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G = D20.39D4order 320 = 26·5

9th non-split extension by D20 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.39D4, M4(2)⋊6D10, Dic10.39D4, (C2×Q8)⋊4D10, C4○D4.7D10, (C5×D4).14D4, C4.106(D4×D5), (C5×Q8).14D4, C8.C221D5, C20.198(C2×D4), C54(D4.9D4), D207C411C2, (Q8×C10)⋊4C22, (C22×D5).6D4, C22.37(D4×D5), C10.65C22≀C2, D42Dic58C2, D48D10.2C2, C20.C235C2, C20.23D47C2, D4.11(C5⋊D4), (C2×C20).17C23, Q8.11(C5⋊D4), (C4×Dic5)⋊6C22, C4.Dic59C22, C20.46D411C2, C4○D20.25C22, C2.33(C23⋊D10), (C2×D20).134C22, (C5×M4(2))⋊16C22, C4.54(C2×C5⋊D4), (C2×C10).36(C2×D4), (C5×C8.C22)⋊5C2, (C2×C4).17(C22×D5), (C5×C4○D4).15C22, SmallGroup(320,829)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.39D4
C1C5C10C2×C10C2×C20C2×D20D48D10 — D20.39D4
C5C10C2×C20 — D20.39D4
C1C2C2×C4C8.C22

Generators and relations for D20.39D4
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a13b, dbd=a18b, dcd=a10c-1 >

Subgroups: 750 in 152 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×6], D4, D4 [×9], Q8, Q8 [×3], C23 [×4], D5 [×3], C10, C10 [×2], C42, C22⋊C4 [×2], M4(2), M4(2), SD16 [×2], Q16 [×2], C2×D4 [×5], C2×Q8, C4○D4, C4○D4 [×3], Dic5 [×2], C20 [×2], C20 [×2], D10 [×7], C2×C10, C2×C10, C4.D4, C4≀C2 [×2], C4.4D4, C8.C22, C8.C22, 2+ 1+4, C52C8, C40, Dic10, C4×D5 [×3], D20, D20 [×4], C2×Dic5, C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×2], C22×D5 [×2], C22×D5 [×2], D4.9D4, C4.Dic5, C4×Dic5, D10⋊C4 [×2], Q8⋊D5, C5⋊Q16, C5×M4(2), C5×SD16, C5×Q16, C2×D20, C2×D20, C4○D20, C4○D20, D4×D5 [×3], Q82D5, Q8×C10, C5×C4○D4, C20.46D4, D207C4, D42Dic5, C20.C23, C20.23D4, C5×C8.C22, D48D10, D20.39D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×2], C22×D5, D4.9D4, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, D20.39D4

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B8A8B10A10B10C10D10E10F20A20B20C20D20E···20J40A40B40C40D
order1222222444444455881010101010102020202020···2040404040
size112420202022482020202284022448844448···88888

38 irreducible representations

dim11111111222222222224448
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D4D5D10D10D10C5⋊D4C5⋊D4D4.9D4D4×D5D4×D5D20.39D4
kernelD20.39D4C20.46D4D207C4D42Dic5C20.C23C20.23D4C5×C8.C22D48D10Dic10D20C5×D4C5×Q8C22×D5C8.C22M4(2)C2×Q8C4○D4D4Q8C5C4C22C1
# reps11111111111122222442222

Matrix representation of D20.39D4 in GL6(𝔽41)

010000
40340000
0004009
0010320
000001
0000400
,
010000
100000
0004009
0010320
0001801
00230400
,
4000000
710000
00092121
003202021
0000320
0000032
,
4000000
710000
000100
001000
000001
000010

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,34,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,32,0,40,0,0,9,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,23,0,0,40,0,18,0,0,0,0,32,0,40,0,0,9,0,1,0],[40,7,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,9,0,0,0,0,0,21,20,32,0,0,0,21,21,0,32],[40,7,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D20.39D4 in GAP, Magma, Sage, TeX

D_{20}._{39}D_4
% in TeX

G:=Group("D20.39D4");
// GroupNames label

G:=SmallGroup(320,829);
// by ID

G=gap.SmallGroup(320,829);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,184,570,1684,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^13*b,d*b*d=a^18*b,d*c*d=a^10*c^-1>;
// generators/relations

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