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## G = D4×D7order 112 = 24·7

### Direct product of D4 and D7

Aliases: D4×D7, C41D14, C28⋊C22, D283C2, C221D14, D142C22, C14.5C23, Dic71C22, C72(C2×D4), (C4×D7)⋊1C2, (C7×D4)⋊2C2, C7⋊D41C2, (C2×C14)⋊C22, (C22×D7)⋊2C2, C2.6(C22×D7), SmallGroup(112,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D4×D7
 Chief series C1 — C7 — C14 — D14 — C22×D7 — D4×D7
 Lower central C7 — C14 — D4×D7
 Upper central C1 — C2 — D4

Generators and relations for D4×D7
G = < a,b,c,d | a4=b2=c7=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 220 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2 [×6], C4, C4, C22 [×2], C22 [×7], C7, C2×C4, D4, D4 [×3], C23 [×2], D7 [×2], D7 [×2], C14, C14 [×2], C2×D4, Dic7, C28, D14, D14 [×2], D14 [×4], C2×C14 [×2], C4×D7, D28, C7⋊D4 [×2], C7×D4, C22×D7 [×2], D4×D7
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D7, C2×D4, D14 [×3], C22×D7, D4×D7

Character table of D4×D7

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 7A 7B 7C 14A 14B 14C 14D 14E 14F 14G 14H 14I 28A 28B 28C size 1 1 2 2 7 7 14 14 2 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 0 0 0 0 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 0 0 2 -2 0 0 0 0 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 2 0 0 0 0 -2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ76-ζ7 -ζ74-ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 orthogonal lifted from D14 ρ12 2 2 2 2 0 0 0 0 2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ13 2 2 2 -2 0 0 0 0 -2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ75+ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 orthogonal lifted from D14 ρ14 2 2 -2 2 0 0 0 0 -2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ75-ζ72 -ζ76-ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 orthogonal lifted from D14 ρ15 2 2 -2 -2 0 0 0 0 2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ76-ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D14 ρ16 2 2 2 -2 0 0 0 0 -2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ74+ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 orthogonal lifted from D14 ρ17 2 2 2 -2 0 0 0 0 -2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ76+ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal lifted from D14 ρ18 2 2 2 2 0 0 0 0 2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ19 2 2 -2 2 0 0 0 0 -2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ75-ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal lifted from D14 ρ20 2 2 -2 -2 0 0 0 0 2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ75-ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D14 ρ21 2 2 2 2 0 0 0 0 2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ22 2 2 -2 -2 0 0 0 0 2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ74-ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D14 ρ23 4 -4 0 0 0 0 0 0 0 0 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ24 4 -4 0 0 0 0 0 0 0 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ25 4 -4 0 0 0 0 0 0 0 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of D4×D7
On 28 points - transitive group 28T18
Generators in S28
(1 20 13 27)(2 21 14 28)(3 15 8 22)(4 16 9 23)(5 17 10 24)(6 18 11 25)(7 19 12 26)
(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)
(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)
(1 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)

G:=sub<Sym(28)| (1,20,13,27)(2,21,14,28)(3,15,8,22)(4,16,9,23)(5,17,10,24)(6,18,11,25)(7,19,12,26), (15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)>;

G:=Group( (1,20,13,27)(2,21,14,28)(3,15,8,22)(4,16,9,23)(5,17,10,24)(6,18,11,25)(7,19,12,26), (15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27) );

G=PermutationGroup([(1,20,13,27),(2,21,14,28),(3,15,8,22),(4,16,9,23),(5,17,10,24),(6,18,11,25),(7,19,12,26)], [(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28)], [(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)], [(1,7),(2,6),(3,5),(8,10),(11,14),(12,13),(15,17),(18,21),(19,20),(22,24),(25,28),(26,27)])

G:=TransitiveGroup(28,18);

D4×D7 is a maximal subgroup of
D8⋊D7  D56⋊C2  D46D14  D48D14  C28⋊D6  D6⋊D14
D4×D7 is a maximal quotient of
C22⋊Dic14  Dic74D4  C22⋊D28  D14.D4  D14⋊D4  Dic7.D4  C28⋊Q8  D28⋊C4  D14.5D4  C4⋊D28  D14⋊Q8  D8⋊D7  D83D7  D56⋊C2  SD16⋊D7  SD163D7  Q16⋊D7  Q8.D14  C23⋊D14  C282D4  Dic7⋊D4  C28⋊D4  C28⋊D6  D6⋊D14

Matrix representation of D4×D7 in GL4(𝔽29) generated by

 1 0 0 0 0 1 0 0 0 0 28 17 0 0 5 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 24 28
,
 22 1 0 0 27 25 0 0 0 0 1 0 0 0 0 1
,
 25 28 0 0 15 4 0 0 0 0 28 0 0 0 0 28
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,28,5,0,0,17,1],[1,0,0,0,0,1,0,0,0,0,1,24,0,0,0,28],[22,27,0,0,1,25,0,0,0,0,1,0,0,0,0,1],[25,15,0,0,28,4,0,0,0,0,28,0,0,0,0,28] >;

D4×D7 in GAP, Magma, Sage, TeX

D_4\times D_7
% in TeX

G:=Group("D4xD7");
// GroupNames label

G:=SmallGroup(112,31);
// by ID

G=gap.SmallGroup(112,31);
# by ID

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

G:=Group<a,b,c,d|a^4=b^2=c^7=d^2=1,b*a*b=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

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