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G = D20⋊19D4  order 320 = 26·5

7th semidirect product of D20 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20⋊19D4, C10.1142+ (1+4), C5⋊3(D42), C4⋊3(D4×D5), C20⋊6(C2×D4), C5⋊D4⋊1D4, C4⋊C4⋊23D10, D10⋊7(C2×D4), C22⋊2(D4×D5), (C2×D4)⋊23D10, C4⋊D4⋊12D5, Dic5⋊4(C2×D4), C20⋊D4⋊17C2, C4⋊2D20⋊22C2, C22⋊C4⋊11D10, (C22×C4)⋊17D10, C22⋊D20⋊13C2, D10⋊D4⋊20C2, C23⋊D10⋊10C2, D20⋊8C4⋊20C2, (D4×C10)⋊13C22, (C2×D20)⋊46C22, (C22×D20)⋊15C2, (C2×C20).40C23, C10.68(C22×D4), (C2×C10).153C24, (C22×C20)⋊21C22, (C4×Dic5)⋊22C22, (C23×D5)⋊10C22, C23.D5⋊51C22, C2.28(D4⋊8D10), D10⋊C4⋊53C22, C10.D4⋊52C22, (C22×D5).64C23, C23.181(C22×D5), C22.174(C23×D5), (C22×C10).188C23, (C2×Dic5).237C23, (C2×D4×D5)⋊11C2, C2.41(C2×D4×D5), (C2×C10)⋊3(C2×D4), (C4×C5⋊D4)⋊16C2, (C2×C4×D5)⋊14C22, (C5×C4⋊D4)⋊15C2, (C5×C4⋊C4)⋊11C22, (C2×C5⋊D4)⋊15C22, (C5×C22⋊C4)⋊13C22, (C2×C4).176(C22×D5), SmallGroup(320,1281)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — D20⋊19D4
Chief series
C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — D20⋊19D4
Lower central
C5 — C2×C10 — D20⋊19D4

Subgroups: 2038 in 428 conjugacy classes, 115 normal (43 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×42], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×11], D4 [×34], C23, C23 [×2], C23 [×25], D5 [×8], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4, C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×29], C24 [×4], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×3], D10 [×6], D10 [×28], C2×C10, C2×C10 [×2], C2×C10 [×8], C4×D4 [×2], C22≀C2 [×4], C4⋊D4, C4⋊D4 [×3], C4⋊1D4, C22×D4 [×4], C4×D5 [×6], D20 [×4], D20 [×10], C2×Dic5 [×3], C5⋊D4 [×4], C5⋊D4 [×10], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×D5, C22×D5 [×4], C22×D5 [×20], C22×C10, C22×C10 [×2], D42, C4×Dic5, C10.D4, D10⋊C4, D10⋊C4 [×4], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×C4×D5 [×2], C2×D20 [×2], C2×D20 [×4], C2×D20 [×4], D4×D5 [×12], C2×C5⋊D4, C2×C5⋊D4 [×6], C22×C20, D4×C10, D4×C10 [×2], C23×D5 [×4], C22⋊D20 [×2], D10⋊D4 [×2], D20⋊8C4, C4⋊2D20, C4×C5⋊D4, C23⋊D10 [×2], C20⋊D4, C5×C4⋊D4, C22×D20, C2×D4×D5, C2×D4×D5 [×2], D20⋊19D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C24, D10 [×7], C22×D4 [×2], 2+ (1+4), C22×D5 [×7], D42, D4×D5 [×4], C23×D5, C2×D4×D5 [×2], D4⋊8D10, D20⋊19D4

Generators and relations
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=dad=a11, cbc-1=dbd=a10b, dcd=c-1 >

Smallest permutation representation
►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 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 60)(58 59)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)

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

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

Matrix representation ►G ⊆ GL6(𝔽41)

4000000
0400000
000100
0040600
000001
0000400
,
4000000
0400000
000100
001000
000001
000010
,
010000
4000000
0040000
0004000
0000400
000001
,
100000
0400000
001000
000100
0000400
000001

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;
 

53 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2M2N2O4A4B4C4D4E4F4G4H4I5A5B10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order122222222···2224444444445510···10101010101010101020···2020202020
size1111224410···1020202244410102020222···2444488884···48888

53 irreducible representations

dim1111111111122222224444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2D4D4D5D10D10D10D102+ (1+4)D4×D5D4×D5D4⋊8D10
kernelD20⋊19D4C22⋊D20D10⋊D4D20⋊8C4C4⋊2D20C4×C5⋊D4C23⋊D10C20⋊D4C5×C4⋊D4C22×D20C2×D4×D5D20C5⋊D4C4⋊D4C22⋊C4C4⋊C4C22×C4C2×D4C10C4C22C2
# reps1221112111344242261444

In GAP, Magma, Sage, TeX 

D_{20}\rtimes_{19}D_4
 
% in TeX
 
G:=Group("D20:19D4");
 
// GroupNames label
 
G:=SmallGroup(320,1281);
 
// by ID
 
G=gap.SmallGroup(320,1281);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,1571,297,192,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;
 
// generators/relations
 

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