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G = C23.9D6order 96 = 25·3

4th non-split extension by C23 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6.4D4, C23.9D6, D6⋊C45C2, (C2×C4).6D6, C2.8(S3×D4), C22⋊C43S3, C4⋊Dic34C2, C6.19(C2×D4), C6.8(C4○D4), Dic3⋊C410C2, C6.D44C2, (C2×C6).24C23, C2.10(C4○D12), C2.8(D42S3), (C2×C12).52C22, C31(C22.D4), C22.42(C22×S3), (C22×C6).13C22, (C2×Dic3).6C22, (C22×S3).17C22, (S3×C2×C4)⋊10C2, (C3×C22⋊C4)⋊5C2, (C2×C3⋊D4).3C2, SmallGroup(96,90)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C23.9D6
C1C3C6C2×C6C22×S3S3×C2×C4 — C23.9D6
C3C2×C6 — C23.9D6
C1C22C22⋊C4

Generators and relations for C23.9D6
 G = < a,b,c,d,e | a2=b2=c2=1, d6=e2=b, ab=ba, dad-1=ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >

Subgroups: 186 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], S3 [×2], C6 [×3], C6, C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, Dic3 [×3], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×3], C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, C4×S3 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×C6, C22.D4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, C23.9D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C22×S3, C22.D4, C4○D12, S3×D4, D42S3, C23.9D6

Character table of C23.9D6

 class 12A2B2C2D2E2F34A4B4C4D4E4F4G6A6B6C6D6E12A12B12C12D
 size 11114662224661212222444444
ρ1111111111111111111111111    trivial
ρ21111-1-1-1111-1-1-111111-1-1-11-11    linear of order 2
ρ311111-1-11111-1-1-1-1111111111    linear of order 2
ρ41111-111111-111-1-1111-1-1-11-11    linear of order 2
ρ51111-1-1-11-1-1111-11111-1-11-11-1    linear of order 2
ρ611111111-1-1-1-1-1-1111111-1-1-1-1    linear of order 2
ρ71111-1111-1-11-1-11-1111-1-11-11-1    linear of order 2
ρ811111-1-11-1-1-1111-111111-1-1-1-1    linear of order 2
ρ92222-200-1-2-220000-1-1-111-11-11    orthogonal lifted from D6
ρ102-22-20-2220000000-22-2000000    orthogonal lifted from D4
ρ112222-200-122-20000-1-1-1111-11-1    orthogonal lifted from D6
ρ122222200-12220000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ132222200-1-2-2-20000-1-1-1-1-11111    orthogonal lifted from D6
ρ142-22-202-220000000-22-2000000    orthogonal lifted from D4
ρ152-2-220002000-2i2i002-2-2000000    complex lifted from C4○D4
ρ162-2-2200020002i-2i002-2-2000000    complex lifted from C4○D4
ρ1722-2-200022i-2i00000-2-220002i0-2i    complex lifted from C4○D4
ρ1822-2-20002-2i2i00000-2-22000-2i02i    complex lifted from C4○D4
ρ1922-2-2000-12i-2i0000011-1--3-3-3-i3i    complex lifted from C4○D12
ρ2022-2-2000-12i-2i0000011-1-3--33-i-3i    complex lifted from C4○D12
ρ2122-2-2000-1-2i2i0000011-1--3-33i-3-i    complex lifted from C4○D12
ρ2222-2-2000-1-2i2i0000011-1-3--3-3i3-i    complex lifted from C4○D12
ρ234-44-4000-200000002-22000000    orthogonal lifted from S3×D4
ρ244-4-44000-20000000-222000000    symplectic lifted from D42S3, Schur index 2

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

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

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

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

C23.9D6 is a maximal subgroup of
C24.38D6  C24.41D6  C24.42D6  C42.93D6  C42.94D6  C42.95D6  C42.99D6  C4214D6  D1224D4  C4218D6  C42.229D6  C42.114D6  C42.116D6  C42.118D6  C42.119D6  C247D6  C24.44D6  C24.46D6  C24.47D6  C6.342+ 1+4  C6.372+ 1+4  C4⋊C421D6  C6.732- 1+4  D1220D4  C6.442+ 1+4  C6.472+ 1+4  C6.492+ 1+4  C6.162- 1+4  D1222D4  C6.202- 1+4  C6.212- 1+4  C6.242- 1+4  C6.252- 1+4  C6.592+ 1+4  S3×C22.D4  C6.822- 1+4  C6.1222+ 1+4  C6.622+ 1+4  C6.632+ 1+4  C6.642+ 1+4  C6.652+ 1+4  C6.662+ 1+4  C6.852- 1+4  C6.682+ 1+4  C6.692+ 1+4  C42.137D6  C42.141D6  D1210D4  C4222D6  C4223D6  C42.234D6  C42.143D6  C42.145D6  C4225D6  C4226D6  C42.189D6  C42.161D6  C42.162D6  C42.165D6  C23.9D18  C62.23C23  C62.24C23  D6.9D12  C62.75C23  C62.111C23  C62.117C23  C62.227C23  D30.34D4  D30.D4  D10⋊C4⋊S3  D6.9D20  (S3×C10).D4  D30.16D4  D30.28D4
C23.9D6 is a maximal quotient of
C6.(C4×D4)  C2.(C4×Dic6)  C6.(C4⋊Q8)  (C2×Dic3).9D4  C22.58(S3×D4)  D6⋊C4⋊C4  (C2×C4).21D12  C6.(C4⋊D4)  D6.D8  D6.SD16  D6⋊C811C2  C241C4⋊C2  D6.1SD16  D6.Q16  D6⋊C8.C2  C8⋊Dic3⋊C2  C24.15D6  C24.18D6  C24.19D6  C24.20D6  C24.23D6  C24.25D6  C24.27D6  C23.9D18  C62.23C23  C62.24C23  D6.9D12  C62.75C23  C62.111C23  C62.117C23  C62.227C23  D30.34D4  D30.D4  D10⋊C4⋊S3  D6.9D20  (S3×C10).D4  D30.16D4  D30.28D4

Matrix representation of C23.9D6 in GL6(𝔽13)

100000
010000
000100
001000
000001
000010
,
100000
010000
0012000
0001200
000010
000001
,
100000
010000
001000
000100
0000120
0000012
,
110000
1200000
005000
000500
000010
0000012
,
110000
0120000
005000
000800
000010
0000012

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

C23.9D6 in GAP, Magma, Sage, TeX

C_2^3._9D_6
% in TeX

G:=Group("C2^3.9D6");
// GroupNames label

G:=SmallGroup(96,90);
// by ID

G=gap.SmallGroup(96,90);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,55,218,188,2309]);
// Polycyclic

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

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Character table of C23.9D6 in TeX

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