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G = D20.37C23order 320 = 26·5

18th non-split extension by D20 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.16C25, C20.51C24, D10.9C24, D20.37C23, 2+ 1+45D5, Dic5.11C24, Dic10.38C23, C4○D411D10, (C2×D4)⋊32D10, (D4×D5)⋊14C22, (C2×C10).7C24, D46D1010C2, (Q8×D5)⋊16C22, C4.48(C23×D5), C2.17(D5×C24), C5⋊D4.3C23, C4○D2013C22, (D4×C10)⋊26C22, C52(C2.C25), D4.31(C22×D5), (C5×D4).31C23, (C4×D5).20C23, Q8.32(C22×D5), (C5×Q8).32C23, D42D516C22, C22.4(C23×D5), (C2×C20).122C23, Q82D519C22, (C5×2+ 1+4)⋊5C2, D4.10D1011C2, C23.71(C22×D5), (C2×Dic10)⋊44C22, (C22×C10).79C23, (C2×Dic5).169C23, (C22×Dic5)⋊39C22, (C22×D5).143C23, (D5×C4○D4)⋊8C2, (C2×C4×D5)⋊37C22, (C2×D42D5)⋊31C2, (C5×C4○D4)⋊11C22, (C2×C5⋊D4)⋊33C22, (C2×C4).106(C22×D5), SmallGroup(320,1623)

Series: Derived Chief Lower central Upper central

C1C10 — D20.37C23
C1C5C10D10C22×D5C2×C4×D5D5×C4○D4 — D20.37C23
C5C10 — D20.37C23
C1C22+ 1+4

Generators and relations for D20.37C23
 G = < a,b,c,d,e | a20=b2=c2=d2=1, e2=a10, bab=a-1, ac=ca, ad=da, eae-1=a9, cbc=dbd=a10b, ebe-1=a18b, dcd=ece-1=a10c, ede-1=a10d >

Subgroups: 2350 in 810 conjugacy classes, 443 normal (8 characteristic)
C1, C2, C2 [×15], C4 [×6], C4 [×10], C22 [×9], C22 [×21], C5, C2×C4 [×9], C2×C4 [×51], D4 [×18], D4 [×42], Q8 [×2], Q8 [×18], C23 [×6], C23 [×9], D5 [×6], C10, C10 [×9], C22×C4 [×15], C2×D4 [×9], C2×D4 [×36], C2×Q8 [×15], C4○D4 [×6], C4○D4 [×74], Dic5, Dic5 [×9], C20 [×6], D10 [×6], D10 [×9], C2×C10 [×9], C2×C10 [×6], C2×C4○D4 [×15], 2+ 1+4, 2+ 1+4 [×9], 2- 1+4 [×6], Dic10 [×18], C4×D5 [×24], D20 [×6], C2×Dic5 [×27], C5⋊D4 [×36], C2×C20 [×9], C5×D4 [×18], C5×Q8 [×2], C22×D5 [×9], C22×C10 [×6], C2.C25, C2×Dic10 [×9], C2×C4×D5 [×9], C4○D20 [×18], D4×D5 [×18], D42D5 [×54], Q8×D5 [×6], Q82D5 [×2], C22×Dic5 [×6], C2×C5⋊D4 [×18], D4×C10 [×9], C5×C4○D4 [×6], C2×D42D5 [×9], D46D10 [×9], D5×C4○D4 [×6], D4.10D10 [×6], C5×2+ 1+4, D20.37C23
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], D5, C24 [×31], D10 [×15], C25, C22×D5 [×35], C2.C25, C23×D5 [×15], D5×C24, D20.37C23

Smallest permutation representation of D20.37C23
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 48)(42 47)(43 46)(44 45)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 61)(19 62)(20 63)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(40 53)
(1 64 11 74)(2 73 12 63)(3 62 13 72)(4 71 14 61)(5 80 15 70)(6 69 16 79)(7 78 17 68)(8 67 18 77)(9 76 19 66)(10 65 20 75)(21 46 31 56)(22 55 32 45)(23 44 33 54)(24 53 34 43)(25 42 35 52)(26 51 36 41)(27 60 37 50)(28 49 38 59)(29 58 39 48)(30 47 40 57)

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

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

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

68 conjugacy classes

class 1 2A2B···2J2K···2P4A···4F4G4H4I···4Q5A5B10A10B10C···10T20A···20L
order122···22···24···4444···455101010···1020···20
size112···210···102···25510···1022224···44···4

68 irreducible representations

dim11111122248
type+++++++++-
imageC1C2C2C2C2C2D5D10D10C2.C25D20.37C23
kernelD20.37C23C2×D42D5D46D10D5×C4○D4D4.10D10C5×2+ 1+42+ 1+4C2×D4C4○D4C5C1
# reps1996612181222

Matrix representation of D20.37C23 in GL6(𝔽41)

4010000
5350000
00104040
000001
00214040
0004000
,
4000000
510000
00404001
0000040
00394011
0004000
,
4000000
0400000
00323290
000009
00233299
0003200
,
4000000
0400000
0040000
00214040
0000040
0000400
,
610000
6350000
00404001
00214040
000001
0000400

G:=sub<GL(6,GF(41))| [40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,0,2,0,0,0,0,0,1,40,0,0,40,0,40,0,0,0,40,1,40,0],[40,5,0,0,0,0,0,1,0,0,0,0,0,0,40,0,39,0,0,0,40,0,40,40,0,0,0,0,1,0,0,0,1,40,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,23,0,0,0,32,0,32,32,0,0,9,0,9,0,0,0,0,9,9,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,2,0,0,0,0,0,1,0,0,0,0,0,40,0,40,0,0,0,40,40,0],[6,6,0,0,0,0,1,35,0,0,0,0,0,0,40,2,0,0,0,0,40,1,0,0,0,0,0,40,0,40,0,0,1,40,1,0] >;

D20.37C23 in GAP, Magma, Sage, TeX

D_{20}._{37}C_2^3
% in TeX

G:=Group("D20.37C2^3");
// GroupNames label

G:=SmallGroup(320,1623);
// by ID

G=gap.SmallGroup(320,1623);
# by ID

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

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

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