000404050
00002332310
,
10000000
01000000
00100000
00010000
000040060
000004040
00000010
00000001
,
30028220000
361122230000
34375360000
372030360000
0000992338
00002332839
000000940
0000003932

G:=sub<GL(8,GF(41))| [28,28,32,0,0,0,0,0,13,19,19,32,0,0,0,0,0,14,22,13,0,0,0,0,27,0,28,13,0,0,0,0,0,0,0,0,3,2,1,0,0,0,0,0,3,2,1,9,0,0,0,0,25,33,36,23,0,0,0,0,0,9,0,0],[28,22,22,9,0,0,0,0,13,13,9,0,0,0,0,0,0,27,19,28,0,0,0,0,27,16,18,22,0,0,0,0,0,0,0,0,38,39,40,23,0,0,0,0,38,39,40,32,0,0,0,0,16,8,5,31,0,0,0,0,9,32,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,6,4,1,0,0,0,0,0,0,0,0,1],[30,36,34,37,0,0,0,0,0,11,37,20,0,0,0,0,28,22,5,30,0,0,0,0,22,23,36,36,0,0,0,0,0,0,0,0,9,23,0,0,0,0,0,0,9,32,0,0,0,0,0,0,23,8,9,39,0,0,0,0,38,39,40,32] >;

D40⋊C22 in GAP, Magma, Sage, TeX

D_{40}\rtimes C_2^2
% in TeX

G:=Group("D40:C2^2");
// GroupNames label

G:=SmallGroup(320,1449);
// by ID

G=gap.SmallGroup(320,1449);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,185,136,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^40=b^2=c^2=d^2=1,b*a*b=a^-1,c*a*c=a^21,d*a*d=a^29,c*b*c=a^20*b,d*b*d=a^8*b,c*d=d*c>;
// generators/relations

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