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G = D4.D7  order 112 = 24·7

The non-split extension by D4 of D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.D7, C7⋊2SD16, C4.2D14, C14.8D4, Dic14⋊2C2, C28.2C22, C7⋊C8⋊2C2, (C7×D4).1C2, C2.5(C7⋊D4), SmallGroup(112,15)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C28 — D4.D7
Chief series
C1 — C7 — C14 — C28 — Dic14 — D4.D7
Lower central
C7 — C14 — C28 — D4.D7
Upper central
C1 — C2 — C4 — D4

Generators and relations for D4.D7
 G = < a,b,c,d | a4=b2=c7=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

C1
C2
4C2
C7
C4
2C22
14C4
C14
4C14
D4
7C8
7Q8
C28
2Dic7
2C2×C14
7SD16
Dic14
C7⋊C8
C7×D4
D4.D7

Character table of D4.D7

 class 12A2B4A4B7A7B7C8A8B14A14B14C14D14E14F14G14H14I28A28B28C
 size 1142282221414222444444444
ρ11111111111111111111111    trivial
ρ211-111111-1-1111-1-1-1-1-1-1111    linear of order 2
ρ31111-1111-1-1111111111111    linear of order 2
ρ411-11-111111111-1-1-1-1-1-1111    linear of order 2
ρ5220-2022200222000000-2-2-2    orthogonal lifted from D4
ρ622-220ζ76+ζ7ζ75+ζ72ζ74+ζ7300ζ75+ζ72ζ74+ζ73ζ76+ζ7-ζ74-ζ73-ζ76-ζ7-ζ76-ζ7-ζ74-ζ73-ζ75-ζ72-ζ75-ζ72ζ74+ζ73ζ76+ζ7ζ75+ζ72    orthogonal lifted from D14
ρ722-220ζ74+ζ73ζ76+ζ7ζ75+ζ7200ζ76+ζ7ζ75+ζ72ζ74+ζ73-ζ75-ζ72-ζ74-ζ73-ζ74-ζ73-ζ75-ζ72-ζ76-ζ7-ζ76-ζ7ζ75+ζ72ζ74+ζ73ζ76+ζ7    orthogonal lifted from D14
ρ822220ζ74+ζ73ζ76+ζ7ζ75+ζ7200ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ76+ζ7    orthogonal lifted from D7
ρ922-220ζ75+ζ72ζ74+ζ73ζ76+ζ700ζ74+ζ73ζ76+ζ7ζ75+ζ72-ζ76-ζ7-ζ75-ζ72-ζ75-ζ72-ζ76-ζ7-ζ74-ζ73-ζ74-ζ73ζ76+ζ7ζ75+ζ72ζ74+ζ73    orthogonal lifted from D14
ρ1022220ζ76+ζ7ζ75+ζ72ζ74+ζ7300ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ75+ζ72    orthogonal lifted from D7
ρ1122220ζ75+ζ72ζ74+ζ73ζ76+ζ700ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ74+ζ73    orthogonal lifted from D7
ρ12220-20ζ76+ζ7ζ75+ζ72ζ74+ζ7300ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ74-ζ73ζ76-ζ7-ζ76+ζ7-ζ74+ζ73ζ75-ζ72-ζ75+ζ72-ζ74-ζ73-ζ76-ζ7-ζ75-ζ72    complex lifted from C7⋊D4
ρ13220-20ζ76+ζ7ζ75+ζ72ζ74+ζ7300ζ75+ζ72ζ74+ζ73ζ76+ζ7-ζ74+ζ73-ζ76+ζ7ζ76-ζ7ζ74-ζ73-ζ75+ζ72ζ75-ζ72-ζ74-ζ73-ζ76-ζ7-ζ75-ζ72    complex lifted from C7⋊D4
ρ14220-20ζ74+ζ73ζ76+ζ7ζ75+ζ7200ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ75-ζ72ζ74-ζ73-ζ74+ζ73-ζ75+ζ72-ζ76+ζ7ζ76-ζ7-ζ75-ζ72-ζ74-ζ73-ζ76-ζ7    complex lifted from C7⋊D4
ρ15220-20ζ75+ζ72ζ74+ζ73ζ76+ζ700ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ76-ζ7-ζ75+ζ72ζ75-ζ72-ζ76+ζ7ζ74-ζ73-ζ74+ζ73-ζ76-ζ7-ζ75-ζ72-ζ74-ζ73    complex lifted from C7⋊D4
ρ16220-20ζ74+ζ73ζ76+ζ7ζ75+ζ7200ζ76+ζ7ζ75+ζ72ζ74+ζ73-ζ75+ζ72-ζ74+ζ73ζ74-ζ73ζ75-ζ72ζ76-ζ7-ζ76+ζ7-ζ75-ζ72-ζ74-ζ73-ζ76-ζ7    complex lifted from C7⋊D4
ρ17220-20ζ75+ζ72ζ74+ζ73ζ76+ζ700ζ74+ζ73ζ76+ζ7ζ75+ζ72-ζ76+ζ7ζ75-ζ72-ζ75+ζ72ζ76-ζ7-ζ74+ζ73ζ74-ζ73-ζ76-ζ7-ζ75-ζ72-ζ74-ζ73    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
ρ204-40002ζ74+2ζ732ζ76+2ζ72ζ75+2ζ7200-2ζ76-2ζ7-2ζ75-2ζ72-2ζ74-2ζ73000000000    symplectic faithful, Schur index 2
ρ214-40002ζ76+2ζ72ζ75+2ζ722ζ74+2ζ7300-2ζ75-2ζ72-2ζ74-2ζ73-2ζ76-2ζ7000000000    symplectic faithful, Schur index 2
ρ224-40002ζ75+2ζ722ζ74+2ζ732ζ76+2ζ700-2ζ74-2ζ73-2ζ76-2ζ7-2ζ75-2ζ72000000000    symplectic faithful, Schur index 2

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

D4.D7 is a maximal subgroup of
 D8⋊D7  D8⋊3D7  D7×SD16  SD16⋊D7  D4.D14  D4.8D14  D4.9D14  D4.F7  C42.D4  D12.D7  D4.D21
D4.D7 is a maximal quotient of
 C4.Dic14  C14.Q16  D4⋊Dic7  C42.D4  D12.D7  D4.D21

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

1000
0100
0011284
00781
,
1000
0100
0011284
0001
,
0100
112900
0010
0001
,
889000
912500
008775
0011026
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,112,78,0,0,84,1],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,84,1],[0,112,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[88,91,0,0,90,25,0,0,0,0,87,110,0,0,75,26] >;
 

D4.D7 in GAP, Magma, Sage, TeX 

D_4.D_7
 
% in TeX
 
G:=Group("D4.D7");
 
// GroupNames label
 
G:=SmallGroup(112,15);
 
// by ID
 
G=gap.SmallGroup(112,15);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-2,-7,40,61,182,97,42,2404]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^7=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

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

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