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G = C23.21D6order 96 = 25·3

6th non-split extension by C23 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.21D6, C22.4D12, D6:C4:7C2, C6.6(C2xD4), (C2xC4).9D6, (C2xC6).4D4, C22:C4:6S3, C4:Dic3:5C2, C2.8(C2xD12), C6.23(C4oD4), (C2xC6).27C23, (C2xC12).3C22, (C22xDic3):2C2, C3:2(C22.D4), C2.10(D4:2S3), (C22xS3).5C22, C22.45(C22xS3), (C22xC6).16C22, (C2xDic3).28C22, (C3xC22:C4):4C2, (C2xC3:D4).5C2, SmallGroup(96,93)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C23.21D6
C1C3C6C2xC6C22xS3C2xC3:D4 — C23.21D6
C3C2xC6 — C23.21D6
C1C22C22:C4

Generators and relations for C23.21D6
 G = < a,b,c,d | a2=b2=c12=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 186 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2xC4, C2xC4, D4, C23, C23, Dic3, C12, D6, C2xC6, C2xC6, C2xC6, C22:C4, C22:C4, C4:C4, C22xC4, C2xD4, C2xDic3, C2xDic3, C2xDic3, C3:D4, C2xC12, C22xS3, C22xC6, C22.D4, C4:Dic3, D6:C4, C3xC22:C4, C22xDic3, C2xC3:D4, C23.21D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, D12, C22xS3, C22.D4, C2xD12, D4:2S3, C23.21D6

Character table of C23.21D6

 class 12A2B2C2D2E2F34A4B4C4D4E4F4G6A6B6C6D6E12A12B12C12D
 size 11112212244666612222444444
ρ1111111111111111111111111    trivial
ρ21111-1-1-11-111-11-11111-1-1-1-111    linear of order 2
ρ3111111-1111-1-1-1-1-1111111111    linear of order 2
ρ41111-1-111-11-11-11-1111-1-1-1-111    linear of order 2
ρ51111-1-1111-11-11-1-1111-1-111-1-1    linear of order 2
ρ6111111-11-1-11111-111111-1-1-1-1    linear of order 2
ρ71111-1-1-111-1-11-111111-1-111-1-1    linear of order 2
ρ811111111-1-1-1-1-1-1111111-1-1-1-1    linear of order 2
ρ92222220-1-2-200000-1-1-1-1-11111    orthogonal lifted from D6
ρ102222-2-20-1-2200000-1-1-11111-1-1    orthogonal lifted from D6
ρ112222-2-20-12-200000-1-1-111-1-111    orthogonal lifted from D6
ρ122-2-22-22020000000-22-22-20000    orthogonal lifted from D4
ρ132222220-12200000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ142-2-222-2020000000-22-2-220000    orthogonal lifted from D4
ρ152-2-22-220-100000001-11-113-3-33    orthogonal lifted from D12
ρ162-2-222-20-100000001-111-13-33-3    orthogonal lifted from D12
ρ172-2-222-20-100000001-111-1-33-33    orthogonal lifted from D12
ρ182-2-22-220-100000001-11-11-333-3    orthogonal lifted from D12
ρ1922-2-2000200-2i02i00-2-22000000    complex lifted from C4oD4
ρ202-22-200020002i0-2i02-2-2000000    complex lifted from C4oD4
ρ2122-2-20002002i0-2i00-2-22000000    complex lifted from C4oD4
ρ222-22-20002000-2i02i02-2-2000000    complex lifted from C4oD4
ρ2344-4-4000-2000000022-2000000    symplectic lifted from D4:2S3, Schur index 2
ρ244-44-4000-20000000-222000000    symplectic lifted from D4:2S3, Schur index 2

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

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

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

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

C23.21D6 is a maximal subgroup of
C23.5D12  C23:4D12  C24.42D6  C42:10D6  C42.92D6  C42.96D6  C42.102D6  D4:5D12  D4:6D12  C42.118D6  C24.67D6  C24:7D6  C24.44D6  C6.462+ 1+4  C6.1152+ 1+4  C6.472+ 1+4  C6.482+ 1+4  C4:C4.187D6  C6.532+ 1+4  C6.772- 1+4  C6.782- 1+4  C6.792- 1+4  S3xC22.D4  C6.822- 1+4  C6.1222+ 1+4  C6.662+ 1+4  C6.852- 1+4  C6.692+ 1+4  C42:22D6  C42.143D6  C42.144D6  C42.145D6  C42.161D6  C42.163D6  C42.164D6  C42.165D6  C22.4D36  D6.D12  D6.9D12  C62.57D4  C62.60D4  C62.69D4  D10.16D12  D10.17D12  C10.(C2xD12)  (C2xC10).D12  C22.D60
C23.21D6 is a maximal quotient of
C2.(C4xD12)  (C2xC4).17D12  (C22xC4).85D6  D6:C4:3C4  (C2xC4).21D12  (C2xC12).33D4  C23.39D12  C23.40D12  C23.15D12  C23.43D12  C22.D24  C23.18D12  C24.56D6  C24.58D6  C24.21D6  C24.60D6  C24.27D6  C22.4D36  D6.D12  D6.9D12  C62.57D4  C62.60D4  C62.69D4  D10.16D12  D10.17D12  C10.(C2xD12)  (C2xC10).D12  C22.D60

Matrix representation of C23.21D6 in GL6(F13)

1200000
0120000
001000
000100
000005
000080
,
100000
010000
001000
000100
0000120
0000012
,
930000
340000
001100
0012000
000001
000010
,
930000
840000
001100
0001200
000001
0000120

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,5,0],[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],[9,3,0,0,0,0,3,4,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[9,8,0,0,0,0,3,4,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;

C23.21D6 in GAP, Magma, Sage, TeX

C_2^3._{21}D_6
% in TeX

G:=Group("C2^3.21D6");
// GroupNames label

G:=SmallGroup(96,93);
// by ID

G=gap.SmallGroup(96,93);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,218,188,122,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^12=1,d^2=b,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations

Export

Character table of C23.21D6 in TeX

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