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G = Q8⋊D7order 112 = 24·7

The semidirect product of Q8 and D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8⋊D7, C73SD16, C14.9D4, C4.3D14, D28.2C2, C28.3C22, C7⋊C83C2, (C7×Q8)⋊1C2, C2.6(C7⋊D4), SmallGroup(112,16)

Series: Derived Chief Lower central Upper central

C1C28 — Q8⋊D7
C1C7C14C28D28 — Q8⋊D7
C7C14C28 — Q8⋊D7
C1C2C4Q8

Generators and relations for Q8⋊D7
 G = < a,b,c,d | a4=c7=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >

28C2
2C4
14C22
4D7
7C8
7D4
2D14
2C28
7SD16

Character table of Q8⋊D7

 class 12A2B4A4B7A7B7C8A8B14A14B14C28A28B28C28D28E28F28G28H28I
 size 1128242221414222444444444
ρ11111111111111111111111    trivial
ρ211-111111-1-1111111111111    linear of order 2
ρ311-11-1111111111-1-1-1-1-11-11    linear of order 2
ρ41111-1111-1-11111-1-1-1-1-11-11    linear of order 2
ρ5220-2022200222-200000-20-2    orthogonal lifted from D4
ρ62202-2ζ767ζ7473ζ757200ζ7473ζ767ζ7572ζ767767757275727677473ζ75727473ζ7473    orthogonal lifted from D14
ρ72202-2ζ7572ζ767ζ747300ζ767ζ7572ζ7473ζ75727572747374737572767ζ7473767ζ767    orthogonal lifted from D14
ρ822022ζ767ζ7473ζ757200ζ7473ζ767ζ7572ζ767ζ767ζ7572ζ7572ζ767ζ7473ζ7572ζ7473ζ7473    orthogonal lifted from D7
ρ922022ζ7572ζ767ζ747300ζ767ζ7572ζ7473ζ7572ζ7572ζ7473ζ7473ζ7572ζ767ζ7473ζ767ζ767    orthogonal lifted from D7
ρ102202-2ζ7473ζ7572ζ76700ζ7572ζ7473ζ767ζ7473747376776774737572ζ7677572ζ7572    orthogonal lifted from D14
ρ1122022ζ7473ζ7572ζ76700ζ7572ζ7473ζ767ζ7473ζ7473ζ767ζ767ζ7473ζ7572ζ767ζ7572ζ7572    orthogonal lifted from D7
ρ12220-20ζ7572ζ767ζ747300ζ767ζ7572ζ7473757275727473ζ7473ζ7572ζ7677473767767    complex lifted from C7⋊D4
ρ13220-20ζ7473ζ7572ζ76700ζ7572ζ7473ζ7677473ζ7473ζ7677677473ζ757276775727572    complex lifted from C7⋊D4
ρ14220-20ζ767ζ7473ζ757200ζ7473ζ767ζ7572767ζ7677572ζ7572767ζ7473757274737473    complex lifted from C7⋊D4
ρ15220-20ζ767ζ7473ζ757200ζ7473ζ767ζ7572767767ζ75727572ζ76774737572ζ74737473    complex lifted from C7⋊D4
ρ16220-20ζ7572ζ767ζ747300ζ767ζ7572ζ74737572ζ7572ζ7473747375727677473ζ767767    complex lifted from C7⋊D4
ρ17220-20ζ7473ζ7572ζ76700ζ7572ζ7473ζ76774737473767ζ767ζ74737572767ζ75727572    complex lifted from C7⋊D4
ρ182-2000222-2--2-2-2-2000000000    complex lifted from SD16
ρ192-2000222--2-2-2-2-2000000000    complex lifted from SD16
ρ204-400075+2ζ7276+2ζ774+2ζ7300-2ζ76-2ζ7-2ζ75-2ζ72-2ζ74-2ζ73000000000    orthogonal faithful, Schur index 2
ρ214-400076+2ζ774+2ζ7375+2ζ7200-2ζ74-2ζ73-2ζ76-2ζ7-2ζ75-2ζ72000000000    orthogonal faithful, Schur index 2
ρ224-400074+2ζ7375+2ζ7276+2ζ700-2ζ75-2ζ72-2ζ74-2ζ73-2ζ76-2ζ7000000000    orthogonal 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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 45)(30 44)(31 43)(32 49)(33 48)(34 47)(35 46)(36 52)(37 51)(38 50)(39 56)(40 55)(41 54)(42 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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,45)(30,44)(31,43)(32,49)(33,48)(34,47)(35,46)(36,52)(37,51)(38,50)(39,56)(40,55)(41,54)(42,53)>;

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

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

Q8⋊D7 is a maximal subgroup of
D7×SD16  D56⋊C2  Q16⋊D7  Q8.D14  C28.C23  D4⋊D14  D4.8D14  Q82F7  C28.D6  Dic6⋊D7  Q82D21  Q8⋊D21
Q8⋊D7 is a maximal quotient of
C4.Dic14  C14.D8  Q8⋊Dic7  C28.D6  Dic6⋊D7  Q82D21

Matrix representation of Q8⋊D7 in GL4(𝔽113) generated by

1000
0100
0001
001120
,
1000
0100
00100100
0010013
,
0100
112900
0010
0001
,
0100
1000
0010
000112
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,0,112,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,100,100,0,0,100,13],[0,112,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,112] >;

Q8⋊D7 in GAP, Magma, Sage, TeX

Q_8\rtimes D_7
% in TeX

G:=Group("Q8:D7");
// GroupNames label

G:=SmallGroup(112,16);
// by ID

G=gap.SmallGroup(112,16);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,61,46,182,97,42,2404]);
// Polycyclic

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

Export

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

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