Copied to
clipboard

G = C23.12D6order 96 = 25·3

7th non-split extension by C23 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.17D4, C23.12D6, (C2×D4).6S3, (C6×D4).5C2, (C2×C4).50D6, C6.48(C2×D4), (C4×Dic3)⋊5C2, C33(C4.4D4), C4.7(C3⋊D4), (C2×Dic6)⋊10C2, C6.30(C4○D4), C6.D49C2, (C2×C6).51C23, (C2×C12).33C22, C2.16(D42S3), C22.58(C22×S3), (C22×C6).19C22, (C2×Dic3).18C22, C2.12(C2×C3⋊D4), SmallGroup(96,143)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C23.12D6
C1C3C6C2×C6C2×Dic3C4×Dic3 — C23.12D6
C3C2×C6 — C23.12D6
C1C22C2×D4

Generators and relations for C23.12D6
 G = < a,b,c,d,e | a2=b2=c2=1, d6=b, e2=cb=bc, dad-1=ab=ba, eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >

Subgroups: 162 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×6], C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic6 [×2], C2×Dic3 [×4], C2×C12, C3×D4 [×2], C22×C6 [×2], C4.4D4, C4×Dic3, C6.D4 [×4], C2×Dic6, C6×D4, C23.12D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C4.4D4, D42S3 [×2], C2×C3⋊D4, C23.12D6

Character table of C23.12D6

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C6D6E6F6G12A12B
 size 11114422266661212222444444
ρ1111111111111111111111111    trivial
ρ21111-1-1111-1-1-1-111111-1-1-1-111    linear of order 2
ρ311111-11-1-1-1-1111-11111-1-11-1-1    linear of order 2
ρ41111-111-1-111-1-11-1111-111-1-1-1    linear of order 2
ρ5111111111-1-1-1-1-1-1111111111    linear of order 2
ρ61111-1-11111111-1-1111-1-1-1-111    linear of order 2
ρ711111-11-1-111-1-1-111111-1-11-1-1    linear of order 2
ρ81111-111-1-1-1-111-11111-111-1-1-1    linear of order 2
ρ922-2-20022-2000000-22-200002-2    orthogonal lifted from D4
ρ10222222-122000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1122-2-2002-22000000-22-20000-22    orthogonal lifted from D4
ρ122222-22-1-2-2000000-1-1-11-1-1111    orthogonal lifted from D6
ρ132222-2-2-122000000-1-1-11111-1-1    orthogonal lifted from D6
ρ1422222-2-1-2-2000000-1-1-1-111-111    orthogonal lifted from D6
ρ1522-2-200-12-20000001-11-3--3-3--3-11    complex lifted from C3⋊D4
ρ1622-2-200-12-20000001-11--3-3--3-3-11    complex lifted from C3⋊D4
ρ1722-2-200-1-220000001-11-3-3--3--31-1    complex lifted from C3⋊D4
ρ1822-2-200-1-220000001-11--3--3-3-31-1    complex lifted from C3⋊D4
ρ192-2-2200200002i-2i00-2-22000000    complex lifted from C4○D4
ρ202-2-220020000-2i2i00-2-22000000    complex lifted from C4○D4
ρ212-22-200200-2i2i00002-2-2000000    complex lifted from C4○D4
ρ222-22-2002002i-2i00002-2-2000000    complex lifted from C4○D4
ρ234-4-4400-20000000022-2000000    symplectic lifted from D42S3, Schur index 2
ρ244-44-400-200000000-222000000    symplectic lifted from D42S3, Schur index 2

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

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

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

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

C23.12D6 is a maximal subgroup of
C23.D12  M4(2)⋊D6  C12⋊Q8⋊C2  Dic6.D4  (C2×C8).200D6  D12.D4  (C6×D8).C2  C2411D4  C24.22D4  (C3×Q8).D4  C24.31D4  C24.43D4  D12.38D4  2+ 1+4.4S3  C42.106D6  C42.229D6  C42.114D6  C42.115D6  C24.43D6  C24.46D6  C249D6  Dic619D4  C4⋊C4.178D6  C6.712- 1+4  D1220D4  C6.422+ 1+4  C6.452+ 1+4  C6.492+ 1+4  C6.812- 1+4  C6.622+ 1+4  C6.652+ 1+4  C42.139D6  S3×C4.4D4  C42.141D6  C42.166D6  C4228D6  C42.238D6  C24.52D6  C24.53D6  C6.1052- 1+4  C6.1072- 1+4  (C2×C12)⋊17D4  C36.17D4  C12.27D12  C62.33C23  C62.101C23  C62.254C23  C60.44D4  C60.88D4  C6.(D4×D5)  C60.17D4
C23.12D6 is a maximal quotient of
C24.15D6  C232Dic6  C24.21D6  C4.(D6⋊C4)  (C4×Dic3)⋊9C4  (C2×C12).288D4  C42.62D6  C42.213D6  C12.16D8  C42.72D6  C12.9Q16  C42.77D6  C24.30D6  C24.31D6  C36.17D4  C12.27D12  C62.33C23  C62.101C23  C62.254C23  C60.44D4  C60.88D4  C6.(D4×D5)  C60.17D4

Matrix representation of C23.12D6 in GL4(𝔽13) generated by

1000
01200
0010
00012
,
1000
0100
00120
00012
,
12000
01200
0010
0001
,
9000
0300
00012
0010
,
0300
4000
0050
0005
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,3,0,0,0,0,0,1,0,0,12,0],[0,4,0,0,3,0,0,0,0,0,5,0,0,0,0,5] >;

C23.12D6 in GAP, Magma, Sage, TeX

C_2^3._{12}D_6
% in TeX

G:=Group("C2^3.12D6");
// GroupNames label

G:=SmallGroup(96,143);
// by ID

G=gap.SmallGroup(96,143);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,55,506,116,2309]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=b,e^2=c*b=b*c,d*a*d^-1=a*b=b*a,e*a*e^-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 C23.12D6 in TeX

׿
×
𝔽