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G = C23.3D20order 320 = 26·5

3rd non-split extension by C23 of D20 acting via D20/C5=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.3D20, C54C2≀C4, (C2×D4).5D10, C4.D45D5, (C2×C20).13D4, (C23×D5)⋊2C4, C23.D53C4, C23.3(C4×D5), C23⋊Dic57C2, C23⋊D10.4C2, (C22×C10).12D4, C10.32(C23⋊C4), (D4×C10).170C22, C22.12(D10⋊C4), C2.12(C23.1D10), (C2×C4).1(C5⋊D4), (C5×C4.D4)⋊11C2, (C22×C10).3(C2×C4), (C2×C10).69(C22⋊C4), SmallGroup(320,33)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C23.3D20
C1C5C10C2×C10C2×C20D4×C10C23⋊D10 — C23.3D20
C5C10C2×C10C22×C10 — C23.3D20
C1C2C22C2×D4C4.D4

Generators and relations for C23.3D20
 G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=a, ab=ba, ac=ca, dad-1=abc, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=acd19 >

Subgroups: 606 in 94 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2 [×5], C4 [×3], C22, C22 [×12], C5, C8, C2×C4, C2×C4 [×2], D4 [×3], C23 [×2], C23 [×4], D5 [×2], C10, C10 [×3], C22⋊C4 [×3], M4(2), C2×D4, C2×D4, C24, Dic5 [×2], C20, D10 [×8], C2×C10, C2×C10 [×4], C23⋊C4, C4.D4, C22≀C2, C40, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C5×D4, C22×D5 [×4], C22×C10 [×2], C2≀C4, D10⋊C4, C23.D5, C23.D5, C5×M4(2), C2×C5⋊D4, D4×C10, C23×D5, C23⋊Dic5, C5×C4.D4, C23⋊D10, C23.3D20
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D10, C23⋊C4, C4×D5, D20, C5⋊D4, C2≀C4, D10⋊C4, C23.1D10, C23.3D20

Smallest permutation representation of C23.3D20
On 40 points
Generators in S40
(2 22)(3 23)(6 26)(7 27)(10 30)(11 31)(14 34)(15 35)(18 38)(19 39)
(2 22)(4 24)(6 26)(8 28)(10 30)(12 32)(14 34)(16 36)(18 38)(20 40)
(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)
(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)
(1 20)(2 39 22 19)(3 18 23 38)(4 17)(5 16)(6 35 26 15)(7 14 27 34)(8 13)(9 12)(10 31 30 11)(21 40)(24 37)(25 36)(28 33)(29 32)

G:=sub<Sym(40)| (2,22)(3,23)(6,26)(7,27)(10,30)(11,31)(14,34)(15,35)(18,38)(19,39), (2,22)(4,24)(6,26)(8,28)(10,30)(12,32)(14,34)(16,36)(18,38)(20,40), (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), (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), (1,20)(2,39,22,19)(3,18,23,38)(4,17)(5,16)(6,35,26,15)(7,14,27,34)(8,13)(9,12)(10,31,30,11)(21,40)(24,37)(25,36)(28,33)(29,32)>;

G:=Group( (2,22)(3,23)(6,26)(7,27)(10,30)(11,31)(14,34)(15,35)(18,38)(19,39), (2,22)(4,24)(6,26)(8,28)(10,30)(12,32)(14,34)(16,36)(18,38)(20,40), (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), (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), (1,20)(2,39,22,19)(3,18,23,38)(4,17)(5,16)(6,35,26,15)(7,14,27,34)(8,13)(9,12)(10,31,30,11)(21,40)(24,37)(25,36)(28,33)(29,32) );

G=PermutationGroup([(2,22),(3,23),(6,26),(7,27),(10,30),(11,31),(14,34),(15,35),(18,38),(19,39)], [(2,22),(4,24),(6,26),(8,28),(10,30),(12,32),(14,34),(16,36),(18,38),(20,40)], [(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)], [(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)], [(1,20),(2,39,22,19),(3,18,23,38),(4,17),(5,16),(6,35,26,15),(7,14,27,34),(8,13),(9,12),(10,31,30,11),(21,40),(24,37),(25,36),(28,33),(29,32)])

35 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D5A5B8A8B10A10B10C10D10E10F10G10H20A20B20C20D40A···40H
order12222224444558810101010101010102020202040···40
size112442020440404022882244888844448···8

35 irreducible representations

dim11111122222224448
type++++++++++++
imageC1C2C2C2C4C4D4D4D5D10C5⋊D4C4×D5D20C23⋊C4C2≀C4C23.1D10C23.3D20
kernelC23.3D20C23⋊Dic5C5×C4.D4C23⋊D10C23.D5C23×D5C2×C20C22×C10C4.D4C2×D4C2×C4C23C23C10C5C2C1
# reps11112211224441242

Matrix representation of C23.3D20 in GL6(𝔽41)

4000000
0400000
001000
0004000
000010
00104040
,
100000
010000
001000
000100
0000400
0011040
,
100000
010000
0040000
0004000
0000400
0000040
,
3230000
21240000
000010
00114039
0004000
0001040
,
17230000
7240000
000010
00114039
001000
0001040

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,1,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[3,21,0,0,0,0,23,24,0,0,0,0,0,0,0,1,0,0,0,0,0,1,40,1,0,0,1,40,0,0,0,0,0,39,0,40],[17,7,0,0,0,0,23,24,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,1,0,0,1,40,0,0,0,0,0,39,0,40] >;

C23.3D20 in GAP, Magma, Sage, TeX

C_2^3._3D_{20}
% in TeX

G:=Group("C2^3.3D20");
// GroupNames label

G:=SmallGroup(320,33);
// by ID

G=gap.SmallGroup(320,33);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,422,184,346,297,851,12550]);
// Polycyclic

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

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