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G = D42D7order 112 = 24·7

The semidirect product of D4 and D7 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D42D7, D4Dic7, C4.5D14, Dic143C2, C28.5C22, C14.6C23, C22.1D14, D14.2C22, Dic7.4C22, (C4×D7)⋊2C2, (C7×D4)⋊3C2, C72(C4○D4), C7⋊D42C2, (C2×C14).C22, (C2×Dic7)⋊3C2, C2.7(C22×D7), SmallGroup(112,32)

Series: Derived Chief Lower central Upper central

C1C14 — D42D7
C1C7C14D14C4×D7 — D42D7
C7C14 — D42D7
C1C2D4

Generators and relations for D42D7
 G = < a,b,c,d | a4=b2=c7=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

2C2
2C2
14C2
7C4
7C22
7C4
7C4
2C14
2C14
2D7
7C2×C4
7D4
7D4
7C2×C4
7Q8
7C2×C4
7C4○D4

Character table of D42D7

 class 12A2B2C2D4A4B4C4D4E7A7B7C14A14B14C14D14E14F14G14H14I28A28B28C
 size 1122142771414222222444444444
ρ11111111111111111111111111    trivial
ρ21111-11-1-1-1-1111111111111111    linear of order 2
ρ311-111-1-1-11-1111111-1111-1-1-1-1-1    linear of order 2
ρ411-11-1-111-11111111-1111-1-1-1-1-1    linear of order 2
ρ511-1-11111-1-1111111-1-1-1-1-1-1111    linear of order 2
ρ611-1-1-11-1-111111111-1-1-1-1-1-1111    linear of order 2
ρ7111-11-1-1-1-111111111-1-1-111-1-1-1    linear of order 2
ρ8111-1-1-1111-11111111-1-1-111-1-1-1    linear of order 2
ρ92222020000ζ7473ζ767ζ7572ζ7473ζ767ζ7572ζ7473ζ7572ζ7473ζ767ζ767ζ7572ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ1022-220-20000ζ7572ζ7473ζ767ζ7572ζ7473ζ7677572ζ767ζ7572ζ7473747376776775727473    orthogonal lifted from D14
ρ112222020000ζ767ζ7572ζ7473ζ767ζ7572ζ7473ζ767ζ7473ζ767ζ7572ζ7572ζ7473ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ122222020000ζ7572ζ7473ζ767ζ7572ζ7473ζ767ζ7572ζ767ζ7572ζ7473ζ7473ζ767ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ13222-20-20000ζ7572ζ7473ζ767ζ7572ζ7473ζ767ζ757276775727473ζ7473ζ76776775727473    orthogonal lifted from D14
ρ1422-2-2020000ζ767ζ7572ζ7473ζ767ζ7572ζ74737677473767757275727473ζ7473ζ767ζ7572    orthogonal lifted from D14
ρ1522-2-2020000ζ7572ζ7473ζ767ζ7572ζ7473ζ7677572767757274737473767ζ767ζ7572ζ7473    orthogonal lifted from D14
ρ1622-2-2020000ζ7473ζ767ζ7572ζ7473ζ767ζ75727473757274737677677572ζ7572ζ7473ζ767    orthogonal lifted from D14
ρ17222-20-20000ζ7473ζ767ζ7572ζ7473ζ767ζ7572ζ747375727473767ζ767ζ757275727473767    orthogonal lifted from D14
ρ18222-20-20000ζ767ζ7572ζ7473ζ767ζ7572ζ7473ζ76774737677572ζ7572ζ747374737677572    orthogonal lifted from D14
ρ1922-220-20000ζ767ζ7572ζ7473ζ767ζ7572ζ7473767ζ7473ζ767ζ75727572747374737677572    orthogonal lifted from D14
ρ2022-220-20000ζ7473ζ767ζ7572ζ7473ζ767ζ75727473ζ7572ζ7473ζ767767757275727473767    orthogonal lifted from D14
ρ212-20000-2i2i00222-2-2-2000000000    complex lifted from C4○D4
ρ222-200002i-2i00222-2-2-2000000000    complex lifted from C4○D4
ρ234-40000000074+2ζ7376+2ζ775+2ζ72-2ζ74-2ζ73-2ζ76-2ζ7-2ζ75-2ζ72000000000    symplectic faithful, Schur index 2
ρ244-40000000076+2ζ775+2ζ7274+2ζ73-2ζ76-2ζ7-2ζ75-2ζ72-2ζ74-2ζ73000000000    symplectic faithful, Schur index 2
ρ254-40000000075+2ζ7274+2ζ7376+2ζ7-2ζ75-2ζ72-2ζ74-2ζ73-2ζ76-2ζ7000000000    symplectic faithful, Schur index 2

Smallest permutation representation of D42D7
On 56 points
Generators in S56
(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 48)(2 49)(3 43)(4 44)(5 45)(6 46)(7 47)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(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 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)

G:=sub<Sym(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), (1,48)(2,49)(3,43)(4,44)(5,45)(6,46)(7,47)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(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)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)>;

G:=Group( (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), (1,48)(2,49)(3,43)(4,44)(5,45)(6,46)(7,47)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(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)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53) );

G=PermutationGroup([(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)], [(1,48),(2,49),(3,43),(4,44),(5,45),(6,46),(7,47),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(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),(29,31),(32,35),(33,34),(36,38),(39,42),(40,41),(43,52),(44,51),(45,50),(46,56),(47,55),(48,54),(49,53)])

D42D7 is a maximal subgroup of
D8⋊D7  D83D7  SD16⋊D7  SD163D7  D46D14  D7×C4○D4  D4.10D14  D42F7  D12⋊D7  D125D7  C42.C23  Dic3.D14  D42D21
D42D7 is a maximal quotient of
C23.11D14  C22⋊Dic14  C23.D14  Dic74D4  D14.D4  Dic7.D4  C22.D28  Dic73Q8  Dic7.Q8  C28.3Q8  C4⋊C47D7  D142Q8  C4⋊C4⋊D7  D4×Dic7  C23.18D14  C28.17D4  C282D4  Dic7⋊D4  D12⋊D7  D125D7  C42.C23  Dic3.D14  D42D21

Matrix representation of D42D7 in GL4(𝔽29) generated by

28000
02800
00170
00012
,
1000
0100
00012
00170
,
0100
28700
0010
0001
,
0100
1000
0010
00028
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,17,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,0,17,0,0,12,0],[0,28,0,0,1,7,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,28] >;

D42D7 in GAP, Magma, Sage, TeX

D_4\rtimes_2D_7
% in TeX

G:=Group("D4:2D7");
// GroupNames label

G:=SmallGroup(112,32);
// by ID

G=gap.SmallGroup(112,32);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,46,182,97,2404]);
// 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,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of D42D7 in TeX
Character table of D42D7 in TeX

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