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G = Q8⋊F7order 336 = 24·3·7

The semidirect product of Q8 and F7 acting via F7/D7=C3

non-abelian, soluble

Aliases: Q8⋊F7, D14.A4, D7⋊SL2(𝔽3), (Q8×D7)⋊2C3, (C7×Q8)⋊2C6, C14.A42C2, C14.2(C2×A4), C7⋊(C2×SL2(𝔽3)), C2.3(D7⋊A4), SmallGroup(336,135)

Series: Derived Chief Lower central Upper central

C1C2C7×Q8 — Q8⋊F7
C1C2C14C7×Q8C14.A4 — Q8⋊F7
C7×Q8 — Q8⋊F7
C1C2

Generators and relations for Q8⋊F7
 G = < a,b,c,d | a4=c7=d6=1, b2=a2, bab-1=a-1, ac=ca, dad-1=b, bc=cb, dbd-1=ab, dcd-1=c5 >

7C2
7C2
28C3
3C4
7C22
21C4
28C6
28C6
28C6
4C7⋊C3
21C2×C4
21Q8
28C2×C6
3Dic7
3C28
4F7
4C2×C7⋊C3
4F7
7C2×Q8
7SL2(𝔽3)
3C4×D7
3Dic14
4C2×F7
7C2×SL2(𝔽3)

Character table of Q8⋊F7

 class 12A2B2C3A3B4A4B6A6B6C6D6E6F71428A28B28C
 size 1177282864228282828282866121212
ρ11111111111111111111    trivial
ρ211-1-1111-1-11-1-1-1111111    linear of order 2
ρ31111ζ3ζ3211ζ3ζ32ζ32ζ32ζ3ζ311111    linear of order 3
ρ411-1-1ζ32ζ31-1ζ6ζ3ζ65ζ65ζ6ζ3211111    linear of order 6
ρ51111ζ32ζ311ζ32ζ3ζ3ζ3ζ32ζ3211111    linear of order 3
ρ611-1-1ζ3ζ321-1ζ65ζ32ζ6ζ6ζ65ζ311111    linear of order 6
ρ72-22-2-1-100-111-1112-2000    symplectic lifted from SL2(𝔽3), Schur index 2
ρ82-2-22-1-10011-11-112-2000    symplectic lifted from SL2(𝔽3), Schur index 2
ρ92-2-22ζ65ζ600ζ3ζ32ζ6ζ32ζ65ζ32-2000    complex lifted from SL2(𝔽3)
ρ102-2-22ζ6ζ6500ζ32ζ3ζ65ζ3ζ6ζ322-2000    complex lifted from SL2(𝔽3)
ρ112-22-2ζ6ζ6500ζ6ζ3ζ3ζ65ζ32ζ322-2000    complex lifted from SL2(𝔽3)
ρ122-22-2ζ65ζ600ζ65ζ32ζ32ζ6ζ3ζ32-2000    complex lifted from SL2(𝔽3)
ρ1333-3-300-1100000033-1-1-1    orthogonal lifted from C2×A4
ρ14333300-1-100000033-1-1-1    orthogonal lifted from A4
ρ1566000060000000-1-1-1-1-1    orthogonal lifted from F7
ρ16660000-20000000-1-175+2ζ72+174+2ζ73+176+2ζ7+1    orthogonal lifted from D7⋊A4
ρ17660000-20000000-1-176+2ζ7+175+2ζ72+174+2ζ73+1    orthogonal lifted from D7⋊A4
ρ18660000-20000000-1-174+2ζ73+176+2ζ7+175+2ζ72+1    orthogonal lifted from D7⋊A4
ρ1912-12000000000000-22000    symplectic faithful, Schur index 2

Smallest permutation representation of Q8⋊F7
On 56 points
Generators in S56
(1 25 17 31)(2 26 18 32)(3 27 19 33)(4 28 20 34)(5 22 21 35)(6 23 15 29)(7 24 16 30)(8 43 51 37)(9 44 52 38)(10 45 53 39)(11 46 54 40)(12 47 55 41)(13 48 56 42)(14 49 50 36)
(1 39 17 45)(2 40 18 46)(3 41 19 47)(4 42 20 48)(5 36 21 49)(6 37 15 43)(7 38 16 44)(8 23 51 29)(9 24 52 30)(10 25 53 31)(11 26 54 32)(12 27 55 33)(13 28 56 34)(14 22 50 35)
(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 4 3 7 5 6)(8 46 34 12 44 35)(9 49 29 11 48 33)(10 45 31)(13 47 30 14 43 32)(15 18 20 19 16 21)(22 51 40 28 55 38)(23 54 42 27 52 36)(24 50 37 26 56 41)(25 53 39)

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

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

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

Matrix representation of Q8⋊F7 in GL8(𝔽337)

0336000000
10000000
002500195219219195
001425514202424
00313118311183130
00031311831118313
002424014255142
001952192191950250
,
208209000000
209129000000
00310118313313118
002192502190195195
001422455241420
000142245524142
001951950219250219
00118313313118031
,
10000000
01000000
00336336336336336336
00100000
00010000
00001000
00000100
00000010
,
3360000000
208209000000
00100000
00000001
00000100
00010000
00336336336336336336
00000010

G:=sub<GL(8,GF(337))| [0,1,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,250,142,313,0,24,195,0,0,0,55,118,313,24,219,0,0,195,142,31,118,0,219,0,0,219,0,118,31,142,195,0,0,219,24,313,118,55,0,0,0,195,24,0,313,142,250],[208,209,0,0,0,0,0,0,209,129,0,0,0,0,0,0,0,0,31,219,142,0,195,118,0,0,0,250,24,142,195,313,0,0,118,219,55,24,0,313,0,0,313,0,24,55,219,118,0,0,313,195,142,24,250,0,0,0,118,195,0,142,219,31],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,336,1,0,0,0,0,0,0,336,0,1,0,0,0,0,0,336,0,0,1,0,0,0,0,336,0,0,0,1,0,0,0,336,0,0,0,0,1,0,0,336,0,0,0,0,0],[336,208,0,0,0,0,0,0,0,209,0,0,0,0,0,0,0,0,1,0,0,0,336,0,0,0,0,0,0,1,336,0,0,0,0,0,0,0,336,0,0,0,0,0,1,0,336,0,0,0,0,0,0,0,336,1,0,0,0,1,0,0,336,0] >;

Q8⋊F7 in GAP, Magma, Sage, TeX

Q_8\rtimes F_7
% in TeX

G:=Group("Q8:F7");
// GroupNames label

G:=SmallGroup(336,135);
// by ID

G=gap.SmallGroup(336,135);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-7,-2,116,518,225,735,357,4324,1450]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^7=d^6=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=b,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^5>;
// generators/relations

Export

Subgroup lattice of Q8⋊F7 in TeX
Character table of Q8⋊F7 in TeX

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