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G = D4⋊D7order 112 = 24·7

The semidirect product of D4 and D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4⋊D7, C72D8, D282C2, C4.1D14, C14.7D4, C28.1C22, C7⋊C81C2, (C7×D4)⋊1C2, C2.4(C7⋊D4), SmallGroup(112,14)

Series: Derived Chief Lower central Upper central

C1C28 — D4⋊D7
C1C7C14C28D28 — D4⋊D7
C7C14C28 — D4⋊D7
C1C2C4D4

Generators and relations for D4⋊D7
 G = < a,b,c,d | a4=b2=c7=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >

4C2
28C2
2C22
14C22
4D7
4C14
7C8
7D4
2D14
2C2×C14
7D8

Character table of D4⋊D7

 class 12A2B2C47A7B7C8A8B14A14B14C14D14E14F14G14H14I28A28B28C
 size 1142822221414222444444444
ρ11111111111111111111111    trivial
ρ211-111111-1-1111-1-1-1-1-1-1111    linear of order 2
ρ311-1-1111111111-1-1-1-1-1-1111    linear of order 2
ρ4111-11111-1-1111111111111    linear of order 2
ρ52200-222200222000000-2-2-2    orthogonal lifted from D4
ρ62-20002222-2-2-2-2000000000    orthogonal lifted from D8
ρ722202ζ7473ζ767ζ757200ζ7572ζ7473ζ767ζ7473ζ767ζ767ζ7473ζ7572ζ7572ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ822202ζ767ζ7572ζ747300ζ7473ζ767ζ7572ζ767ζ7572ζ7572ζ767ζ7473ζ7473ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ922-202ζ767ζ7572ζ747300ζ7473ζ767ζ75727677572757276774737473ζ767ζ7572ζ7473    orthogonal lifted from D14
ρ1022-202ζ7572ζ7473ζ76700ζ767ζ7572ζ74737572747374737572767767ζ7572ζ7473ζ767    orthogonal lifted from D14
ρ1122202ζ7572ζ7473ζ76700ζ767ζ7572ζ7473ζ7572ζ7473ζ7473ζ7572ζ767ζ767ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ1222-202ζ7473ζ767ζ757200ζ7572ζ7473ζ7677473767767747375727572ζ7473ζ767ζ7572    orthogonal lifted from D14
ρ132-2000222-22-2-2-2000000000    orthogonal lifted from D8
ρ142200-2ζ767ζ7572ζ747300ζ7473ζ767ζ7572ζ7677572ζ7572767ζ7473747376775727473    complex lifted from C7⋊D4
ρ152200-2ζ7473ζ767ζ757200ζ7572ζ7473ζ767ζ7473ζ7677677473ζ7572757274737677572    complex lifted from C7⋊D4
ρ162200-2ζ767ζ7572ζ747300ζ7473ζ767ζ7572767ζ75727572ζ7677473ζ747376775727473    complex lifted from C7⋊D4
ρ172200-2ζ7473ζ767ζ757200ζ7572ζ7473ζ7677473767ζ767ζ74737572ζ757274737677572    complex lifted from C7⋊D4
ρ182200-2ζ7572ζ7473ζ76700ζ767ζ7572ζ7473ζ7572ζ747374737572767ζ76775727473767    complex lifted from C7⋊D4
ρ192200-2ζ7572ζ7473ζ76700ζ767ζ7572ζ747375727473ζ7473ζ7572ζ76776775727473767    complex lifted from C7⋊D4
ρ204-400074+2ζ7376+2ζ775+2ζ7200-2ζ75-2ζ72-2ζ74-2ζ73-2ζ76-2ζ7000000000    orthogonal faithful
ρ214-400076+2ζ775+2ζ7274+2ζ7300-2ζ74-2ζ73-2ζ76-2ζ7-2ζ75-2ζ72000000000    orthogonal faithful
ρ224-400075+2ζ7274+2ζ7376+2ζ700-2ζ76-2ζ7-2ζ75-2ζ72-2ζ74-2ζ73000000000    orthogonal faithful

Smallest permutation representation of D4⋊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 55)(2 56)(3 50)(4 51)(5 52)(6 53)(7 54)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)
(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), (1,55)(2,56)(3,50)(4,51)(5,52)(6,53)(7,54)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42), (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), (1,55)(2,56)(3,50)(4,51)(5,52)(6,53)(7,54)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42), (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)], [(1,55),(2,56),(3,50),(4,51),(5,52),(6,53),(7,54),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,41),(28,42)], [(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)]])

D4⋊D7 is a maximal subgroup of
D7×D8  D8⋊D7  D56⋊C2  SD163D7  D4.D14  D4⋊D14  D4.8D14  D4⋊F7  C21⋊D8  C7⋊D24  D4⋊D21
D4⋊D7 is a maximal quotient of
C28.Q8  C14.D8  C7⋊D16  D8.D7  C7⋊SD32  C7⋊Q32  D4⋊Dic7  C21⋊D8  C7⋊D24  D4⋊D21

Matrix representation of D4⋊D7 in GL4(𝔽113) generated by

0100
112000
0010
0001
,
823100
313100
0010
0001
,
1000
0100
001121
008725
,
1000
011200
001120
00871
G:=sub<GL(4,GF(113))| [0,112,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[82,31,0,0,31,31,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,112,87,0,0,1,25],[1,0,0,0,0,112,0,0,0,0,112,87,0,0,0,1] >;

D4⋊D7 in GAP, Magma, Sage, TeX

D_4\rtimes D_7
% in TeX

G:=Group("D4:D7");
// GroupNames label

G:=SmallGroup(112,14);
// by ID

G=gap.SmallGroup(112,14);
# by ID

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

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

Export

Subgroup lattice of D4⋊D7 in TeX
Character table of D4⋊D7 in TeX

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