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G = D10.C23order 160 = 25·5

11st non-split extension by D10 of C23 acting via C23/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.11C23, C4⋊F53C2, (C2×C4)⋊4F5, (C2×C20)⋊4C4, (C4×D5)⋊6C4, (C4×F5)⋊4C2, C22⋊F5.C2, C5⋊(C42⋊C2), C4.13(C2×F5), C20.21(C2×C4), (C2×Dic5)⋊9C4, C2.7(C22×F5), C22.7(C2×F5), D5.1(C4○D4), D10.17(C2×C4), C10.6(C22×C4), (C2×F5).2C22, Dic5.17(C2×C4), (C4×D5).36C22, (C22×D5).38C22, (C2×C4×D5).16C2, (C2×C10).18(C2×C4), SmallGroup(160,205)

Series: Derived Chief Lower central Upper central

C1C10 — D10.C23
C1C5D5D10C2×F5C4×F5 — D10.C23
C5C10 — D10.C23
C1C4C2×C4

Generators and relations for D10.C23
 G = < a,b,c,d,e | a10=b2=e2=1, c2=a-1b, d2=a5, bab=a-1, cac-1=a3, ad=da, ae=ea, cbc-1=a2b, bd=db, be=eb, cd=dc, ece=a5c, de=ed >

Subgroups: 244 in 76 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×6], C22, C22 [×4], C5, C2×C4, C2×C4 [×9], C23, D5 [×2], D5, C10, C10, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×2], C2×C10, C42⋊C2, C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C22×D5, C4×F5 [×2], C4⋊F5 [×2], C22⋊F5 [×2], C2×C4×D5, D10.C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C4○D4 [×2], F5, C42⋊C2, C2×F5 [×3], C22×F5, D10.C23

Character table of D10.C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N510A10B10C20A20B20C20D
 size 11255101125510101010101010101044444444
ρ11111111111111111111111111111    trivial
ρ211-111-111-111-1-111-1-111-11-11-11-1-11    linear of order 2
ρ311-111-111-1111-1-1-111-1-111-11-11-1-11    linear of order 2
ρ411111111111-11-1-1-1-1-1-1-111111111    linear of order 2
ρ511-111-1-1-11-1-1-11-1-11111-11-11-1-111-1    linear of order 2
ρ6111111-1-1-1-1-11-1-1-1-1-11111111-1-1-1-1    linear of order 2
ρ711-111-1-1-11-1-11111-1-1-1-111-11-1-111-1    linear of order 2
ρ8111111-1-1-1-1-1-1-11111-1-1-11111-1-1-1-1    linear of order 2
ρ911-1-1-11-1-1111i-1-iii-ii-i-i1-11-1-111-1    linear of order 4
ρ10111-1-1-1111-1-1-i-1i-ii-ii-ii11111111    linear of order 4
ρ1111-1-1-1111-1-1-1-i1-iii-i-iii1-11-11-1-11    linear of order 4
ρ12111-1-1-1-1-1-111i1i-ii-i-ii-i1111-1-1-1-1    linear of order 4
ρ1311-1-1-1111-1-1-1i1i-i-iii-i-i1-11-11-1-11    linear of order 4
ρ14111-1-1-1-1-1-111-i1-ii-iii-ii1111-1-1-1-1    linear of order 4
ρ1511-1-1-11-1-1111-i-1i-i-ii-iii1-11-1-111-1    linear of order 4
ρ16111-1-1-1111-1-1i-1-ii-ii-ii-i11111111    linear of order 4
ρ172-202-20-2i2i02i-2i00000000020-202i00-2i    complex lifted from C4○D4
ρ182-202-202i-2i0-2i2i00000000020-20-2i002i    complex lifted from C4○D4
ρ192-20-220-2i2i0-2i2i00000000020-202i00-2i    complex lifted from C4○D4
ρ202-20-2202i-2i02i-2i00000000020-20-2i002i    complex lifted from C4○D4
ρ2144-4000-4-4400000000000-11-111-1-11    orthogonal lifted from C2×F5
ρ2244400044400000000000-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ2344-400044-400000000000-11-11-111-1    orthogonal lifted from C2×F5
ρ24444000-4-4-400000000000-1-1-1-11111    orthogonal lifted from C2×F5
ρ254-400004i-4i000000000000-1-515i--5-5-i    complex faithful
ρ264-40000-4i4i000000000000-1-515-i-5--5i    complex faithful
ρ274-400004i-4i000000000000-151-5i-5--5-i    complex faithful
ρ284-40000-4i4i000000000000-151-5-i--5-5i    complex faithful

Smallest permutation representation of D10.C23
On 40 points
Generators in S40
(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 5)(2 4)(6 10)(7 9)(11 19)(12 18)(13 17)(14 16)(21 29)(22 28)(23 27)(24 26)(31 39)(32 38)(33 37)(34 36)
(1 13 6 18)(2 20 5 11)(3 17 4 14)(7 15 10 16)(8 12 9 19)(21 39 30 32)(22 36 29 35)(23 33 28 38)(24 40 27 31)(25 37 26 34)
(1 28 6 23)(2 29 7 24)(3 30 8 25)(4 21 9 26)(5 22 10 27)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)
(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)

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

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

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

D10.C23 is a maximal subgroup of
C426F5  C423F5  (C2×C8)⋊F5  C20.24C42  M4(2)⋊3F5  M4(2)⋊4F5  C20.12C42  (C2×C8)⋊6F5  M4(2)⋊1F5  M4(2)⋊5F5  C23⋊F55C2  (D4×C10)⋊C4  (C2×D4)⋊6F5  (C2×D4)⋊8F5  (C2×Q8)⋊4F5  (C2×Q8)⋊6F5  (C2×Q8)⋊7F5  C4○D20⋊C4  D4⋊F5⋊C2  D10.C24  D5.2- 1+4  C4○D4×F5  D5.2+ 1+4  C4⋊F53S3  (C4×S3)⋊F5  C22⋊F5.S3  (C2×C12)⋊6F5
D10.C23 is a maximal quotient of
C42.6F5  C42.12F5  C42.15F5  C42.7F5  C424F5  C4×C4⋊F5  C429F5  C425F5  C23.(C2×F5)  C10.(C4×D4)  C4⋊C4.7F5  C4⋊C45F5  Dic5.12M4(2)  C20.34M4(2)  Dic5.13M4(2)  C20.30M4(2)  C4×C22⋊F5  (C22×C4)⋊7F5  D106(C4⋊C4)  C4⋊F53S3  (C4×S3)⋊F5  C22⋊F5.S3  (C2×C12)⋊6F5

Matrix representation of D10.C23 in GL6(𝔽41)

4000000
0400000
000001
0040404040
001000
000100
,
4000000
0400000
000001
000010
000100
001000
,
090000
3200000
001000
000001
000100
0040404040
,
3200000
0320000
001000
000100
000010
000001
,
010000
100000
001000
000100
000010
000001

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

D10.C23 in GAP, Magma, Sage, TeX

D_{10}.C_2^3
% in TeX

G:=Group("D10.C2^3");
// GroupNames label

G:=SmallGroup(160,205);
// by ID

G=gap.SmallGroup(160,205);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,362,2309,599]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=b^2=e^2=1,c^2=a^-1*b,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,a*e=e*a,c*b*c^-1=a^2*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^5*c,d*e=e*d>;
// generators/relations

Export

Character table of D10.C23 in TeX

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