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## G = Q8×F8order 448 = 26·7

### Direct product of Q8 and F8

Aliases: Q8×F8, C4.(C2×F8), (C4×F8).C2, (Q8×C23)⋊C7, C23⋊(C7×Q8), (C23×C4).C14, C2.3(C22×F8), C24.3(C2×C14), (C2×F8).3C22, SmallGroup(448,1364)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — Q8×F8
 Chief series C1 — C23 — C24 — C2×F8 — C4×F8 — Q8×F8
 Lower central C23 — C24 — Q8×F8
 Upper central C1 — C2 — Q8

Generators and relations for Q8×F8
G = < a,b,c,d,e,f | a4=c2=d2=e2=f7=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=ed=de, fdf-1=c, fef-1=d >

Subgroups: 479 in 83 conjugacy classes, 18 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C7, C2×C4, Q8, Q8, C23, C23, C14, C22×C4, C2×Q8, C24, C28, C23×C4, C22×Q8, C7×Q8, F8, Q8×C23, C2×F8, C4×F8, Q8×F8
Quotients: C1, C2, C22, C7, Q8, C14, C2×C14, C7×Q8, F8, C2×F8, C22×F8, Q8×F8

Smallest permutation representation of Q8×F8
On 56 points
Generators in S56
(1 31 17 24)(2 32 18 25)(3 33 19 26)(4 34 20 27)(5 35 21 28)(6 29 15 22)(7 30 16 23)(8 41 55 48)(9 42 56 49)(10 36 50 43)(11 37 51 44)(12 38 52 45)(13 39 53 46)(14 40 54 47)
(1 45 17 38)(2 46 18 39)(3 47 19 40)(4 48 20 41)(5 49 21 42)(6 43 15 36)(7 44 16 37)(8 34 55 27)(9 35 56 28)(10 29 50 22)(11 30 51 23)(12 31 52 24)(13 32 53 25)(14 33 54 26)
(1 17)(4 20)(6 15)(7 16)(8 55)(10 50)(11 51)(12 52)(22 29)(23 30)(24 31)(27 34)(36 43)(37 44)(38 45)(41 48)
(1 17)(2 18)(5 21)(7 16)(9 56)(11 51)(12 52)(13 53)(23 30)(24 31)(25 32)(28 35)(37 44)(38 45)(39 46)(42 49)
(1 17)(2 18)(3 19)(6 15)(10 50)(12 52)(13 53)(14 54)(22 29)(24 31)(25 32)(26 33)(36 43)(38 45)(39 46)(40 47)
(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)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 7A ··· 7F 14A ··· 14F 28A ··· 28R order 1 2 2 2 4 4 4 4 4 4 7 ··· 7 14 ··· 14 28 ··· 28 size 1 1 7 7 2 2 2 14 14 14 8 ··· 8 8 ··· 8 16 ··· 16

40 irreducible representations

 dim 1 1 1 1 14 2 2 7 7 type + + - - + + image C1 C2 C7 C14 Q8×F8 Q8 C7×Q8 F8 C2×F8 kernel Q8×F8 C4×F8 Q8×C23 C23×C4 C1 F8 C23 Q8 C4 # reps 1 3 6 18 1 1 6 1 3

Matrix representation of Q8×F8 in GL9(𝔽29)

 15 21 0 0 0 0 0 0 0 21 14 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 0 28
,
 0 28 0 0 0 0 0 0 0 1 0 0 0 0 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 0 1 0 0 0 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 0 1 0 0 0 0 0 0 0 0 0 1
,
 1 0 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 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 28 28 28 28 28 28 28 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 28 28 28 28 28 28 28 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 28 28 28 28 28 28 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 0 0 0 25 0 0 0 0 0 0 0 0 0 25 0 0 0 0 0 0 4 4 4 4 4 4 4 0 0 0 0 0 25 0 0 0 0 0 0 0 0 0 25 0 0

G:=sub<GL(9,GF(29))| [15,21,0,0,0,0,0,0,0,21,14,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,28],[0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,28,0,0,0,0,0,0,0,0,28,1,0,0,0,0,0,0,0,28,0,1,0,0,1,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,1,0,0,28,0,0,0,0,0,0,1,0,28,0,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,28,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,28,0,0,0,0,0,0,0,0,28,0,0,0,1,0,0,1,0,28,0,0,0,0,0,0,0,0,28,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,1,0,0,28,0,0,0,0,1,0,0,0,28,0,0,0,1,0,0,0,0,28,0,0,1,0,0,0,0,0,28,0,1,0,0,0,0,0,0,28,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,25,0,0,0,4,0,0,0,0,0,0,25,0,4,0,0,0,0,0,0,0,25,4,0,0,0,0,0,0,0,0,4,25,0,0,0,0,0,0,0,4,0,25,0,0,0,0,0,0,4,0,0,0,0,0,25,0,0,4,0,0] >;

Q8×F8 in GAP, Magma, Sage, TeX

Q_8\times F_8
% in TeX

G:=Group("Q8xF8");
// GroupNames label

G:=SmallGroup(448,1364);
// by ID

G=gap.SmallGroup(448,1364);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,2,2,196,421,204,998,2371,3450]);
// Polycyclic

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

׿
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