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G = C23.D20order 320 = 26·5

1st non-split extension by C23 of D20 acting via D20/C5=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.1D20, (C4×Dic5)⋊1C4, (C2×D4).3D10, C23⋊C4.2D5, C53(C423C4), (C2×Dic10)⋊4C4, (D4×C10).3C22, (C22×C10).10D4, C23.3(C5⋊D4), C23⋊Dic5.1C2, C10.30(C23⋊C4), C20.17D4.1C2, C22.10(D10⋊C4), C2.10(C23.1D10), (C2×C4).1(C4×D5), (C2×C20).1(C2×C4), (C5×C23⋊C4).2C2, (C2×C10).67(C22⋊C4), SmallGroup(320,31)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C23.D20
C1C5C10C2×C10C22×C10D4×C10C20.17D4 — C23.D20
C5C10C2×C10C2×C20 — C23.D20
C1C2C22C2×D4C23⋊C4

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

Subgroups: 334 in 70 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2 [×3], C4 [×5], C22, C22 [×4], C5, C2×C4, C2×C4 [×4], D4, Q8, C23 [×2], C10, C10 [×3], C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×4], C23⋊C4, C23⋊C4, C4.4D4, Dic10, C2×Dic5 [×3], C2×C20, C2×C20, C5×D4, C22×C10 [×2], C423C4, C4×Dic5, C23.D5 [×3], C5×C22⋊C4, C2×Dic10, D4×C10, C23⋊Dic5, C5×C23⋊C4, C20.17D4, C23.D20
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D10, C23⋊C4, C4×D5, D20, C5⋊D4, C423C4, D10⋊C4, C23.1D10, C23.D20

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

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

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

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

35 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H5A5B10A10B10C···10H10I10J20A···20J
order122224444444455101010···10101020···20
size11244488202040404022224···4888···8

35 irreducible representations

dim1111112222224448
type+++++++++-
imageC1C2C2C2C4C4D4D5D10C4×D5D20C5⋊D4C23⋊C4C423C4C23.1D10C23.D20
kernelC23.D20C23⋊Dic5C5×C23⋊C4C20.17D4C4×Dic5C2×Dic10C22×C10C23⋊C4C2×D4C2×C4C23C23C10C5C2C1
# reps1111222224441242

Matrix representation of C23.D20 in GL8(𝔽41)

400000000
040000000
120100000
1224010000
000011800
000004000
000036331336
00002712928
,
400000000
040000000
004000000
000400000
000011800
000004000
000000285
0000003213
,
10000000
01000000
00100000
00010000
000040000
000004000
000000400
000000040
,
3726490000
03328130000
293017280000
1389360000
00009039
000040322740
0000370320
00001539279
,
2438000000
117000000
14184000000
331510000
000011832
000094039
00000371336
00000383428

G:=sub<GL(8,GF(41))| [40,0,12,12,0,0,0,0,0,40,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,36,27,0,0,0,0,18,40,33,12,0,0,0,0,0,0,13,9,0,0,0,0,0,0,36,28],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,40,0,0,0,0,0,0,0,0,28,32,0,0,0,0,0,0,5,13],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[37,0,29,13,0,0,0,0,26,33,30,8,0,0,0,0,4,28,17,9,0,0,0,0,9,13,28,36,0,0,0,0,0,0,0,0,9,40,37,15,0,0,0,0,0,32,0,39,0,0,0,0,3,27,32,27,0,0,0,0,9,40,0,9],[24,1,14,3,0,0,0,0,38,17,18,31,0,0,0,0,0,0,40,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,18,40,37,38,0,0,0,0,3,3,13,34,0,0,0,0,2,9,36,28] >;

C23.D20 in GAP, Magma, Sage, TeX

C_2^3.D_{20}
% in TeX

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

G:=SmallGroup(320,31);
// by ID

G=gap.SmallGroup(320,31);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,422,1123,794,297,136,851,12550]);
// Polycyclic

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

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