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G = D20.38D4order 320 = 26·5

8th non-split extension by D20 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.38D4, Dic10.38D4, M4(2).14D10, C8⋊C223D5, C4○D4.6D10, (C5×D4).13D4, C4.105(D4×D5), (C5×Q8).13D4, (C2×D4).81D10, C20.197(C2×D4), C54(D4.8D4), (C2×Dic5).5D4, D207C410C2, C22.36(D4×D5), C10.64C22≀C2, D42Dic57C2, D4.D106C2, C20.17D47C2, D4.10(C5⋊D4), (C2×C20).16C23, Q8.10(C5⋊D4), D4.10D102C2, C4.12D2010C2, C4○D20.24C22, C2.32(C23⋊D10), (D4×C10).106C22, (C4×Dic5).64C22, C4.Dic5.26C22, (C5×M4(2)).24C22, (C2×Dic10).139C22, (C5×C8⋊C22)⋊7C2, C4.53(C2×C5⋊D4), (C2×C10).35(C2×D4), (C2×C4).16(C22×D5), (C5×C4○D4).14C22, SmallGroup(320,828)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.38D4
C1C5C10C20C2×C20C4○D20D4.10D10 — D20.38D4
C5C10C2×C20 — D20.38D4
C1C2C2×C4C8⋊C22

Generators and relations for D20.38D4
 G = < a,b,c,d | a20=b2=1, c4=d2=a10, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd-1=c3 >

Subgroups: 590 in 146 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×5], C22, C22 [×5], C5, C8 [×2], C2×C4, C2×C4 [×9], D4, D4 [×7], Q8, Q8 [×5], C23, D5, C10, C10 [×3], C42, C22⋊C4 [×2], M4(2), M4(2), D8 [×2], SD16 [×2], C2×D4, C2×Q8 [×3], C4○D4, C4○D4 [×5], Dic5 [×4], C20 [×2], C20, D10, C2×C10, C2×C10 [×4], C4.10D4, C4≀C2 [×2], C4.4D4, C8⋊C22, C8⋊C22, 2- 1+4, C52C8, C40, Dic10, Dic10 [×4], C4×D5 [×3], D20, C2×Dic5 [×2], C2×Dic5 [×3], C5⋊D4 [×3], C2×C20, C2×C20, C5×D4, C5×D4 [×3], C5×Q8, C22×C10, D4.8D4, C4.Dic5, C4×Dic5, D4⋊D5, D4.D5, C23.D5 [×2], C5×M4(2), C5×D8, C5×SD16, C2×Dic10, C2×Dic10, C4○D20, C4○D20, D42D5 [×3], Q8×D5, D4×C10, C5×C4○D4, C4.12D20, D207C4, D42Dic5, D4.D10, C20.17D4, C5×C8⋊C22, D4.10D10, D20.38D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×2], C22×D5, D4.8D4, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, D20.38D4

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D···4H5A5B8A8B10A10B10C10D10E···10J20A20B20C20D20E20F40A40B40C40D
order1222224444···455881010101010···1020202020202040404040
size112482022420···202284022448···84444888888

38 irreducible representations

dim11111111222222222224448
type+++++++++++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D4D4D4D5D10D10D10C5⋊D4C5⋊D4D4.8D4D4×D5D4×D5D20.38D4
kernelD20.38D4C4.12D20D207C4D42Dic5D4.D10C20.17D4C5×C8⋊C22D4.10D10Dic10D20C2×Dic5C5×D4C5×Q8C8⋊C22M4(2)C2×D4C4○D4D4Q8C5C4C22C1
# reps11111111112112222442222

Matrix representation of D20.38D4 in GL6(𝔽41)

2500000
0230000
0032000
005900
0000320
009099
,
0230000
2500000
000163223
0003205
0091600
0002509
,
100000
0400000
000204039
00040037
00322500
0093901
,
4000000
0400000
000010
00040037
0040000
0002101

G:=sub<GL(6,GF(41))| [25,0,0,0,0,0,0,23,0,0,0,0,0,0,32,5,0,9,0,0,0,9,0,0,0,0,0,0,32,9,0,0,0,0,0,9],[0,25,0,0,0,0,23,0,0,0,0,0,0,0,0,0,9,0,0,0,16,32,16,25,0,0,32,0,0,0,0,0,23,5,0,9],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,32,9,0,0,20,40,25,39,0,0,40,0,0,0,0,0,39,37,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,21,0,0,1,0,0,0,0,0,0,37,0,1] >;

D20.38D4 in GAP, Magma, Sage, TeX

D_{20}._{38}D_4
% in TeX

G:=Group("D20.38D4");
// GroupNames label

G:=SmallGroup(320,828);
// by ID

G=gap.SmallGroup(320,828);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,1123,297,136,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=1,c^4=d^2=a^10,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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