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G = C23.8D6order 96 = 25·3

3rd non-split extension by C23 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.8D6, C4⋊Dic33C2, (C2×C4).27D6, Dic3⋊C48C2, C6.7(C4○D4), C22⋊C4.2S3, C32(C422C2), (C4×Dic3)⋊10C2, C2.9(C4○D12), (C2×C6).20C23, (C2×C12).2C22, C2.7(D42S3), C6.D4.3C2, (C22×C6).9C22, C22.40(C22×S3), (C2×Dic3).26C22, (C3×C22⋊C4).2C2, SmallGroup(96,86)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C23.8D6
C1C3C6C2×C6C2×Dic3C4×Dic3 — C23.8D6
C3C2×C6 — C23.8D6
C1C22C22⋊C4

Generators and relations for C23.8D6
 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=d5 >

Subgroups: 122 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], C23, Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×3], C42, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×3], C2×Dic3 [×4], C2×C12 [×2], C22×C6, C422C2, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C3×C22⋊C4, C23.8D6
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], C4○D4 [×3], C22×S3, C422C2, C4○D12, D42S3 [×2], C23.8D6

Character table of C23.8D6

 class 12A2B2C2D34A4B4C4D4E4F4G4H4I6A6B6C6D6E12A12B12C12D
 size 11114222466661212222444444
ρ1111111111111111111111111    trivial
ρ21111-11-1-1111-1-11-1111-1-11-11-1    linear of order 2
ρ3111111-1-1-111-1-1-1111111-1-1-1-1    linear of order 2
ρ41111-1111-11111-1-1111-1-1-11-11    linear of order 2
ρ5111111-1-1-1-1-1111-111111-1-1-1-1    linear of order 2
ρ61111-1111-1-1-1-1-111111-1-1-11-11    linear of order 2
ρ7111111111-1-1-1-1-1-1111111111    linear of order 2
ρ81111-11-1-11-1-111-11111-1-11-11-1    linear of order 2
ρ92222-2-122-2000000-1-1-1111-11-1    orthogonal lifted from D6
ρ1022222-1-2-2-2000000-1-1-1-1-11111    orthogonal lifted from D6
ρ112222-2-1-2-22000000-1-1-111-11-11    orthogonal lifted from D6
ρ1222222-1222000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ132-22-202-2i2i0000000-2-22000-2i02i    complex lifted from C4○D4
ρ142-2-2202000-2i2i00002-2-2000000    complex lifted from C4○D4
ρ152-22-2022i-2i0000000-2-220002i0-2i    complex lifted from C4○D4
ρ1622-2-20200000-2i2i00-22-2000000    complex lifted from C4○D4
ρ172-2-22020002i-2i00002-2-2000000    complex lifted from C4○D4
ρ1822-2-202000002i-2i00-22-2000000    complex lifted from C4○D4
ρ192-22-20-12i-2i000000011-1-3--33-i-3i    complex lifted from C4○D12
ρ202-22-20-12i-2i000000011-1--3-3-3-i3i    complex lifted from C4○D12
ρ212-22-20-1-2i2i000000011-1--3-33i-3-i    complex lifted from C4○D12
ρ222-22-20-1-2i2i000000011-1-3--3-3i3-i    complex lifted from C4○D12
ρ2344-4-40-20000000002-22000000    symplectic lifted from D42S3, Schur index 2
ρ244-4-440-2000000000-222000000    symplectic lifted from D42S3, Schur index 2

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

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

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

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

C23.8D6 is a maximal subgroup of
C24.41D6  C24.42D6  C42.89D6  C42.93D6  C42.94D6  C42.98D6  C42.102D6  C42.104D6  C42.105D6  C42.106D6  C42.229D6  C42.113D6  C42.115D6  C42.118D6  C24.43D6  C24.46D6  C24.47D6  C4⋊C4.178D6  C6.342+ 1+4  C6.702- 1+4  C6.712- 1+4  C6.422+ 1+4  C6.432+ 1+4  C6.1152+ 1+4  C6.482+ 1+4  C6.152- 1+4  C6.202- 1+4  C6.212- 1+4  C6.222- 1+4  C6.232- 1+4  C6.252- 1+4  C4⋊C4.197D6  C6.802- 1+4  C6.812- 1+4  C6.612+ 1+4  C6.622+ 1+4  C6.632+ 1+4  C6.642+ 1+4  C6.652+ 1+4  C6.852- 1+4  C42.137D6  C42.139D6  C42.140D6  C4222D6  C4223D6  C42.234D6  C42.144D6  C42.159D6  C42.160D6  S3×C422C2  C4226D6  C42.162D6  C42.165D6  C23.8D18  C62.28C23  C62.29C23  C62.31C23  C62.32C23  C62.98C23  C62.223C23  (C2×C12).D10  (C2×C60).C22  (C4×Dic3)⋊D5  (C4×Dic15)⋊C2  C23.13(S3×D5)  C23.14(S3×D5)  C23.8D30
C23.8D6 is a maximal quotient of
C3⋊(C425C4)  C6.(C4×D4)  C2.(C4×D12)  Dic3⋊C4⋊C4  (C2×Dic3).9D4  (C2×C4).17D12  (C2×C4).Dic6  (C22×C4).30D6  C24.14D6  C24.15D6  C24.17D6  C24.19D6  C24.20D6  C24.21D6  C23.8D18  C62.28C23  C62.29C23  C62.31C23  C62.32C23  C62.98C23  C62.223C23  (C2×C12).D10  (C2×C60).C22  (C4×Dic3)⋊D5  (C4×Dic15)⋊C2  C23.13(S3×D5)  C23.14(S3×D5)  C23.8D30

Matrix representation of C23.8D6 in GL4(𝔽13) generated by

1000
01200
0010
001212
,
12000
01200
00120
00012
,
1000
0100
00120
00012
,
6000
0200
00510
0008
,
0200
6000
001211
0001
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,12,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[6,0,0,0,0,2,0,0,0,0,5,0,0,0,10,8],[0,6,0,0,2,0,0,0,0,0,12,0,0,0,11,1] >;

C23.8D6 in GAP, Magma, Sage, TeX

C_2^3._8D_6
% in TeX

G:=Group("C2^3.8D6");
// GroupNames label

G:=SmallGroup(96,86);
// by ID

G=gap.SmallGroup(96,86);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,55,506,188,2309]);
// 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=d^5>;
// generators/relations

Export

Character table of C23.8D6 in TeX

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