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## G = Q8×F7order 336 = 24·3·7

### Direct product of Q8 and F7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — Q8×F7
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C2×F7 — C4×F7 — Q8×F7
 Lower central C7 — C14 — Q8×F7
 Upper central C1 — C2 — Q8

Generators and relations for Q8×F7
G = < a,b,c,d | a4=c7=d6=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 268 in 76 conjugacy classes, 44 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C7, C2×C4, Q8, Q8, C12, C2×C6, D7, C14, C2×Q8, C7⋊C3, C2×C12, C3×Q8, Dic7, C28, D14, F7, C2×C7⋊C3, C6×Q8, Dic14, C4×D7, C7×Q8, C7⋊C12, C4×C7⋊C3, C2×F7, Q8×D7, C4.F7, C4×F7, Q8×C7⋊C3, Q8×F7
Quotients: C1, C2, C3, C22, C6, Q8, C23, C2×C6, C2×Q8, C3×Q8, C22×C6, F7, C6×Q8, C2×F7, C22×F7, Q8×F7

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

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A ··· 6F 7 12A ··· 12L 14 28A 28B 28C order 1 2 2 2 3 3 4 4 4 4 4 4 6 ··· 6 7 12 ··· 12 14 28 28 28 size 1 1 7 7 7 7 2 2 2 14 14 14 7 ··· 7 6 14 ··· 14 6 12 12 12

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 12 2 2 6 6 type + + + + - - + + image C1 C2 C2 C2 C3 C6 C6 C6 Q8×F7 Q8 C3×Q8 F7 C2×F7 kernel Q8×F7 C4.F7 C4×F7 Q8×C7⋊C3 Q8×D7 Dic14 C4×D7 C7×Q8 C1 F7 D7 Q8 C4 # reps 1 3 3 1 2 6 6 2 1 2 4 1 3

Matrix representation of Q8×F7 in GL8(𝔽337)

 0 1 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 138 294 0 0 0 0 0 0 294 199 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 336
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 336 336 336 336 336 336 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 209 0 0 0 0 0 0 0 0 209 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 336 336 336 336 336 336 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(337))| [0,336,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[138,294,0,0,0,0,0,0,294,199,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[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],[209,0,0,0,0,0,0,0,0,209,0,0,0,0,0,0,0,0,1,0,0,0,336,0,0,0,0,0,0,1,336,0,0,0,0,0,0,0,336,0,0,0,0,0,1,0,336,0,0,0,0,0,0,0,336,1,0,0,0,1,0,0,336,0] >;

Q8×F7 in GAP, Magma, Sage, TeX

Q_8\times F_7
% in TeX

G:=Group("Q8xF7");
// GroupNames label

G:=SmallGroup(336,127);
// by ID

G=gap.SmallGroup(336,127);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,151,260,122,10373,887]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations

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