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G = C23.11D6order 96 = 25·3

6th non-split extension by C23 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic3.1D4, C23.11D6, D6⋊C46C2, (C2×C4).8D6, C22⋊C45S3, C6.21(C2×D4), C2.10(S3×D4), (C2×Dic6)⋊3C2, C32(C4.4D4), (C4×Dic3)⋊12C2, C6.10(C4○D4), C6.D45C2, (C2×C6).26C23, C2.12(C4○D12), C2.9(D42S3), (C2×C12).54C22, (C22×S3).4C22, (C22×C6).15C22, C22.44(C22×S3), (C2×Dic3).27C22, (C3×C22⋊C4)⋊7C2, (C2×C3⋊D4).4C2, SmallGroup(96,92)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C23.11D6
C1C3C6C2×C6C22×S3D6⋊C4 — C23.11D6
C3C2×C6 — C23.11D6
C1C22C22⋊C4

Generators and relations for C23.11D6
 G = < a,b,c,d,e | a2=b2=c2=1, d6=b, e2=cb=bc, eae-1=ab=ba, dad-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd5 >

Subgroups: 186 in 76 conjugacy classes, 31 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×6], C22, C22 [×6], S3, C6 [×3], C6, C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×2], C23, C23, Dic3 [×2], Dic3 [×2], C12 [×2], D6 [×3], C2×C6, C2×C6 [×3], C42, C22⋊C4, C22⋊C4 [×3], C2×D4, C2×Q8, Dic6 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×C6, C4.4D4, C4×Dic3, D6⋊C4 [×2], C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, C23.11D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C22×S3, C4.4D4, C4○D12, S3×D4, D42S3, C23.11D6

Character table of C23.11D6

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C6D6E12A12B12C12D
 size 11114122224666612222444444
ρ1111111111111111111111111    trivial
ρ21111-1-11-1-11-11-111111-1-111-1-1    linear of order 2
ρ311111-11-1-1-11-11-1111111-1-1-1-1    linear of order 2
ρ41111-11111-1-1-1-1-11111-1-1-1-111    linear of order 2
ρ511111-11111-1-1-1-1-1111111111    linear of order 2
ρ61111-111-1-111-11-1-1111-1-111-1-1    linear of order 2
ρ71111111-1-1-1-11-11-111111-1-1-1-1    linear of order 2
ρ81111-1-1111-11111-1111-1-1-1-111    linear of order 2
ρ92-2-220020000-2020-2-22000000    orthogonal lifted from D4
ρ10222220-122200000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ112222-20-1-2-2200000-1-1-111-1-111    orthogonal lifted from D6
ρ122-2-22002000020-20-2-22000000    orthogonal lifted from D4
ρ132222-20-122-200000-1-1-11111-1-1    orthogonal lifted from D6
ρ14222220-1-2-2-200000-1-1-1-1-11111    orthogonal lifted from D6
ρ1522-2-2002-2i2i000000-22-200002i-2i    complex lifted from C4○D4
ρ162-22-20020002i0-2i002-2-2000000    complex lifted from C4○D4
ρ1722-2-20022i-2i000000-22-20000-2i2i    complex lifted from C4○D4
ρ182-22-2002000-2i02i002-2-2000000    complex lifted from C4○D4
ρ1922-2-200-12i-2i0000001-11--3-3-33i-i    complex lifted from C4○D12
ρ2022-2-200-12i-2i0000001-11-3--33-3i-i    complex lifted from C4○D12
ρ2122-2-200-1-2i2i0000001-11--3-33-3-ii    complex lifted from C4○D12
ρ2222-2-200-1-2i2i0000001-11-3--3-33-ii    complex lifted from C4○D12
ρ234-4-4400-20000000022-2000000    orthogonal lifted from S3×D4
ρ244-44-400-200000000-222000000    symplectic lifted from D42S3, Schur index 2

Smallest permutation representation of C23.11D6
On 48 points
Generators in S48
(1 22)(2 30)(3 24)(4 32)(5 14)(6 34)(7 16)(8 36)(9 18)(10 26)(11 20)(12 28)(13 47)(15 37)(17 39)(19 41)(21 43)(23 45)(25 40)(27 42)(29 44)(31 46)(33 48)(35 38)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 32)(14 33)(15 34)(16 35)(17 36)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)
(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)
(1 6 38 43)(2 42 39 5)(3 4 40 41)(7 12 44 37)(8 48 45 11)(9 10 46 47)(13 24 26 25)(14 36 27 23)(15 22 28 35)(16 34 29 21)(17 20 30 33)(18 32 31 19)

G:=sub<Sym(48)| (1,22)(2,30)(3,24)(4,32)(5,14)(6,34)(7,16)(8,36)(9,18)(10,26)(11,20)(12,28)(13,47)(15,37)(17,39)(19,41)(21,43)(23,45)(25,40)(27,42)(29,44)(31,46)(33,48)(35,38), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,44)(2,45)(3,46)(4,47)(5,48)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,32)(14,33)(15,34)(16,35)(17,36)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31), (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), (1,6,38,43)(2,42,39,5)(3,4,40,41)(7,12,44,37)(8,48,45,11)(9,10,46,47)(13,24,26,25)(14,36,27,23)(15,22,28,35)(16,34,29,21)(17,20,30,33)(18,32,31,19)>;

G:=Group( (1,22)(2,30)(3,24)(4,32)(5,14)(6,34)(7,16)(8,36)(9,18)(10,26)(11,20)(12,28)(13,47)(15,37)(17,39)(19,41)(21,43)(23,45)(25,40)(27,42)(29,44)(31,46)(33,48)(35,38), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,44)(2,45)(3,46)(4,47)(5,48)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,32)(14,33)(15,34)(16,35)(17,36)(18,25)(19,26)(20,27)(21,28)(22,29)(23,30)(24,31), (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), (1,6,38,43)(2,42,39,5)(3,4,40,41)(7,12,44,37)(8,48,45,11)(9,10,46,47)(13,24,26,25)(14,36,27,23)(15,22,28,35)(16,34,29,21)(17,20,30,33)(18,32,31,19) );

G=PermutationGroup([(1,22),(2,30),(3,24),(4,32),(5,14),(6,34),(7,16),(8,36),(9,18),(10,26),(11,20),(12,28),(13,47),(15,37),(17,39),(19,41),(21,43),(23,45),(25,40),(27,42),(29,44),(31,46),(33,48),(35,38)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,32),(14,33),(15,34),(16,35),(17,36),(18,25),(19,26),(20,27),(21,28),(22,29),(23,30),(24,31)], [(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)], [(1,6,38,43),(2,42,39,5),(3,4,40,41),(7,12,44,37),(8,48,45,11),(9,10,46,47),(13,24,26,25),(14,36,27,23),(15,22,28,35),(16,34,29,21),(17,20,30,33),(18,32,31,19)])

C23.11D6 is a maximal subgroup of
C24.38D6  C24.41D6  C24.42D6  C42.93D6  C42.97D6  C42.98D6  C42.99D6  C42.102D6  C42.228D6  Dic623D4  C4218D6  C42.114D6  C4219D6  C42.115D6  C42.117D6  C24.44D6  C24.45D6  C24.46D6  C249D6  C12⋊(C4○D4)  Dic619D4  C6.382+ 1+4  C6.402+ 1+4  C6.422+ 1+4  C6.452+ 1+4  C6.462+ 1+4  C6.492+ 1+4  C6.162- 1+4  Dic622D4  C6.222- 1+4  C6.232- 1+4  C6.242- 1+4  C6.252- 1+4  C6.592+ 1+4  C6.792- 1+4  C4⋊C4.197D6  C6.1212+ 1+4  C6.612+ 1+4  C6.1222+ 1+4  C6.652+ 1+4  C6.672+ 1+4  C6.692+ 1+4  C42.137D6  C42.138D6  S3×C4.4D4  Dic610D4  C42.143D6  C42.144D6  C4224D6  C42.160D6  C4225D6  C4226D6  C42.189D6  C42.164D6  C42.165D6  Dic9.D4  Dic3.D12  C62.77C23  C62.83C23  C62.85C23  C62.95C23  C62.101C23  C62.229C23  Dic3.D20  (C2×Dic6)⋊D5  Dic15.10D4  Dic15.31D4  C23.D5⋊S3  Dic15.19D4  C23.11D30
C23.11D6 is a maximal quotient of
(C2×C12)⋊Q8  C3⋊(C428C4)  (C2×Dic3).9D4  (C2×C4).Dic6  D6⋊C45C4  D6⋊C43C4  (C22×S3)⋊Q8  C6.(C4⋊D4)  Dic3.SD16  C4⋊C4.D6  C12⋊Q8⋊C2  (C2×C8).200D6  Dic3.1Q16  (C2×C8).D6  (C2×Q8).36D6  Q8⋊C4⋊S3  C24.14D6  C232Dic6  C24.19D6  C24.20D6  C24.24D6  C24.25D6  C24.27D6  Dic9.D4  Dic3.D12  C62.77C23  C62.83C23  C62.85C23  C62.95C23  C62.101C23  C62.229C23  Dic3.D20  (C2×Dic6)⋊D5  Dic15.10D4  Dic15.31D4  C23.D5⋊S3  Dic15.19D4  C23.11D30

Matrix representation of C23.11D6 in GL6(𝔽13)

100000
010000
0011100
0001200
000010
0000012
,
100000
010000
0012000
0001200
0000120
0000012
,
100000
010000
001000
000100
0000120
0000012
,
110000
1200000
005000
000500
000001
0000120
,
110000
0120000
005000
005800
0000012
0000120

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,11,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,5,5,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

C23.11D6 in GAP, Magma, Sage, TeX

C_2^3._{11}D_6
% in TeX

G:=Group("C2^3.11D6");
// GroupNames label

G:=SmallGroup(96,92);
// by ID

G=gap.SmallGroup(96,92);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,55,506,188,2309]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=b,e^2=c*b=b*c,e*a*e^-1=a*b=b*a,d*a*d^-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=c*d^5>;
// generators/relations

Export

Character table of C23.11D6 in TeX

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