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G = Q8.Dic6order 192 = 26·3

1st non-split extension by Q8 of Dic6 acting via Dic6/C4=S3

non-abelian, soluble

Aliases: Q8.1Dic6, SL2(𝔽3).Q8, (C2×C4).2S4, (C4×Q8).5S3, (C2×Q8).5D6, C2.4(A4⋊Q8), Q8⋊Dic3.1C2, C22.33(C2×S4), C2.4(C4.6S4), C2.4(Q8.D6), (C4×SL2(𝔽3)).2C2, (C2×SL2(𝔽3)).5C22, SmallGroup(192,948)

Series: Derived Chief Lower central Upper central

C1C2Q8C2×SL2(𝔽3) — Q8.Dic6
C1C2Q8SL2(𝔽3)C2×SL2(𝔽3)Q8⋊Dic3 — Q8.Dic6
SL2(𝔽3)C2×SL2(𝔽3) — Q8.Dic6
C1C22C2×C4

Generators and relations for Q8.Dic6
 G = < a,b,c,d | a4=c12=1, b2=a2, d2=a2c6, bab-1=dbd-1=a-1, cac-1=b, dad-1=a2b, cbc-1=ab, dcd-1=a2c-1 >

4C3
2C4
3C4
3C4
6C4
12C4
12C4
4C6
4C6
4C6
3Q8
3C2×C4
3C2×C4
6C8
6C8
6C2×C4
6C2×C4
4C2×C6
4Dic3
4Dic3
8C12
8Dic3
3C4⋊C4
3C2×C8
3C4⋊C4
3C2×C8
3C42
3C4⋊C4
6C4⋊C4
6C4⋊C4
4C2×Dic3
4C2×Dic3
4C2×C12
3C4.Q8
3Q8⋊C4
3C2.D8
3C4⋊C8
3C42.C2
3Q8⋊C4
4Dic3⋊C4
3Q8.Q8

Character table of Q8.Dic6

 class 12A2B2C34A4B4C4D4E4F4G6A6B6C8A8B8C8D12A12B12C12D
 size 111182266122424888121212128888
ρ111111111111111111111111    trivial
ρ211111-1-111-1-11111-11-11-1-1-1-1    linear of order 2
ρ311111-1-111-11-11111-11-1-1-1-1-1    linear of order 2
ρ41111111111-1-1111-1-1-1-11111    linear of order 2
ρ52222-1-2-222-200-1-1-100001111    orthogonal lifted from D6
ρ62222-12222200-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ722-2-2200-22000-2-2200000000    symplectic lifted from Q8, Schur index 2
ρ822-2-2-100-2200011-1000033-3-3    symplectic lifted from Dic6, Schur index 2
ρ922-2-2-100-2200011-10000-3-333    symplectic lifted from Dic6, Schur index 2
ρ102-22-2-1-2i2i000001-11-2--22-2-ii-ii    complex lifted from C4.6S4
ρ112-22-2-12i-2i000001-112--2-2-2i-ii-i    complex lifted from C4.6S4
ρ122-22-2-1-2i2i000001-112-2-2--2-ii-ii    complex lifted from C4.6S4
ρ132-22-2-12i-2i000001-11-2-22--2i-ii-i    complex lifted from C4.6S4
ρ143333033-1-1-1-1-100011110000    orthogonal lifted from S4
ρ1533330-3-3-1-11-110001-11-10000    orthogonal lifted from C2×S4
ρ1633330-3-3-1-111-1000-11-110000    orthogonal lifted from C2×S4
ρ173333033-1-1-111000-1-1-1-10000    orthogonal lifted from S4
ρ184-4-44-20000000-22200000000    symplectic lifted from Q8.D6, Schur index 2
ρ194-4-44100000001-1-10000-3--3--3-3    complex lifted from Q8.D6
ρ204-4-44100000001-1-10000--3-3-3--3    complex lifted from Q8.D6
ρ214-44-414i-4i00000-11-10000-ii-ii    complex lifted from C4.6S4
ρ224-44-41-4i4i00000-11-10000i-ii-i    complex lifted from C4.6S4
ρ2366-6-60002-200000000000000    symplectic lifted from A4⋊Q8, Schur index 2

Smallest permutation representation of Q8.Dic6
On 64 points
Generators in S64
(1 23 12 33)(2 20 9 30)(3 17 10 39)(4 26 11 36)(5 60 16 41)(6 57 13 50)(7 54 14 47)(8 63 15 44)(18 22 40 32)(19 37 29 27)(21 25 31 35)(24 28 34 38)(42 46 61 53)(43 58 62 51)(45 49 64 56)(48 52 55 59)
(1 19 12 29)(2 28 9 38)(3 25 10 35)(4 22 11 32)(5 56 16 49)(6 53 13 46)(7 62 14 43)(8 59 15 52)(17 21 39 31)(18 36 40 26)(20 24 30 34)(23 27 33 37)(41 45 60 64)(42 57 61 50)(44 48 63 55)(47 51 54 58)
(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 49 50 51 52)(53 54 55 56 57 58 59 60 61 62 63 64)
(1 13 10 8)(2 5 11 14)(3 15 12 6)(4 7 9 16)(17 59 33 46)(18 51 34 64)(19 57 35 44)(20 49 36 62)(21 55 37 42)(22 47 38 60)(23 53 39 52)(24 45 40 58)(25 63 29 50)(26 43 30 56)(27 61 31 48)(28 41 32 54)

G:=sub<Sym(64)| (1,23,12,33)(2,20,9,30)(3,17,10,39)(4,26,11,36)(5,60,16,41)(6,57,13,50)(7,54,14,47)(8,63,15,44)(18,22,40,32)(19,37,29,27)(21,25,31,35)(24,28,34,38)(42,46,61,53)(43,58,62,51)(45,49,64,56)(48,52,55,59), (1,19,12,29)(2,28,9,38)(3,25,10,35)(4,22,11,32)(5,56,16,49)(6,53,13,46)(7,62,14,43)(8,59,15,52)(17,21,39,31)(18,36,40,26)(20,24,30,34)(23,27,33,37)(41,45,60,64)(42,57,61,50)(44,48,63,55)(47,51,54,58), (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)(41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64), (1,13,10,8)(2,5,11,14)(3,15,12,6)(4,7,9,16)(17,59,33,46)(18,51,34,64)(19,57,35,44)(20,49,36,62)(21,55,37,42)(22,47,38,60)(23,53,39,52)(24,45,40,58)(25,63,29,50)(26,43,30,56)(27,61,31,48)(28,41,32,54)>;

G:=Group( (1,23,12,33)(2,20,9,30)(3,17,10,39)(4,26,11,36)(5,60,16,41)(6,57,13,50)(7,54,14,47)(8,63,15,44)(18,22,40,32)(19,37,29,27)(21,25,31,35)(24,28,34,38)(42,46,61,53)(43,58,62,51)(45,49,64,56)(48,52,55,59), (1,19,12,29)(2,28,9,38)(3,25,10,35)(4,22,11,32)(5,56,16,49)(6,53,13,46)(7,62,14,43)(8,59,15,52)(17,21,39,31)(18,36,40,26)(20,24,30,34)(23,27,33,37)(41,45,60,64)(42,57,61,50)(44,48,63,55)(47,51,54,58), (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)(41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64), (1,13,10,8)(2,5,11,14)(3,15,12,6)(4,7,9,16)(17,59,33,46)(18,51,34,64)(19,57,35,44)(20,49,36,62)(21,55,37,42)(22,47,38,60)(23,53,39,52)(24,45,40,58)(25,63,29,50)(26,43,30,56)(27,61,31,48)(28,41,32,54) );

G=PermutationGroup([[(1,23,12,33),(2,20,9,30),(3,17,10,39),(4,26,11,36),(5,60,16,41),(6,57,13,50),(7,54,14,47),(8,63,15,44),(18,22,40,32),(19,37,29,27),(21,25,31,35),(24,28,34,38),(42,46,61,53),(43,58,62,51),(45,49,64,56),(48,52,55,59)], [(1,19,12,29),(2,28,9,38),(3,25,10,35),(4,22,11,32),(5,56,16,49),(6,53,13,46),(7,62,14,43),(8,59,15,52),(17,21,39,31),(18,36,40,26),(20,24,30,34),(23,27,33,37),(41,45,60,64),(42,57,61,50),(44,48,63,55),(47,51,54,58)], [(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),(41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64)], [(1,13,10,8),(2,5,11,14),(3,15,12,6),(4,7,9,16),(17,59,33,46),(18,51,34,64),(19,57,35,44),(20,49,36,62),(21,55,37,42),(22,47,38,60),(23,53,39,52),(24,45,40,58),(25,63,29,50),(26,43,30,56),(27,61,31,48),(28,41,32,54)]])

Matrix representation of Q8.Dic6 in GL4(𝔽73) generated by

1000
0100
006071
001213
,
1000
0100
00161
006172
,
59700
666600
006814
004632
,
207100
185300
001113
00262
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,60,12,0,0,71,13],[1,0,0,0,0,1,0,0,0,0,1,61,0,0,61,72],[59,66,0,0,7,66,0,0,0,0,68,46,0,0,14,32],[20,18,0,0,71,53,0,0,0,0,11,2,0,0,13,62] >;

Q8.Dic6 in GAP, Magma, Sage, TeX

Q_8.{\rm Dic}_6
% in TeX

G:=Group("Q8.Dic6");
// GroupNames label

G:=SmallGroup(192,948);
// by ID

G=gap.SmallGroup(192,948);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,28,85,708,451,1684,655,172,1013,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^12=1,b^2=a^2,d^2=a^2*c^6,b*a*b^-1=d*b*d^-1=a^-1,c*a*c^-1=b,d*a*d^-1=a^2*b,c*b*c^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations

Export

Subgroup lattice of Q8.Dic6 in TeX
Character table of Q8.Dic6 in TeX

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