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G = Q8.14D6order 96 = 25·3

4th non-split extension by Q8 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.9D6, C12.50D4, Q8.14D6, C12.19C23, Dic6.12C22, D4.S36C2, C4○D4.4S3, C3⋊Q166C2, (C2×C6).10D4, (C2×C4).23D6, C6.61(C2×D4), C3⋊C8.4C22, C35(C8.C22), (C2×Dic6)⋊11C2, C4.25(C3⋊D4), C4.Dic310C2, C4.19(C22×S3), (C3×D4).9C22, (C3×Q8).9C22, (C2×C12).44C22, C22.6(C3⋊D4), (C3×C4○D4).3C2, C2.25(C2×C3⋊D4), SmallGroup(96,158)

Series: Derived Chief Lower central Upper central

C1C12 — Q8.14D6
C1C3C6C12Dic6C2×Dic6 — Q8.14D6
C3C6C12 — Q8.14D6
C1C2C2×C4C4○D4

Generators and relations for Q8.14D6
 G = < a,b,c,d | a4=c6=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab, dcd-1=c-1 >

Subgroups: 122 in 60 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×3], Dic3 [×2], C12 [×2], C12, C2×C6, C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3⋊C8 [×2], Dic6 [×2], Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C8.C22, C4.Dic3, D4.S3 [×2], C3⋊Q16 [×2], C2×Dic6, C3×C4○D4, Q8.14D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C3⋊D4 [×2], C22×S3, C8.C22, C2×C3⋊D4, Q8.14D6

Character table of Q8.14D6

 class 12A2B2C34A4B4C4D4E6A6B6C6D8A8B12A12B12C12D12E
 size 1124222412122444121222444
ρ1111111111111111111111    trivial
ρ211-111-11-1-111-111-11-1-1-1-11    linear of order 2
ρ311-1-11-111-111-1-1-11-1-1-1111    linear of order 2
ρ411-1-11-1111-11-1-1-1-11-1-1111    linear of order 2
ρ511-111-11-11-11-1111-1-1-1-1-11    linear of order 2
ρ6111-1111-11111-1-1-1-111-1-11    linear of order 2
ρ7111-1111-1-1-111-1-11111-1-11    linear of order 2
ρ811111111-1-11111-1-111111    linear of order 2
ρ9222-2-122-200-1-11100-1-111-1    orthogonal lifted from D6
ρ1022-22-1-22-200-11-1-1001111-1    orthogonal lifted from D6
ρ112222-122200-1-1-1-100-1-1-1-1-1    orthogonal lifted from S3
ρ1222202-2-2000220000-2-200-2    orthogonal lifted from D4
ρ1322-2022-20002-200002200-2    orthogonal lifted from D4
ρ1422-2-2-1-22200-11110011-1-1-1    orthogonal lifted from D6
ρ1522-20-12-2000-11--3-300-1-1-3--31    complex lifted from C3⋊D4
ρ162220-1-2-2000-1-1--3-30011--3-31    complex lifted from C3⋊D4
ρ1722-20-12-2000-11-3--300-1-1--3-31    complex lifted from C3⋊D4
ρ182220-1-2-2000-1-1-3--30011-3--31    complex lifted from C3⋊D4
ρ194-400400000-40000000000    symplectic lifted from C8.C22, Schur index 2
ρ204-400-200000200000-2323000    symplectic faithful, Schur index 2
ρ214-400-20000020000023-23000    symplectic faithful, Schur index 2

Smallest permutation representation of Q8.14D6
On 48 points
Generators in S48
(1 30 19 7)(2 25 20 8)(3 26 21 9)(4 27 22 10)(5 28 23 11)(6 29 24 12)(13 34 38 47)(14 35 39 48)(15 36 40 43)(16 31 41 44)(17 32 42 45)(18 33 37 46)
(1 22 19 4)(2 5 20 23)(3 24 21 6)(7 10 30 27)(8 28 25 11)(9 12 26 29)(13 44 38 31)(14 32 39 45)(15 46 40 33)(16 34 41 47)(17 48 42 35)(18 36 37 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 46 19 33)(2 45 20 32)(3 44 21 31)(4 43 22 36)(5 48 23 35)(6 47 24 34)(7 18 30 37)(8 17 25 42)(9 16 26 41)(10 15 27 40)(11 14 28 39)(12 13 29 38)

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

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

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

Q8.14D6 is a maximal subgroup of
C425D6  Q8.14D12  Q8.8D12  Q8.10D12  M4(2).13D6  M4(2).16D6  2+ 1+4.4S3  2- 1+4.2S3  D811D6  D8.10D6  D84D6  S3×C8.C22  C12.C24  D12.33C23  D12.35C23  D4.D18  C12.3S4  D12.32D6  D12.29D6  Dic6.19D6  D12.11D6  C62.75D4  C12.6S4  D20.37D6  C60.63D4  C60.8C23  D20.13D6  D4.9D30
Q8.14D6 is a maximal quotient of
C4⋊C4.232D6  C4⋊C4.233D6  C4⋊C4.237D6  C12.50D8  C42.51D6  D4.2D12  Q84Dic6  C42.59D6  C127Q16  (C2×C6).D8  Dic617D4  C3⋊C85D4  (C2×Q8).49D6  Dic6.37D4  C3⋊C8.6D4  C42.61D6  C42.62D6  C42.65D6  Dic6.4Q8  C42.68D6  C42.71D6  C4○D43Dic3  (C3×D4).32D4  D4.D18  D12.32D6  D12.29D6  Dic6.19D6  D12.11D6  C62.75D4  D20.37D6  C60.63D4  C60.8C23  D20.13D6  D4.9D30

Matrix representation of Q8.14D6 in GL6(𝔽73)

100000
010000
0011000
00297200
003214171
001614172
,
7200000
0720000
00525130
000461459
002313219
0023134846
,
800000
71640000
00525130
004846014
0035132154
0029132127
,
72280000
010000
004014036
006941110
0056383233
002451633

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,29,32,16,0,0,10,72,14,14,0,0,0,0,1,1,0,0,0,0,71,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,52,0,23,23,0,0,51,46,13,13,0,0,3,14,2,48,0,0,0,59,19,46],[8,71,0,0,0,0,0,64,0,0,0,0,0,0,52,48,35,29,0,0,51,46,13,13,0,0,3,0,21,21,0,0,0,14,54,27],[72,0,0,0,0,0,28,1,0,0,0,0,0,0,40,69,56,24,0,0,14,41,38,5,0,0,0,11,32,16,0,0,36,0,33,33] >;

Q8.14D6 in GAP, Magma, Sage, TeX

Q_8._{14}D_6
% in TeX

G:=Group("Q8.14D6");
// GroupNames label

G:=SmallGroup(96,158);
// by ID

G=gap.SmallGroup(96,158);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,218,188,579,159,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^6=1,b^2=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of Q8.14D6 in TeX

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