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G = D20.31D4order 320 = 26·5

1st non-split extension by D20 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.31D4, C23.36D20, (C2×C8)⋊15D10, C22⋊C88D5, (C2×C10)⋊1SD16, C4.119(D4×D5), (C2×C20).42D4, (C2×C4).31D20, D205C48C2, (C2×C40)⋊14C22, C10.7C22≀C2, C20.331(C2×D4), C51(C22⋊SD16), C4⋊Dic51C22, C10.7(C2×SD16), C20.48D41C2, C10.8(C8⋊C22), C223(C40⋊C2), (C22×D20).3C2, (C22×C4).81D10, (C22×C10).51D4, C2.11(C8⋊D10), (C2×C20).741C23, (C2×Dic10)⋊1C22, C22.104(C2×D20), C2.10(C22⋊D20), (C2×D20).197C22, (C22×C20).50C22, (C2×C40⋊C2)⋊9C2, (C5×C22⋊C8)⋊10C2, C2.10(C2×C40⋊C2), (C2×C10).124(C2×D4), (C2×C4).686(C22×D5), SmallGroup(320,358)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.31D4
C1C5C10C20C2×C20C2×D20C22×D20 — D20.31D4
C5C10C2×C20 — D20.31D4
C1C22C22×C4C22⋊C8

Generators and relations for D20.31D4
 G = < a,b,c,d | a20=b2=d2=1, c4=a10, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd=a15c3 >

Subgroups: 1070 in 188 conjugacy classes, 47 normal (25 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×18], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], D4 [×10], Q8 [×2], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4 [×7], C2×Q8, C24, Dic5 [×2], C20 [×2], C20, D10 [×16], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, D4⋊C4 [×2], C22⋊Q8, C2×SD16 [×2], C22×D4, C40 [×2], Dic10 [×2], D20 [×4], D20 [×6], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5 [×10], C22×C10, C22⋊SD16, C40⋊C2 [×4], C10.D4, C4⋊Dic5, C23.D5, C2×C40 [×2], C2×Dic10, C2×D20 [×2], C2×D20 [×5], C22×C20, C23×D5, D205C4 [×2], C5×C22⋊C8, C2×C40⋊C2 [×2], C20.48D4, C22×D20, D20.31D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×SD16, C8⋊C22, D20 [×2], C22×D5, C22⋊SD16, C40⋊C2 [×2], C2×D20, D4×D5 [×2], C22⋊D20, C2×C40⋊C2, C8⋊D10, D20.31D4

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

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222222224444455888810···101010101020···202020202040···40
size1111222020202022440402244442···244442···244444···4

59 irreducible representations

dim1111112222222222444
type+++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5SD16D10D10D20D20C40⋊C2C8⋊C22D4×D5C8⋊D10
kernelD20.31D4D205C4C5×C22⋊C8C2×C40⋊C2C20.48D4C22×D20D20C2×C20C22×C10C22⋊C8C2×C10C2×C8C22×C4C2×C4C23C22C10C4C2
# reps12121141124424416144

Matrix representation of D20.31D4 in GL6(𝔽41)

1050000
29310000
0035100
0054000
0000400
0000040
,
1050000
13310000
00404000
000100
0000400
0000241
,
2970000
1610000
0040000
0004000
00002236
00003119
,
100000
010000
001000
000100
000010
00001740

G:=sub<GL(6,GF(41))| [10,29,0,0,0,0,5,31,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[10,13,0,0,0,0,5,31,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,40,24,0,0,0,0,0,1],[29,16,0,0,0,0,7,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,22,31,0,0,0,0,36,19],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,40] >;

D20.31D4 in GAP, Magma, Sage, TeX

D_{20}._{31}D_4
% in TeX

G:=Group("D20.31D4");
// GroupNames label

G:=SmallGroup(320,358);
// by ID

G=gap.SmallGroup(320,358);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,58,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=d^2=1,c^4=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d=a^15*c^3>;
// generators/relations

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