Copied to
clipboard

## G = Q8×D7order 112 = 24·7

### Direct product of Q8 and D7

Aliases: Q8×D7, C4.6D14, Dic144C2, C28.6C22, C14.7C23, D14.5C22, Dic7.3C22, C72(C2×Q8), (C7×Q8)⋊2C2, (C4×D7).1C2, C2.8(C22×D7), SmallGroup(112,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — Q8×D7
 Chief series C1 — C7 — C14 — D14 — C4×D7 — Q8×D7
 Lower central C7 — C14 — Q8×D7
 Upper central C1 — C2 — Q8

Generators and relations for Q8×D7
G = < a,b,c,d | a4=c7=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Character table of Q8×D7

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 7A 7B 7C 14A 14B 14C 28A 28B 28C 28D 28E 28F 28G 28H 28I size 1 1 7 7 2 2 2 14 14 14 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 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 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 0 0 2 2 2 0 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ10 2 2 0 0 2 2 2 0 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ11 2 2 0 0 -2 -2 2 0 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ75-ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 orthogonal lifted from D14 ρ12 2 2 0 0 2 -2 -2 0 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ76+ζ7 ζ74+ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 orthogonal lifted from D14 ρ13 2 2 0 0 2 -2 -2 0 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ75+ζ72 ζ76+ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 orthogonal lifted from D14 ρ14 2 2 0 0 2 2 2 0 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ15 2 2 0 0 -2 2 -2 0 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ76-ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ74-ζ73 -ζ75-ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D14 ρ16 2 2 0 0 -2 2 -2 0 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ75-ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 -ζ74-ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D14 ρ17 2 2 0 0 -2 2 -2 0 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ75-ζ72 -ζ76-ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D14 ρ18 2 2 0 0 -2 -2 2 0 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ76-ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal lifted from D14 ρ19 2 2 0 0 2 -2 -2 0 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ74+ζ73 ζ75+ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal lifted from D14 ρ20 2 2 0 0 -2 -2 2 0 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 orthogonal lifted from D14 ρ21 2 -2 2 -2 0 0 0 0 0 0 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ22 2 -2 -2 2 0 0 0 0 0 0 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ23 4 -4 0 0 0 0 0 0 0 0 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ24 4 -4 0 0 0 0 0 0 0 0 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ25 4 -4 0 0 0 0 0 0 0 0 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of Q8×D7
On 56 points
Generators in S56
(1 27 13 20)(2 28 14 21)(3 22 8 15)(4 23 9 16)(5 24 10 17)(6 25 11 18)(7 26 12 19)(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 41 13 34)(2 42 14 35)(3 36 8 29)(4 37 9 30)(5 38 10 31)(6 39 11 32)(7 40 12 33)(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 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)

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

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

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

Q8×D7 is a maximal subgroup of
SD16⋊D7  Q16⋊D7  Q8.10D14  D4.10D14  Q8⋊F7  D21⋊Q8
Q8×D7 is a maximal quotient of
Dic73Q8  C28⋊Q8  Dic7.Q8  D14⋊Q8  D142Q8  Dic7⋊Q8  D143Q8  D21⋊Q8

Matrix representation of Q8×D7 in GL4(𝔽29) generated by

 28 0 0 0 0 28 0 0 0 0 0 1 0 0 28 0
,
 1 0 0 0 0 1 0 0 0 0 11 9 0 0 9 18
,
 0 1 0 0 28 7 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 28 0 0 0 0 28
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,0,28,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,11,9,0,0,9,18],[0,28,0,0,1,7,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,28,0,0,0,0,28] >;

Q8×D7 in GAP, Magma, Sage, TeX

Q_8\times D_7
% in TeX

G:=Group("Q8xD7");
// GroupNames label

G:=SmallGroup(112,33);
// by ID

G=gap.SmallGroup(112,33);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,46,97,42,2404]);
// Polycyclic

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

Export

׿
×
𝔽