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G = C23.2D20order 320 = 26·5

2nd non-split extension by C23 of D20 acting via D20/C5=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.2D20, (C2×D20)⋊4C4, C23⋊C42D5, (C4×Dic5)⋊2C4, (C2×D4).4D10, C53(C42⋊C4), C20⋊D4.1C2, C23⋊Dic51C2, (D4×C10).4C22, (C22×C10).11D4, C23.4(C5⋊D4), C10.31(C23⋊C4), C22.11(D10⋊C4), C2.11(C23.1D10), (C2×C4).2(C4×D5), (C5×C23⋊C4)⋊2C2, (C2×C20).2(C2×C4), (C2×C10).68(C22⋊C4), SmallGroup(320,32)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C23.2D20
C1C5C10C2×C10C22×C10D4×C10C20⋊D4 — C23.2D20
C5C10C2×C10C2×C20 — C23.2D20
C1C2C22C2×D4C23⋊C4

Generators and relations for C23.2D20
 G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=a, dad-1=ab=ba, ac=ca, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=ad-1 >

Subgroups: 526 in 86 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2 [×4], C4 [×5], C22, C22 [×7], C5, C2×C4, C2×C4 [×3], D4 [×6], C23 [×2], C23, D5, C10, C10 [×3], C42, C22⋊C4 [×2], C2×D4, C2×D4 [×3], Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×4], C23⋊C4, C23⋊C4, C41D4, D20, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C2×C20, C5×D4, C22×D5, C22×C10 [×2], C42⋊C4, C4×Dic5, C23.D5, C5×C22⋊C4, C2×D20, C2×C5⋊D4 [×2], D4×C10, C23⋊Dic5, C5×C23⋊C4, C20⋊D4, C23.2D20
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D10, C23⋊C4, C4×D5, D20, C5⋊D4, C42⋊C4, D10⋊C4, C23.1D10, C23.2D20

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

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G5A5B10A10B10C···10H10I10J20A···20J
order122222444444455101010···10101020···20
size11244404882020404022224···4888···8

35 irreducible representations

dim1111112222224448
type+++++++++++
imageC1C2C2C2C4C4D4D5D10C4×D5D20C5⋊D4C23⋊C4C42⋊C4C23.1D10C23.2D20
kernelC23.2D20C23⋊Dic5C5×C23⋊C4C20⋊D4C4×Dic5C2×D20C22×C10C23⋊C4C2×D4C2×C4C23C23C10C5C2C1
# reps1111222224441242

Matrix representation of C23.2D20 in GL6(𝔽41)

4000000
0400000
00403200
000100
00184001
00184010
,
100000
010000
001000
000100
00230400
00230040
,
100000
010000
0040000
0004000
0000400
0000040
,
0320000
9220000
0040090
0000401
000010
00184010
,
0320000
3200000
0040090
002304040
00184010
000010

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,18,18,0,0,32,1,40,40,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,23,23,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[0,9,0,0,0,0,32,22,0,0,0,0,0,0,40,0,0,18,0,0,0,0,0,40,0,0,9,40,1,1,0,0,0,1,0,0],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,40,23,18,0,0,0,0,0,40,0,0,0,9,40,1,1,0,0,0,40,0,0] >;

C23.2D20 in GAP, Magma, Sage, TeX

C_2^3._2D_{20}
% in TeX

G:=Group("C2^3.2D20");
// GroupNames label

G:=SmallGroup(320,32);
// by ID

G=gap.SmallGroup(320,32);
# by ID

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

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=a,d*a*d^-1=a*b=b*a,a*c=c*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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