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## G = D20.3D4order 320 = 26·5

### 3rd non-split extension by D20 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.3D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D20.2C4 — D20.3D4
 Lower central C5 — C10 — C2×C20 — D20.3D4
 Upper central C1 — C2 — C2×C4 — C4.D4

Generators and relations for D20.3D4
G = < a,b,c,d | a20=b2=1, c4=a10, d2=a15, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, dbd-1=a15b, dcd-1=a5c3 >

Subgroups: 510 in 108 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C4.D4, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C52C8, C52C8, C40, Dic10, C4×D5, D20, D20, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, D4.4D4, C8×D5, C8⋊D5, C40⋊C2, D40, C2×C52C8, C4.Dic5, D4⋊D5, D4.D5, C5×M4(2), C2×D20, C4○D20, D4×C10, C20.53D4, C20.46D4, C5×C4.D4, D20.2C4, C8⋊D10, C2×D4⋊D5, D4.D10, D20.3D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C22×D5, D4.4D4, C4○D20, D4×D5, D10⋊D4, D20.3D4

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

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 5A 5B 8A 8B 8C 8D 8E 8F 8G 10A 10B 10C 10D 10E 10F 10G 10H 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 4 4 4 5 5 8 8 8 8 8 8 8 10 10 10 10 10 10 10 10 20 20 20 20 40 ··· 40 size 1 1 2 8 20 40 2 2 20 2 2 4 4 8 10 10 20 40 2 2 4 4 8 8 8 8 4 4 4 4 8 ··· 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 8 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D10 C4○D20 D4.4D4 D4×D5 D20.3D4 kernel D20.3D4 C20.53D4 C20.46D4 C5×C4.D4 D20.2C4 C8⋊D10 C2×D4⋊D5 D4.D10 C5⋊2C8 Dic10 D20 C4.D4 C2×C10 M4(2) C2×D4 C22 C5 C4 C1 # reps 1 1 1 1 1 1 1 1 2 1 1 2 2 4 2 8 2 4 2

Matrix representation of D20.3D4 in GL6(𝔽41)

 34 1 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 4 0 40 0 0 37 0 1 0
,
 1 34 0 0 0 0 0 40 0 0 0 0 0 0 29 29 17 17 0 0 15 26 24 17 0 0 35 40 15 12 0 0 13 35 26 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 36 39 0 0 0 36 4 0 39 0 0 0 0 37 5 0 0 1 0 5 37
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 15 26 24 17 0 0 29 29 17 17 0 0 13 35 26 12 0 0 35 40 15 12

G:=sub<GL(6,GF(41))| [34,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,37,0,0,1,0,4,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,0,0,0,0,0,34,40,0,0,0,0,0,0,29,15,35,13,0,0,29,26,40,35,0,0,17,24,15,26,0,0,17,17,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,36,0,1,0,0,36,4,0,0,0,0,39,0,37,5,0,0,0,39,5,37],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,29,13,35,0,0,26,29,35,40,0,0,24,17,26,15,0,0,17,17,12,12] >;

D20.3D4 in GAP, Magma, Sage, TeX

D_{20}._3D_4
% in TeX

G:=Group("D20.3D4");
// GroupNames label

G:=SmallGroup(320,376);
// by ID

G=gap.SmallGroup(320,376);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,297,136,1684,851,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^15,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^10*b,d*b*d^-1=a^15*b,d*c*d^-1=a^5*c^3>;
// generators/relations

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