Copied to
clipboard

G = D4.12D20order 320 = 26·5

2nd non-split extension by D4 of D20 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.12D20, Q8.12D20, D4012C22, C40.11C23, C20.62C24, M4(2)⋊21D10, D20.25C23, Dic2021C22, Dic10.25C23, C8○D44D5, (C2×C8)⋊7D10, C51(D4○D8), (C2×D40)⋊15C2, (C5×D4).24D4, C20.74(C2×D4), C4.28(C2×D20), (C5×Q8).24D4, D48D104C2, C8⋊D1012C2, (C2×C40)⋊10C22, C4○D4.39D10, C4○D202C22, D407C212C2, C22.4(C2×D20), C8.53(C22×D5), C4.59(C23×D5), (C2×D20)⋊31C22, C40⋊C212C22, C10.29(C22×D4), C2.31(C22×D20), (C2×C20).516C23, (C5×M4(2))⋊23C22, (C5×C8○D4)⋊4C2, (C2×C10).9(C2×D4), (C5×C4○D4).46C22, (C2×C4).227(C22×D5), SmallGroup(320,1424)

Series: Derived Chief Lower central Upper central

C1C20 — D4.12D20
C1C5C10C20D20C2×D20D48D10 — D4.12D20
C5C10C20 — D4.12D20
C1C2C4○D4C8○D4

Generators and relations for D4.12D20
 G = < a,b,c,d | a4=b2=d2=1, c20=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c19 >

Subgroups: 1286 in 268 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2 [×9], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C5, C8, C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×18], Q8, Q8 [×2], C23 [×6], D5 [×6], C10, C10 [×3], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×12], C4○D4, C4○D4 [×8], Dic5 [×2], C20, C20 [×3], D10 [×12], C2×C10 [×3], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ 1+4 [×2], C40, C40 [×3], Dic10 [×2], C4×D5 [×6], D20 [×6], D20 [×6], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×6], D4○D8, C40⋊C2 [×6], D40 [×9], Dic20, C2×C40 [×3], C5×M4(2) [×3], C2×D20 [×6], C4○D20 [×6], D4×D5 [×6], Q82D5 [×2], C5×C4○D4, C2×D40 [×3], D407C2 [×3], C8⋊D10 [×6], C5×C8○D4, D48D10 [×2], D4.12D20
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, D20 [×4], C22×D5 [×7], D4○D8, C2×D20 [×6], C23×D5, C22×D20, D4.12D20

Smallest permutation representation of D4.12D20
On 80 points
Generators in S80
(1 57 21 77)(2 58 22 78)(3 59 23 79)(4 60 24 80)(5 61 25 41)(6 62 26 42)(7 63 27 43)(8 64 28 44)(9 65 29 45)(10 66 30 46)(11 67 31 47)(12 68 32 48)(13 69 33 49)(14 70 34 50)(15 71 35 51)(16 72 36 52)(17 73 37 53)(18 74 38 54)(19 75 39 55)(20 76 40 56)
(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 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 72)(42 71)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)(73 80)(74 79)(75 78)(76 77)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E···2J4A4B4C4D4E4F5A5B8A8B8C8D8E10A10B10C···10H20A20B20C20D20E···20J40A···40H40I···40T
order122222···24444445588888101010···102020202020···2040···4040···40
size1122220···20222220202222444224···422224···42···24···4

62 irreducible representations

dim1111112222222244
type++++++++++++++++
imageC1C2C2C2C2C2D4D4D5D10D10D10D20D20D4○D8D4.12D20
kernelD4.12D20C2×D40D407C2C8⋊D10C5×C8○D4D48D10C5×D4C5×Q8C8○D4C2×C8M4(2)C4○D4D4Q8C5C1
# reps13361231266212428

Matrix representation of D4.12D20 in GL6(𝔽41)

100000
010000
0010200
003802740
0040400
00231360
,
4000000
0400000
001000
000100
0040400
00200040
,
0400000
1350000
00401300
0081800
0037263526
002371223
,
010000
100000
00401300
000100
0037263526
00217166

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,38,4,23,0,0,0,0,0,1,0,0,20,27,40,36,0,0,0,40,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,4,20,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,35,0,0,0,0,0,0,40,8,37,23,0,0,13,18,26,7,0,0,0,0,35,12,0,0,0,0,26,23],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,37,21,0,0,13,1,26,7,0,0,0,0,35,16,0,0,0,0,26,6] >;

D4.12D20 in GAP, Magma, Sage, TeX

D_4._{12}D_{20}
% in TeX

G:=Group("D4.12D20");
// GroupNames label

G:=SmallGroup(320,1424);
// by ID

G=gap.SmallGroup(320,1424);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,192,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^20=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c^19>;
// generators/relations

׿
×
𝔽