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G = D20.33C23order 320 = 26·5

14th non-split extension by D20 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.38C24, D20.33C23, 2+ 1+44D5, Dic10.33C23, (C5×D4).37D4, C56(D4○SD16), (C5×Q8).37D4, D4⋊D521C22, C4○D4.16D10, C20.270(C2×D4), Q8⋊D522C22, C4.38(C23×D5), (C2×D4).118D10, D4.8D109C2, C4○D2011C22, D4.19(C5⋊D4), D4.Dic511C2, C52C8.17C23, D4.D521C22, Q8.19(C5⋊D4), C5⋊Q1618C22, D4.26(C22×D5), (C5×D4).26C23, D4.10D109C2, D4.9D1011C2, D4.D1012C2, Q8.26(C22×D5), (C5×Q8).26C23, (C2×C20).119C23, C10.172(C22×D4), C4.Dic517C22, (C5×2+ 1+4)⋊3C2, (C2×Dic10)⋊43C22, (D4×C10).169C22, C4.76(C2×C5⋊D4), (C2×D4.D5)⋊32C2, (C2×C10).86(C2×D4), C22.7(C2×C5⋊D4), (C2×C52C8)⋊25C22, C2.45(C22×C5⋊D4), (C5×C4○D4).29C22, (C2×C4).103(C22×D5), SmallGroup(320,1508)

Series: Derived Chief Lower central Upper central

C1C20 — D20.33C23
C1C5C10C20D20C4○D20D4.10D10 — D20.33C23
C5C10C20 — D20.33C23
C1C2C4○D42+ 1+4

Generators and relations for D20.33C23
 G = < a,b,c,d,e | a20=b2=1, c2=d2=e2=a10, bab=a-1, ac=ca, ad=da, eae-1=a11, bc=cb, bd=db, ebe-1=a15b, dcd-1=a10c, ce=ec, de=ed >

Subgroups: 790 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×7], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], C5, C8 [×4], C2×C4 [×3], C2×C4 [×9], D4 [×6], D4 [×10], Q8 [×2], Q8 [×6], C23 [×3], D5, C10, C10 [×6], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×3], C2×D4 [×3], C2×Q8 [×4], C4○D4, C4○D4 [×3], C4○D4 [×7], Dic5 [×3], C20, C20 [×3], C20, D10, C2×C10 [×3], C2×C10 [×6], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ 1+4, 2- 1+4, C52C8, C52C8 [×3], Dic10 [×3], Dic10 [×3], C4×D5 [×3], D20, C2×Dic5 [×3], C5⋊D4 [×3], C2×C20 [×3], C2×C20 [×3], C5×D4 [×6], C5×D4 [×6], C5×Q8 [×2], C22×C10 [×3], D4○SD16, C2×C52C8 [×3], C4.Dic5 [×3], D4⋊D5 [×3], D4.D5 [×9], Q8⋊D5, C5⋊Q16 [×3], C2×Dic10 [×3], C4○D20 [×3], D42D5 [×3], Q8×D5, D4×C10 [×3], D4×C10 [×3], C5×C4○D4, C5×C4○D4 [×3], C5×C4○D4, D4.D10 [×3], C2×D4.D5 [×3], D4.Dic5, D4.8D10 [×3], D4.9D10 [×3], D4.10D10, C5×2+ 1+4, D20.33C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C5⋊D4 [×4], C22×D5 [×7], D4○SD16, C2×C5⋊D4 [×6], C23×D5, C22×C5⋊D4, D20.33C23

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E10A10B10C···10T20A···20L
order122222222444444445588888101010···1020···20
size112224442022224202020221010202020224···44···4

56 irreducible representations

dim11111111222222248
type+++++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D5D10D10C5⋊D4C5⋊D4D4○SD16D20.33C23
kernelD20.33C23D4.D10C2×D4.D5D4.Dic5D4.8D10D4.9D10D4.10D10C5×2+ 1+4C5×D4C5×Q82+ 1+4C2×D4C4○D4D4Q8C5C1
# reps133133113126812422

Matrix representation of D20.33C23 in GL6(𝔽41)

2300000
35250000
000100
0040000
0000040
000010
,
1490000
33270000
00001526
00001515
00262600
00152600
,
100000
010000
0004000
001000
0000040
000010
,
4000000
0400000
0000400
000001
001000
0004000
,
100000
6400000
000010
000001
0040000
0004000

G:=sub<GL(6,GF(41))| [23,35,0,0,0,0,0,25,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[14,33,0,0,0,0,9,27,0,0,0,0,0,0,0,0,26,15,0,0,0,0,26,26,0,0,15,15,0,0,0,0,26,15,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,1,0,0],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;

D20.33C23 in GAP, Magma, Sage, TeX

D_{20}._{33}C_2^3
% in TeX

G:=Group("D20.33C2^3");
// GroupNames label

G:=SmallGroup(320,1508);
// by ID

G=gap.SmallGroup(320,1508);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,136,1684,235,102,12550]);
// Polycyclic

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

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