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G = Q8×C7⋊C3order 168 = 23·3·7

Direct product of Q8 and C7⋊C3

Aliases: Q8×C7⋊C3, C28.3C6, C72(C3×Q8), (C7×Q8)⋊3C3, C14.8(C2×C6), C4.(C2×C7⋊C3), (C4×C7⋊C3).3C2, C2.3(C22×C7⋊C3), (C2×C7⋊C3).8C22, SmallGroup(168,21)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — Q8×C7⋊C3
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C4×C7⋊C3 — Q8×C7⋊C3
 Lower central C7 — C14 — Q8×C7⋊C3
 Upper central C1 — C2 — Q8

Generators and relations for Q8×C7⋊C3
G = < a,b,c,d | a4=c7=d3=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Character table of Q8×C7⋊C3

 class 1 2 3A 3B 4A 4B 4C 6A 6B 7A 7B 12A 12B 12C 12D 12E 12F 14A 14B 28A 28B 28C 28D 28E 28F size 1 1 7 7 2 2 2 7 7 3 3 14 14 14 14 14 14 3 3 6 6 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 -1 1 -1 1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ4 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 -1 1 -1 linear of order 2 ρ5 1 1 ζ32 ζ3 1 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ6 1 1 ζ32 ζ3 -1 -1 1 ζ3 ζ32 1 1 ζ65 ζ6 ζ65 ζ3 ζ6 ζ32 1 1 -1 1 -1 -1 1 -1 linear of order 6 ρ7 1 1 ζ3 ζ32 -1 -1 1 ζ32 ζ3 1 1 ζ6 ζ65 ζ6 ζ32 ζ65 ζ3 1 1 -1 1 -1 -1 1 -1 linear of order 6 ρ8 1 1 ζ3 ζ32 1 -1 -1 ζ32 ζ3 1 1 ζ32 ζ65 ζ6 ζ6 ζ3 ζ65 1 1 -1 -1 1 1 -1 -1 linear of order 6 ρ9 1 1 ζ32 ζ3 -1 1 -1 ζ3 ζ32 1 1 ζ65 ζ32 ζ3 ζ65 ζ6 ζ6 1 1 1 -1 -1 -1 -1 1 linear of order 6 ρ10 1 1 ζ32 ζ3 1 -1 -1 ζ3 ζ32 1 1 ζ3 ζ6 ζ65 ζ65 ζ32 ζ6 1 1 -1 -1 1 1 -1 -1 linear of order 6 ρ11 1 1 ζ3 ζ32 -1 1 -1 ζ32 ζ3 1 1 ζ6 ζ3 ζ32 ζ6 ζ65 ζ65 1 1 1 -1 -1 -1 -1 1 linear of order 6 ρ12 1 1 ζ3 ζ32 1 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ13 2 -2 2 2 0 0 0 -2 -2 2 2 0 0 0 0 0 0 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 -1-√-3 -1+√-3 0 0 0 1-√-3 1+√-3 2 2 0 0 0 0 0 0 -2 -2 0 0 0 0 0 0 complex lifted from C3×Q8 ρ15 2 -2 -1+√-3 -1-√-3 0 0 0 1+√-3 1-√-3 2 2 0 0 0 0 0 0 -2 -2 0 0 0 0 0 0 complex lifted from C3×Q8 ρ16 3 3 0 0 -3 3 -3 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -1-√-7/2 1+√-7/2 1-√-7/2 1+√-7/2 1-√-7/2 -1+√-7/2 complex lifted from C2×C7⋊C3 ρ17 3 3 0 0 -3 -3 3 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 1+√-7/2 -1-√-7/2 1-√-7/2 1+√-7/2 -1+√-7/2 1-√-7/2 complex lifted from C2×C7⋊C3 ρ18 3 3 0 0 3 3 3 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ19 3 3 0 0 3 -3 -3 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 1+√-7/2 1+√-7/2 -1+√-7/2 -1-√-7/2 1-√-7/2 1-√-7/2 complex lifted from C2×C7⋊C3 ρ20 3 3 0 0 3 -3 -3 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 1-√-7/2 1-√-7/2 -1-√-7/2 -1+√-7/2 1+√-7/2 1+√-7/2 complex lifted from C2×C7⋊C3 ρ21 3 3 0 0 3 3 3 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ22 3 3 0 0 -3 -3 3 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 1-√-7/2 -1+√-7/2 1+√-7/2 1-√-7/2 -1-√-7/2 1+√-7/2 complex lifted from C2×C7⋊C3 ρ23 3 3 0 0 -3 3 -3 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1+√-7/2 1-√-7/2 1+√-7/2 1-√-7/2 1+√-7/2 -1-√-7/2 complex lifted from C2×C7⋊C3 ρ24 6 -6 0 0 0 0 0 0 0 -1-√-7 -1+√-7 0 0 0 0 0 0 1-√-7 1+√-7 0 0 0 0 0 0 complex faithful, Schur index 2 ρ25 6 -6 0 0 0 0 0 0 0 -1+√-7 -1-√-7 0 0 0 0 0 0 1+√-7 1-√-7 0 0 0 0 0 0 complex faithful, Schur index 2

Smallest permutation representation of Q8×C7⋊C3
On 56 points
Generators in S56
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)
(1 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)
(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)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)(37 38 40)(39 42 41)(44 45 47)(46 49 48)(51 52 54)(53 56 55)

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

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

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

Q8×C7⋊C3 is a maximal subgroup of   Q82F7  Q8.2F7  Q83F7

Matrix representation of Q8×C7⋊C3 in GL5(𝔽337)

 336 335 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 284 133 0 0 0 288 53 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336
,
 1 0 0 0 0 0 1 0 0 0 0 0 336 124 1 0 0 0 124 1 0 0 336 125 1
,
 208 0 0 0 0 0 208 0 0 0 0 0 125 1 213 0 0 1 0 0 0 0 1 1 212

G:=sub<GL(5,GF(337))| [336,1,0,0,0,335,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[284,288,0,0,0,133,53,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[1,0,0,0,0,0,1,0,0,0,0,0,336,0,336,0,0,124,124,125,0,0,1,1,1],[208,0,0,0,0,0,208,0,0,0,0,0,125,1,1,0,0,1,0,1,0,0,213,0,212] >;

Q8×C7⋊C3 in GAP, Magma, Sage, TeX

Q_8\times C_7\rtimes C_3
% in TeX

G:=Group("Q8xC7:C3");
// GroupNames label

G:=SmallGroup(168,21);
// by ID

G=gap.SmallGroup(168,21);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-7,60,141,66,314]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^7=d^3=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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