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G = D20.32C23order 320 = 26·5

13rd non-split extension by D20 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.37C24, D20.32C23, 2+ 1+43D5, Dic10.32C23, C55(D4○D8), C4○D45D10, (C2×D4)⋊16D10, (C5×D4).36D4, (C5×Q8).36D4, D48D109C2, D4⋊D520C22, C20.269(C2×D4), Q8⋊D519C22, D4⋊D1011C2, C4.37(C23×D5), D4.8D108C2, D4.18(C5⋊D4), C4○D2010C22, (D4×C10)⋊24C22, (C2×D20)⋊39C22, D4.Dic510C2, C52C8.16C23, D4.D519C22, Q8.18(C5⋊D4), D4.25(C22×D5), C5⋊Q1621C22, (C5×D4).25C23, D4.D1011C2, (C5×Q8).25C23, Q8.25(C22×D5), (C2×C20).118C23, C10.171(C22×D4), C4.Dic516C22, (C5×2+ 1+4)⋊2C2, (C2×D4⋊D5)⋊32C2, C4.75(C2×C5⋊D4), (C2×C10).85(C2×D4), (C5×C4○D4)⋊8C22, C22.6(C2×C5⋊D4), (C2×C52C8)⋊24C22, C2.44(C22×C5⋊D4), (C2×C4).102(C22×D5), SmallGroup(320,1507)

Series: Derived Chief Lower central Upper central

C1C20 — D20.32C23
C1C5C10C20D20C2×D20D48D10 — D20.32C23
C5C10C20 — D20.32C23
C1C2C4○D42+ 1+4

Generators and relations for D20.32C23
 G = < a,b,c,d,e | a20=b2=c2=d2=e2=1, bab=dad=a-1, ac=ca, eae=a11, cbc=a10b, dbd=a18b, ebe=a15b, cd=dc, ce=ec, ede=a5d >

Subgroups: 982 in 268 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×9], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C5, C8 [×4], C2×C4 [×3], C2×C4 [×6], D4 [×6], D4 [×15], Q8 [×2], Q8, C23 [×6], D5 [×3], C10, C10 [×6], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×3], C2×D4 [×9], C4○D4, C4○D4 [×3], C4○D4 [×5], Dic5, C20, C20 [×3], C20, D10 [×6], C2×C10 [×3], C2×C10 [×6], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ 1+4, 2+ 1+4, C52C8, C52C8 [×3], Dic10, C4×D5 [×3], D20 [×3], D20 [×3], C5⋊D4 [×3], C2×C20 [×3], C2×C20 [×3], C5×D4 [×6], C5×D4 [×6], C5×Q8 [×2], C22×D5 [×3], C22×C10 [×3], D4○D8, C2×C52C8 [×3], C4.Dic5 [×3], D4⋊D5 [×9], D4.D5 [×3], Q8⋊D5 [×3], C5⋊Q16, C2×D20 [×3], C4○D20 [×3], D4×D5 [×3], Q82D5, D4×C10 [×3], D4×C10 [×3], C5×C4○D4, C5×C4○D4 [×3], C5×C4○D4, C2×D4⋊D5 [×3], D4.D10 [×3], D4.Dic5, D4⋊D10 [×3], D4.8D10 [×3], D48D10, C5×2+ 1+4, D20.32C23
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○D8, C2×C5⋊D4 [×6], C23×D5, C22×C5⋊D4, D20.32C23

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

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F5A5B8A8B8C8D8E10A10B10C···10T20A···20L
order122222222224444445588888101010···1020···20
size112224442020202222420221010202020224···44···4

56 irreducible representations

dim11111111222222248
type+++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D5D10D10C5⋊D4C5⋊D4D4○D8D20.32C23
kernelD20.32C23C2×D4⋊D5D4.D10D4.Dic5D4⋊D10D4.8D10D48D10C5×2+ 1+4C5×D4C5×Q82+ 1+4C2×D4C4○D4D4Q8C5C1
# reps133133113126812422

Matrix representation of D20.32C23 in GL6(𝔽41)

4010000
3370000
00403900
001100
000012
00004040
,
4000000
3310000
00403900
000100
000010
00004040
,
4000000
0400000
00002424
00002917
00242400
00291700
,
3410000
3470000
0000017
00001224
00242400
0029000
,
100000
010000
000010
000001
001000
000100

G:=sub<GL(6,GF(41))| [40,33,0,0,0,0,1,7,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,1,40,0,0,0,0,2,40],[40,33,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,24,29,0,0,0,0,24,17,0,0,24,29,0,0,0,0,24,17,0,0],[34,34,0,0,0,0,1,7,0,0,0,0,0,0,0,0,24,29,0,0,0,0,24,0,0,0,0,12,0,0,0,0,17,24,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

D20.32C23 in GAP, Magma, Sage, TeX

D_{20}._{32}C_2^3
% in TeX

G:=Group("D20.32C2^3");
// GroupNames label

G:=SmallGroup(320,1507);
// by ID

G=gap.SmallGroup(320,1507);
# by ID

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

G:=Group<a,b,c,d,e|a^20=b^2=c^2=d^2=e^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,e*a*e=a^11,c*b*c=a^10*b,d*b*d=a^18*b,e*b*e=a^15*b,c*d=d*c,c*e=e*c,e*d*e=a^5*d>;
// generators/relations

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