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G = C20.C24order 320 = 26·5

35th non-split extension by C20 of C24 acting via C24/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.35C24, D20.31C23, Dic10.30C23, C4○D416D10, (C2×D4)⋊42D10, (C2×Q8)⋊31D10, D4⋊D519C22, (C2×C20).218D4, C20.427(C2×D4), Q8⋊D518C22, D4⋊D1013C2, C4.35(C23×D5), D4.8D107C2, C4○D2021C22, (D4×C10)⋊46C22, (C2×D20)⋊59C22, C55(D8⋊C22), C52C8.14C23, D4.D517C22, (Q8×C10)⋊38C22, D4.23(C22×D5), (C5×D4).23C23, C5⋊Q1616C22, D4.9D1013C2, D4.D1013C2, (C5×Q8).23C23, Q8.23(C22×D5), C20.C2313C2, (C2×C20).557C23, (C22×C10).124D4, C10.160(C22×D4), (C22×C4).282D10, C23.34(C5⋊D4), C4.Dic537C22, (C2×Dic10)⋊69C22, (C22×C20).292C22, (C2×C4○D4)⋊4D5, (C10×C4○D4)⋊4C2, (C2×C4○D20)⋊31C2, C4.121(C2×C5⋊D4), (C2×C52C8)⋊23C22, (C2×C10).591(C2×D4), (C5×C4○D4)⋊18C22, (C2×C4.Dic5)⋊31C2, C2.33(C22×C5⋊D4), C22.21(C2×C5⋊D4), (C2×C4).203(C5⋊D4), (C2×C4).246(C22×D5), SmallGroup(320,1494)

Series: Derived Chief Lower central Upper central

C1C20 — C20.C24
C1C5C10C20D20C2×D20C2×C4○D20 — C20.C24
C5C10C20 — C20.C24
C1C4C22×C4C2×C4○D4

Generators and relations for C20.C24
 G = < a,b,c,d,e | a20=b2=c2=e2=1, d2=a10, bab=a-1, ac=ca, ad=da, eae=a11, bc=cb, bd=db, ebe=a5b, cd=dc, ece=a10c, de=ed >

Subgroups: 830 in 262 conjugacy classes, 107 normal (45 characteristic)
C1, C2, C2 [×7], C4 [×4], C4 [×4], C22 [×3], C22 [×9], C5, C8 [×4], C2×C4 [×6], C2×C4 [×10], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23, C23 [×2], D5 [×2], C10, C10 [×5], C2×C8 [×2], M4(2) [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×4], C4○D4 [×8], Dic5 [×2], C20 [×4], C20 [×2], D10 [×4], C2×C10 [×3], C2×C10 [×5], C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4, C2×C4○D4, C52C8 [×4], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×6], C2×C20 [×5], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×D5, C22×C10, C22×C10, D8⋊C22, C2×C52C8 [×2], C4.Dic5 [×4], D4⋊D5 [×4], D4.D5 [×4], Q8⋊D5 [×4], C5⋊Q16 [×4], C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], C2×C4.Dic5, D4.D10 [×2], C20.C23 [×2], D4⋊D10 [×2], D4.8D10 [×4], D4.9D10 [×2], C2×C4○D20, C10×C4○D4, C20.C24
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], D8⋊C22, C2×C5⋊D4 [×6], C23×D5, C22×C5⋊D4, C20.C24

Smallest permutation representation of C20.C24
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)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(1 41 11 51)(2 42 12 52)(3 43 13 53)(4 44 14 54)(5 45 15 55)(6 46 16 56)(7 47 17 57)(8 48 18 58)(9 49 19 59)(10 50 20 60)(21 76 31 66)(22 77 32 67)(23 78 33 68)(24 79 34 69)(25 80 35 70)(26 61 36 71)(27 62 37 72)(28 63 38 73)(29 64 39 74)(30 65 40 75)
(1 21)(2 32)(3 23)(4 34)(5 25)(6 36)(7 27)(8 38)(9 29)(10 40)(11 31)(12 22)(13 33)(14 24)(15 35)(16 26)(17 37)(18 28)(19 39)(20 30)(41 76)(42 67)(43 78)(44 69)(45 80)(46 71)(47 62)(48 73)(49 64)(50 75)(51 66)(52 77)(53 68)(54 79)(55 70)(56 61)(57 72)(58 63)(59 74)(60 65)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G···10R20A···20H20I···20T
order12222222244444444455888810···1010···1020···2020···20
size112224420201122244202022202020202···24···42···24···4

62 irreducible representations

dim11111111122222222244
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D5D10D10D10D10C5⋊D4C5⋊D4D8⋊C22C20.C24
kernelC20.C24C2×C4.Dic5D4.D10C20.C23D4⋊D10D4.8D10D4.9D10C2×C4○D20C10×C4○D4C2×C20C22×C10C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C2×C4C23C5C1
# reps112224211312222812428

Matrix representation of C20.C24 in GL6(𝔽41)

0400000
160000
000100
0040000
0000040
000010
,
3560000
160000
001000
0004000
000001
000010
,
4000000
0400000
0040000
0004000
000010
000001
,
100000
010000
0032000
0003200
0000320
0000032
,
100000
010000
000010
000001
001000
000100

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

C20.C24 in GAP, Magma, Sage, TeX

C_{20}.C_2^4
% in TeX

G:=Group("C20.C2^4");
// GroupNames label

G:=SmallGroup(320,1494);
// by ID

G=gap.SmallGroup(320,1494);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,570,1684,235,102,12550]);
// Polycyclic

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

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