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G = C23.5D20order 320 = 26·5

5th non-split extension by C23 of D20 acting via D20/C5=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.5D20, C23⋊C44D5, (C2×C20).7D4, (C2×C4).5D20, C22⋊C42D10, (C2×D4).12D10, (C2×Dic5).1D4, (C22×D5).1D4, C22.25(D4×D5), C22.9(C2×D20), C10.14C22≀C2, D46D10.1C2, (D4×C10).9C22, (C22×C10).18D4, C51(C23.7D4), C23.D52C22, C23.3(C22×D5), C23.1D102C2, C22.D201C2, (C22×C10).3C23, C2.17(C22⋊D20), C23.18D101C2, (C22×Dic5)⋊1C22, (C5×C23⋊C4)⋊5C2, (C2×C10).18(C2×D4), (C5×C22⋊C4)⋊2C22, (C2×C5⋊D4).3C22, SmallGroup(320,369)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C23.5D20
C1C5C10C2×C10C22×C10C2×C5⋊D4D46D10 — C23.5D20
C5C10C22×C10 — C23.5D20
C1C2C23C23⋊C4

Generators and relations for C23.5D20
 G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=c, ab=ba, ac=ca, dad-1=eae-1=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 766 in 160 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2 [×6], C4 [×7], C22, C22 [×2], C22 [×8], C5, C2×C4, C2×C4 [×11], D4 [×9], Q8, C23 [×2], C23 [×3], D5 [×2], C10, C10 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×3], C22×C4, C2×D4, C2×D4 [×5], C4○D4 [×3], Dic5 [×4], C20 [×3], D10 [×5], C2×C10, C2×C10 [×2], C2×C10 [×3], C23⋊C4, C23⋊C4 [×2], C22.D4 [×3], 2+ 1+4, Dic10, C4×D5 [×2], D20, C2×Dic5 [×2], C2×Dic5 [×5], C5⋊D4 [×6], C2×C20, C2×C20 [×2], C5×D4 [×2], C22×D5 [×2], C22×D5, C22×C10 [×2], C23.7D4, C10.D4, C4⋊Dic5 [×2], D10⋊C4 [×2], C23.D5, C23.D5, C5×C22⋊C4 [×2], C4○D20, D4×D5 [×2], D42D5 [×2], C22×Dic5, C2×C5⋊D4 [×2], C2×C5⋊D4, D4×C10, C23.1D10 [×2], C5×C23⋊C4, C22.D20 [×2], C23.18D10, D46D10, C23.5D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, D20 [×2], C22×D5, C23.7D4, C2×D20, D4×D5 [×2], C22⋊D20, C23.5D20

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H5A5B10A10B10C···10H10I10J20A···20J
order122222224444444455101010···10101020···20
size1122242020488202020204022224···4888···8

38 irreducible representations

dim111111222222222448
type++++++++++++++++-
imageC1C2C2C2C2C2D4D4D4D4D5D10D10D20D20C23.7D4D4×D5C23.5D20
kernelC23.5D20C23.1D10C5×C23⋊C4C22.D20C23.18D10D46D10C2×Dic5C2×C20C22×D5C22×C10C23⋊C4C22⋊C4C2×D4C2×C4C23C5C22C1
# reps121211212124244242

Matrix representation of C23.5D20 in GL8(𝔽41)

10000000
01000000
3104000000
1130400000
00001000
00000100
000010400
000010040
,
400000000
040000000
004000000
000400000
000013900
000004000
000004001
000004010
,
10000000
01000000
00100000
00010000
000040000
000004000
000000400
000000040
,
92328280000
181222280000
16829180000
221623320000
0000323299
000032099
000003209
000032009
,
92328280000
13228220000
131718290000
172432230000
0000323299
000003200
000003209
000003290

G:=sub<GL(8,GF(41))| [1,0,3,11,0,0,0,0,0,1,10,3,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,39,40,40,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[9,18,16,22,0,0,0,0,23,12,8,16,0,0,0,0,28,22,29,23,0,0,0,0,28,28,18,32,0,0,0,0,0,0,0,0,32,32,0,32,0,0,0,0,32,0,32,0,0,0,0,0,9,9,0,0,0,0,0,0,9,9,9,9],[9,1,13,17,0,0,0,0,23,32,17,24,0,0,0,0,28,28,18,32,0,0,0,0,28,22,29,23,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,32,32,32,32,0,0,0,0,9,0,0,9,0,0,0,0,9,0,9,0] >;

C23.5D20 in GAP, Magma, Sage, TeX

C_2^3._5D_{20}
% in TeX

G:=Group("C2^3.5D20");
// GroupNames label

G:=SmallGroup(320,369);
// by ID

G=gap.SmallGroup(320,369);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,226,570,1684,438,12550]);
// Polycyclic

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

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