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G = C23.14D6order 96 = 25·3

9th non-split extension by C23 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic33D4, C23.14D6, (C2×C6)⋊3D4, (C2×D4)⋊5S3, (C6×D4)⋊9C2, D6⋊C415C2, C35(C4⋊D4), (C2×C4).19D6, C6.51(C2×D4), C2.27(S3×D4), Dic3⋊C415C2, C6.32(C4○D4), C222(C3⋊D4), (C2×C6).54C23, C6.D412C2, (C2×C12).62C22, (C22×Dic3)⋊6C2, C2.18(D42S3), C22.61(C22×S3), (C22×C6).21C22, (C22×S3).11C22, (C2×Dic3).38C22, (C2×C3⋊D4)⋊6C2, C2.15(C2×C3⋊D4), SmallGroup(96,146)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C23.14D6
C1C3C6C2×C6C22×S3C2×C3⋊D4 — C23.14D6
C3C2×C6 — C23.14D6
C1C22C2×D4

Generators and relations for C23.14D6
 G = < a,b,c,d,e | a2=b2=c2=d6=1, e2=c, ab=ba, dad-1=ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 226 in 94 conjugacy classes, 35 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×5], C22, C22 [×2], C22 [×8], S3, C6 [×3], C6 [×3], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, Dic3 [×2], Dic3 [×2], C12, D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], C2×Dic3 [×3], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6 [×2], C4⋊D4, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4 [×2], C6×D4, C23.14D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, S3×D4, D42S3, C2×C3⋊D4, C23.14D6

Character table of C23.14D6

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F6A6B6C6D6E6F6G12A12B
 size 11112241224666612222444444
ρ1111111111111111111111111    trivial
ρ21111-1-1-1-111-11-111111-1-1-1-111    linear of order 2
ρ3111111-111-1-1-1-1-11111-111-1-1-1    linear of order 2
ρ41111-1-11-11-11-11-111111-1-11-1-1    linear of order 2
ρ51111-1-1111-1-11-11-11111-1-11-1-1    linear of order 2
ρ6111111-1-11-11111-1111-111-1-1-1    linear of order 2
ρ71111-1-1-11111-11-1-1111-1-1-1-111    linear of order 2
ρ81111111-111-1-1-1-1-1111111111    linear of order 2
ρ92222-2-220-1-200000-1-1-1-111-111    orthogonal lifted from D6
ρ1022222220-1200000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ11222222-20-1-200000-1-1-11-1-1111    orthogonal lifted from D6
ρ122222-2-2-20-1200000-1-1-11111-1-1    orthogonal lifted from D6
ρ132-22-2000020020-202-2-2000000    orthogonal lifted from D4
ρ1422-2-22-2002000000-22-202-2000    orthogonal lifted from D4
ρ152-22-20000200-20202-2-2000000    orthogonal lifted from D4
ρ1622-2-2-22002000000-22-20-22000    orthogonal lifted from D4
ρ1722-2-2-2200-10000001-11--31-1-3-3--3    complex lifted from C3⋊D4
ρ1822-2-22-200-10000001-11-3-11--3-3--3    complex lifted from C3⋊D4
ρ1922-2-2-2200-10000001-11-31-1--3--3-3    complex lifted from C3⋊D4
ρ2022-2-22-200-10000001-11--3-11-3--3-3    complex lifted from C3⋊D4
ρ212-2-22000020-2i02i00-2-22000000    complex lifted from C4○D4
ρ222-2-220000202i0-2i00-2-22000000    complex lifted from C4○D4
ρ234-44-40000-2000000-222000000    orthogonal lifted from S3×D4
ρ244-4-440000-200000022-2000000    symplectic lifted from D42S3, Schur index 2

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

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

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

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

C23.14D6 is a maximal subgroup of
C42.102D6  C42.104D6  C4214D6  Dic623D4  C4219D6  C42.118D6  C42.119D6  C24.67D6  C248D6  C24.44D6  C24.45D6  C24.46D6  C24.47D6  C12⋊(C4○D4)  C6.322+ 1+4  Dic619D4  Dic620D4  C6.342+ 1+4  S3×C4⋊D4  C6.372+ 1+4  C6.722- 1+4  C6.402+ 1+4  C6.422+ 1+4  C6.442+ 1+4  C6.452+ 1+4  C6.462+ 1+4  C6.1152+ 1+4  C6.472+ 1+4  C6.482+ 1+4  C6.492+ 1+4  C4⋊C4.197D6  C6.1212+ 1+4  C6.822- 1+4  C6.642+ 1+4  C6.652+ 1+4  C6.662+ 1+4  C6.852- 1+4  C6.692+ 1+4  C42.137D6  C42.138D6  C4222D6  C42.145D6  C4228D6  Dic611D4  C42.168D6  C4230D6  D4×C3⋊D4  C2412D6  C24.53D6  C6.1042- 1+4  C6.1452+ 1+4  (C2×C12)⋊17D4  C6.1082- 1+4  Dic9⋊D4  C62.55C23  Dic3⋊D12  C62.112C23  C62.113C23  C626D4  C627D4  C6214D4  Dic3⋊S4  Dic15⋊D4  Dic3⋊D20  (C6×D5)⋊D4  Dic153D4  (C2×C6)⋊D20  Dic1518D4  Dic1512D4
C23.14D6 is a maximal quotient of
C24.55D6  C24.15D6  C24.17D6  C24.20D6  C24.24D6  C24.60D6  C24.25D6  C24.27D6  Dic3⋊(C4⋊C4)  (C2×C12).54D4  D6⋊C47C4  (C2×C12).56D4  C3⋊C822D4  C4⋊D4⋊S3  C3⋊C823D4  C3⋊C85D4  C3⋊C824D4  C3⋊C86D4  C3⋊C8.29D4  C3⋊C8.6D4  Dic3⋊D8  (C6×D8).C2  Dic33SD16  Dic35SD16  (C3×D4).D4  (C3×Q8).D4  Dic33Q16  (C2×Q16)⋊S3  M4(2).D6  M4(2).13D6  M4(2).15D6  M4(2).16D6  C24.29D6  C24.31D6  C24.32D6  Dic9⋊D4  C62.55C23  Dic3⋊D12  C62.112C23  C62.113C23  C626D4  C627D4  C6214D4  Dic15⋊D4  Dic3⋊D20  (C6×D5)⋊D4  Dic153D4  (C2×C6)⋊D20  Dic1518D4  Dic1512D4

Matrix representation of C23.14D6 in GL4(𝔽13) generated by

1000
01200
00119
0042
,
1000
0100
00120
00012
,
12000
01200
0010
0001
,
0100
1000
001212
0010
,
01200
1000
001212
0001
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,11,4,0,0,9,2],[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],[0,1,0,0,1,0,0,0,0,0,12,1,0,0,12,0],[0,1,0,0,12,0,0,0,0,0,12,0,0,0,12,1] >;

C23.14D6 in GAP, Magma, Sage, TeX

C_2^3._{14}D_6
% in TeX

G:=Group("C2^3.14D6");
// GroupNames label

G:=SmallGroup(96,146);
// by ID

G=gap.SmallGroup(96,146);
# by ID

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

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

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Character table of C23.14D6 in TeX

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