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G = D20.34C23order 320 = 26·5

15th non-split extension by D20 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.39C24, D20.34C23, 2- 1+43D5, Dic10.34C23, C4○D46D10, (C2×Q8)⋊13D10, (C5×D4).38D4, C57(D4○SD16), (C5×Q8).38D4, D4⋊D522C22, C20.271(C2×D4), Q8⋊D520C22, D4⋊D1012C2, D48D1010C2, C4.39(C23×D5), D4.20(C5⋊D4), D4.Dic512C2, C52C8.18C23, D4.D522C22, Q8.20(C5⋊D4), (Q8×C10)⋊23C22, D4.27(C22×D5), C5⋊Q1619C22, (C5×D4).27C23, D4.8D1010C2, Q8.27(C22×D5), (C5×Q8).27C23, C20.C2311C2, (C2×C20).120C23, C4○D20.33C22, C10.173(C22×D4), C4.Dic518C22, (C5×2- 1+4)⋊2C2, (C2×D20).192C22, (C2×Q8⋊D5)⋊32C2, C4.77(C2×C5⋊D4), (C2×C10).87(C2×D4), (C5×C4○D4)⋊9C22, C22.8(C2×C5⋊D4), (C2×C52C8)⋊26C22, C2.46(C22×C5⋊D4), (C2×C4).104(C22×D5), SmallGroup(320,1509)

Series: Derived Chief Lower central Upper central

C1C20 — D20.34C23
C1C5C10C20D20C2×D20D48D10 — D20.34C23
C5C10C20 — D20.34C23
C1C2C4○D42- 1+4

Generators and relations for D20.34C23
 G = < a,b,c,d,e | a20=b2=c2=d2=1, e2=a10, bab=dad=a-1, ac=ca, eae-1=a11, cbc=a10b, dbd=a18b, ebe-1=a15b, cd=dc, ce=ec, ede-1=a5d >

Subgroups: 918 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, D4 [×3], D4 [×12], Q8, Q8 [×3], Q8 [×4], C23 [×3], D5 [×3], C10, C10 [×4], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×6], C2×Q8 [×3], C2×Q8, C4○D4, C4○D4 [×3], C4○D4 [×7], Dic5, C20, C20 [×3], C20 [×3], D10 [×6], C2×C10 [×3], C2×C10, C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 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 [×6], C5×D4, C5×D4 [×3], C5×D4 [×3], C5×Q8, C5×Q8 [×3], C5×Q8 [×3], C22×D5 [×3], D4○SD16, C2×C52C8 [×3], C4.Dic5 [×3], D4⋊D5 [×3], D4.D5, Q8⋊D5 [×9], C5⋊Q16 [×3], C2×D20 [×3], C4○D20 [×3], D4×D5 [×3], Q82D5, Q8×C10 [×3], Q8×C10, C5×C4○D4, C5×C4○D4 [×3], C5×C4○D4 [×3], C2×Q8⋊D5 [×3], C20.C23 [×3], D4.Dic5, D4⋊D10 [×3], D4.8D10 [×3], D48D10, C5×2- 1+4, D20.34C23
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, D20.34C23

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

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E10A10B10C···10L20A···20T
order122222222444444445588888101010···1020···20
size112224202020222244420221010202020224···44···4

56 irreducible representations

dim11111111222222248
type++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D5D10D10C5⋊D4C5⋊D4D4○SD16D20.34C23
kernelD20.34C23C2×Q8⋊D5C20.C23D4.Dic5D4⋊D10D4.8D10D48D10C5×2- 1+4C5×D4C5×Q82- 1+4C2×Q8C4○D4D4Q8C5C1
# reps133133113126812422

Matrix representation of D20.34C23 in GL6(𝔽41)

010000
4060000
0011600
0054000
00004025
0000361
,
010000
100000
0011600
0004000
000010
0000540
,
4000000
0400000
000006
0000340
0003500
007000
,
100000
6400000
00003035
0000340
0003500
00341100
,
100000
010000
000010
000001
0040000
0004000

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,5,0,0,0,0,16,40,0,0,0,0,0,0,40,36,0,0,0,0,25,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,16,40,0,0,0,0,0,0,1,5,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,7,0,0,0,0,35,0,0,0,0,34,0,0,0,0,6,0,0,0],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,34,0,0,0,0,35,11,0,0,30,34,0,0,0,0,35,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;

D20.34C23 in GAP, Magma, Sage, TeX

D_{20}._{34}C_2^3
% in TeX

G:=Group("D20.34C2^3");
// GroupNames label

G:=SmallGroup(320,1509);
// by ID

G=gap.SmallGroup(320,1509);
# by ID

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

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

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