Copied to
clipboard

G = Q8×D7order 112 = 24·7

Direct product of Q8 and D7

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

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

C1C14 — Q8×D7
C1C7C14D14C4×D7 — Q8×D7
C7C14 — Q8×D7
C1C2Q8

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 >

7C2
7C2
7C4
7C4
7C22
7C4
7C2×C4
7C2×C4
7Q8
7Q8
7C2×C4
7Q8
7C2×Q8

Character table of Q8×D7

 class 12A2B2C4A4B4C4D4E4F7A7B7C14A14B14C28A28B28C28D28E28F28G28H28I
 size 1177222141414222222444444444
ρ11111111111111111111111111    trivial
ρ211-1-1-11-11-11111111-1-1-1-1-1-1111    linear of order 2
ρ311-1-11-1-111-11111111-1-1-111-1-1-1    linear of order 2
ρ41111-1-111-1-1111111-1111-1-1-1-1-1    linear of order 2
ρ511111-1-1-1-111111111-1-1-111-1-1-1    linear of order 2
ρ611-1-1-1-11-111111111-1111-1-1-1-1-1    linear of order 2
ρ711-1-1111-1-1-1111111111111111    linear of order 2
ρ81111-11-1-11-1111111-1-1-1-1-1-1111    linear of order 2
ρ92200222000ζ767ζ7473ζ7572ζ7572ζ7473ζ767ζ767ζ767ζ7572ζ7473ζ7473ζ7572ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ102200222000ζ7572ζ767ζ7473ζ7473ζ767ζ7572ζ7572ζ7572ζ7473ζ767ζ767ζ7473ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ112200-2-22000ζ7572ζ767ζ7473ζ7473ζ767ζ75727572ζ7572ζ7473ζ767767747374737677572    orthogonal lifted from D14
ρ1222002-2-2000ζ7572ζ767ζ7473ζ7473ζ767ζ7572ζ757275727473767ζ767ζ747374737677572    orthogonal lifted from D14
ρ1322002-2-2000ζ7473ζ7572ζ767ζ767ζ7572ζ7473ζ747374737677572ζ7572ζ76776775727473    orthogonal lifted from D14
ρ142200222000ζ7473ζ7572ζ767ζ767ζ7572ζ7473ζ7473ζ7473ζ767ζ7572ζ7572ζ767ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ152200-22-2000ζ767ζ7473ζ7572ζ7572ζ7473ζ7677677677572747374737572ζ7572ζ7473ζ767    orthogonal lifted from D14
ρ162200-22-2000ζ7572ζ767ζ7473ζ7473ζ767ζ75727572757274737677677473ζ7473ζ767ζ7572    orthogonal lifted from D14
ρ172200-22-2000ζ7473ζ7572ζ767ζ767ζ7572ζ74737473747376775727572767ζ767ζ7572ζ7473    orthogonal lifted from D14
ρ182200-2-22000ζ767ζ7473ζ7572ζ7572ζ7473ζ767767ζ767ζ7572ζ74737473757275727473767    orthogonal lifted from D14
ρ1922002-2-2000ζ767ζ7473ζ7572ζ7572ζ7473ζ767ζ76776775727473ζ7473ζ757275727473767    orthogonal lifted from D14
ρ202200-2-22000ζ7473ζ7572ζ767ζ767ζ7572ζ74737473ζ7473ζ767ζ7572757276776775727473    orthogonal lifted from D14
ρ212-22-2000000222-2-2-2000000000    symplectic lifted from Q8, Schur index 2
ρ222-2-22000000222-2-2-2000000000    symplectic lifted from Q8, Schur index 2
ρ234-40000000076+2ζ774+2ζ7375+2ζ72-2ζ75-2ζ72-2ζ74-2ζ73-2ζ76-2ζ7000000000    symplectic faithful, Schur index 2
ρ244-40000000075+2ζ7276+2ζ774+2ζ73-2ζ74-2ζ73-2ζ76-2ζ7-2ζ75-2ζ72000000000    symplectic faithful, Schur index 2
ρ254-40000000074+2ζ7375+2ζ7276+2ζ7-2ζ76-2ζ7-2ζ75-2ζ72-2ζ74-2ζ73000000000    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

28000
02800
0001
00280
,
1000
0100
00119
00918
,
0100
28700
0010
0001
,
0100
1000
00280
00028
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

Subgroup lattice of Q8×D7 in TeX
Character table of Q8×D7 in TeX

׿
×
𝔽