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G = C23.30D20order 320 = 26·5

1st non-split extension by C23 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.30D20, C22.2Dic20, C4⋊Dic53C4, C10.20C4≀C2, C22⋊C8.1D5, (C2×C10).1Q16, (C2×Dic10)⋊2C4, (C2×C20).437D4, (C2×C10).1SD16, (C22×C4).54D10, (C22×C10).39D4, C2.6(D207C4), C10.24(C23⋊C4), C20.48D4.1C2, C22.4(C40⋊C2), C54(C23.31D4), C10.13(Q8⋊C4), C2.3(C20.44D4), (C22×C20).40C22, C2.6(C23.1D10), C22.58(D10⋊C4), C10.10C42.21C2, (C2×C4).12(C4×D5), (C5×C22⋊C8).1C2, (C2×C20).197(C2×C4), (C2×C4).208(C5⋊D4), (C2×C10).103(C22⋊C4), SmallGroup(320,25)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C23.30D20
C1C5C10C2×C10C2×C20C22×C20C10.10C42 — C23.30D20
C5C2×C10C2×C20 — C23.30D20
C1C22C22×C4C22⋊C8

Generators and relations for C23.30D20
 G = < a,b,c,d,e | a2=b2=c2=1, d20=cb=bc, e2=b, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=acd19 >

Subgroups: 350 in 80 conjugacy classes, 29 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C22 [×3], C22 [×2], C5, C8, C2×C4 [×2], C2×C4 [×7], Q8, C23, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8, C22×C4, C22×C4, C2×Q8, Dic5 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×2], C2.C42, C22⋊C8, C22⋊Q8, C40, Dic10, C2×Dic5 [×6], C2×C20 [×2], C2×C20, C22×C10, C23.31D4, C10.D4, C4⋊Dic5, C23.D5, C2×C40, C2×Dic10, C22×Dic5, C22×C20, C10.10C42, C5×C22⋊C8, C20.48D4, C23.30D20
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, SD16, Q16, D10, C23⋊C4, Q8⋊C4, C4≀C2, C4×D5, D20, C5⋊D4, C23.31D4, C40⋊C2, Dic20, D10⋊C4, C23.1D10, C20.44D4, D207C4, C23.30D20

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

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222244444444455888810···101010101020···202020202040···40
size1111222242020202040402244442···244442···244444···4

59 irreducible representations

dim111111222222222222444
type+++++++-++-+
imageC1C2C2C2C4C4D4D4D5SD16Q16D10C4≀C2C4×D5C5⋊D4D20C40⋊C2Dic20C23⋊C4C23.1D10D207C4
kernelC23.30D20C10.10C42C5×C22⋊C8C20.48D4C4⋊Dic5C2×Dic10C2×C20C22×C10C22⋊C8C2×C10C2×C10C22×C4C10C2×C4C2×C4C23C22C22C10C2C2
# reps111122112222444488144

Matrix representation of C23.30D20 in GL4(𝔽41) generated by

1000
0100
0010
00140
,
40000
04000
0010
0001
,
1000
0100
00400
00040
,
14400
373100
00139
00540
,
26200
101500
00402
0001
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[14,37,0,0,4,31,0,0,0,0,1,5,0,0,39,40],[26,10,0,0,2,15,0,0,0,0,40,0,0,0,2,1] >;

C23.30D20 in GAP, Magma, Sage, TeX

C_2^3._{30}D_{20}
% in TeX

G:=Group("C2^3.30D20");
// GroupNames label

G:=SmallGroup(320,25);
// by ID

G=gap.SmallGroup(320,25);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,85,92,422,387,268,570,12550]);
// Polycyclic

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

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