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G = C23.20D20order 320 = 26·5

13rd non-split extension by C23 of D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.20D20, M4(2)⋊24D10, C4○D2014C4, (C2×D20)⋊28C4, (C2×C4).55D20, D20.41(C2×C4), C20.446(C2×D4), (C2×C20).178D4, D207C413C2, (C2×Dic10)⋊27C4, (C2×M4(2))⋊15D5, (C4×Dic5)⋊4C22, C22.17(C2×D20), C20.76(C22⋊C4), (C10×M4(2))⋊23C2, C20.129(C22×C4), (C2×C20).420C23, Dic10.43(C2×C4), C56(C42⋊C22), C4○D20.43C22, (C22×C4).146D10, (C22×C10).107D4, C4.30(D10⋊C4), (C5×M4(2))⋊36C22, C23.21D1017C2, (C22×C20).193C22, C22.29(D10⋊C4), C4.55(C2×C4×D5), (C2×C4).56(C4×D5), (C2×C10).33(C2×D4), C4.137(C2×C5⋊D4), (C2×C20).286(C2×C4), (C2×C4○D20).15C2, C2.35(C2×D10⋊C4), (C2×C4).258(C5⋊D4), C10.104(C2×C22⋊C4), (C2×C4).513(C22×D5), (C2×C10).87(C22⋊C4), SmallGroup(320,766)

Series: Derived Chief Lower central Upper central

C1C20 — C23.20D20
C1C5C10C20C2×C20C4○D20C2×C4○D20 — C23.20D20
C5C10C20 — C23.20D20
C1C4C22×C4C2×M4(2)

Generators and relations for C23.20D20
 G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=b, ab=ba, dad-1=eae-1=ac=ca, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd19 >

Subgroups: 622 in 154 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×4], C22 [×3], C22 [×5], C5, C8 [×2], C2×C4 [×6], C2×C4 [×7], D4 [×7], Q8 [×3], C23, C23, D5 [×2], C10, C10 [×3], C42 [×2], C22⋊C4, C4⋊C4, C2×C8, M4(2) [×2], M4(2), C22×C4, C22×C4, C2×D4 [×2], C2×Q8, C4○D4 [×6], Dic5 [×4], C20 [×4], D10 [×4], C2×C10 [×3], C2×C10, C4≀C2 [×4], C42⋊C2, C2×M4(2), C2×C4○D4, C40 [×2], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5 [×3], C5⋊D4 [×4], C2×C20 [×6], C22×D5, C22×C10, C42⋊C22, C4×Dic5 [×2], C4⋊Dic5, C23.D5, C2×C40, C5×M4(2) [×2], C5×M4(2), C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, D207C4 [×4], C23.21D10, C10×M4(2), C2×C4○D20, C23.20D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C42⋊C22, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, C23.20D20

Smallest permutation representation of C23.20D20
On 80 points
Generators in S80
(2 22)(4 24)(6 26)(8 28)(10 30)(12 32)(14 34)(16 36)(18 38)(20 40)(42 62)(44 64)(46 66)(48 68)(50 70)(52 72)(54 74)(56 76)(58 78)(60 80)
(1 55)(2 76)(3 57)(4 78)(5 59)(6 80)(7 61)(8 42)(9 63)(10 44)(11 65)(12 46)(13 67)(14 48)(15 69)(16 50)(17 71)(18 52)(19 73)(20 54)(21 75)(22 56)(23 77)(24 58)(25 79)(26 60)(27 41)(28 62)(29 43)(30 64)(31 45)(32 66)(33 47)(34 68)(35 49)(36 70)(37 51)(38 72)(39 53)(40 74)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 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 74 55 40)(2 19 76 73)(3 52 57 18)(4 37 78 51)(5 70 59 36)(6 15 80 69)(7 48 61 14)(8 33 42 47)(9 66 63 32)(10 11 44 65)(12 29 46 43)(13 62 67 28)(16 25 50 79)(17 58 71 24)(20 21 54 75)(22 39 56 53)(23 72 77 38)(26 35 60 49)(27 68 41 34)(30 31 64 45)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F···4K5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order1222222444444···455888810···101010101020···202020202040···40
size1122220201122220···202244442···244442···244444···4

62 irreducible representations

dim1111111122222222244
type++++++++++++
imageC1C2C2C2C2C4C4C4D4D4D5D10D10C4×D5D20C5⋊D4D20C42⋊C22C23.20D20
kernelC23.20D20D207C4C23.21D10C10×M4(2)C2×C4○D20C2×Dic10C2×D20C4○D20C2×C20C22×C10C2×M4(2)M4(2)C22×C4C2×C4C2×C4C2×C4C23C5C1
# reps1411122431242848428

Matrix representation of C23.20D20 in GL4(𝔽41) generated by

10014
012727
00400
00040
,
2412735
4017835
00235
00118
,
40000
04000
00400
00040
,
011212
40343910
35565
03511
,
1191936
1430172
2729227
1628939
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,27,40,0,14,27,0,40],[24,40,0,0,1,17,0,0,27,8,23,1,35,35,5,18],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,40,35,0,1,34,5,35,12,39,6,1,12,10,5,1],[11,14,27,16,9,30,29,28,19,17,2,9,36,2,27,39] >;

C23.20D20 in GAP, Magma, Sage, TeX

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

G:=Group("C2^3.20D20");
// GroupNames label

G:=SmallGroup(320,766);
// by ID

G=gap.SmallGroup(320,766);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,387,58,136,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^19>;
// generators/relations

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