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G = D8⋊D7order 224 = 25·7

2nd semidirect product of D8 and D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D82D7, C82D14, D42D14, C564C22, D14.6D4, C28.2C23, Dic7.8D4, D28.1C22, Dic141C22, D4⋊D72C2, (C7×D8)⋊4C2, (D4×D7)⋊2C2, C7⋊C81C22, C8⋊D73C2, C56⋊C23C2, C72(C8⋊C22), D4.D71C2, C2.16(D4×D7), D42D71C2, C14.28(C2×D4), (C7×D4)⋊2C22, C4.2(C22×D7), (C4×D7).1C22, SmallGroup(224,106)

Series: Derived Chief Lower central Upper central

C1C28 — D8⋊D7
C1C7C14C28C4×D7D4×D7 — D8⋊D7
C7C14C28 — D8⋊D7
C1C2C4D8

Generators and relations for D8⋊D7
 G = < a,b,c,d | a8=b2=c7=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >

Subgroups: 350 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×4], C4, C4 [×2], C22 [×6], C7, C8, C8, C2×C4 [×2], D4 [×2], D4 [×3], Q8, C23, D7 [×2], C14, C14 [×2], M4(2), D8, D8, SD16 [×2], C2×D4, C4○D4, Dic7, Dic7, C28, D14, D14 [×3], C2×C14 [×2], C8⋊C22, C7⋊C8, C56, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4 [×2], C7×D4 [×2], C22×D7, C8⋊D7, C56⋊C2, D4⋊D7, D4.D7, C7×D8, D4×D7, D42D7, D8⋊D7
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D7, C2×D4, D14 [×3], C8⋊C22, C22×D7, D4×D7, D8⋊D7

Smallest permutation representation of D8⋊D7
On 56 points
Generators in S56
(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)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(18 24)(19 23)(20 22)(25 27)(28 32)(29 31)(34 40)(35 39)(36 38)(42 48)(43 47)(44 46)(49 55)(50 54)(51 53)
(1 26 37 10 21 52 41)(2 27 38 11 22 53 42)(3 28 39 12 23 54 43)(4 29 40 13 24 55 44)(5 30 33 14 17 56 45)(6 31 34 15 18 49 46)(7 32 35 16 19 50 47)(8 25 36 9 20 51 48)
(1 41)(2 46)(3 43)(4 48)(5 45)(6 42)(7 47)(8 44)(9 13)(11 15)(17 33)(18 38)(19 35)(20 40)(21 37)(22 34)(23 39)(24 36)(25 55)(26 52)(27 49)(28 54)(29 51)(30 56)(31 53)(32 50)

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

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

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

D8⋊D7 is a maximal subgroup of
D813D14  D810D14  D811D14  D7×C8⋊C22  SD16⋊D14  D85D14  D86D14
D8⋊D7 is a maximal quotient of
D4.D7⋊C4  D4⋊Dic14  Dic142D4  D4.Dic14  C4⋊C4.D14  C28⋊Q8⋊C2  (D4×D7)⋊C4  D4⋊(C4×D7)  D4.6D28  D14.SD16  C8⋊Dic7⋊C2  C7⋊C81D4  D43D28  C7⋊C8⋊D4  D4⋊D7⋊C4  D28.D4  Dic142Q8  C564Q8  C56⋊(C2×C4)  D14.2Q16  C2.D8⋊D7  C83D28  C56⋊C2⋊C4  D28.2Q8  Dic7⋊D8  D8⋊Dic7  (C2×D8).D7  C5611D4  D28⋊D4  Dic14⋊D4  C5612D4

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C7A7B7C8A8B14A14B14C14D···14I28A28B28C56A···56F
order1222224447778814141414···1428282856···56
size11441428214282224282228···84444···4

32 irreducible representations

dim1111111122222444
type+++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D7D14D14C8⋊C22D4×D7D8⋊D7
kernelD8⋊D7C8⋊D7C56⋊C2D4⋊D7D4.D7C7×D8D4×D7D42D7Dic7D14D8C8D4C7C2C1
# reps1111111111336136

Matrix representation of D8⋊D7 in GL4(𝔽113) generated by

004349
00715
178210395
751011810
,
1000
0100
51391120
7400112
,
34100
11110300
0001
0011224
,
10311200
991000
0001
0010
G:=sub<GL(4,GF(113))| [0,0,17,75,0,0,82,101,43,71,103,18,49,5,95,10],[1,0,51,74,0,1,39,0,0,0,112,0,0,0,0,112],[34,111,0,0,1,103,0,0,0,0,0,112,0,0,1,24],[103,99,0,0,112,10,0,0,0,0,0,1,0,0,1,0] >;

D8⋊D7 in GAP, Magma, Sage, TeX

D_8\rtimes D_7
% in TeX

G:=Group("D8:D7");
// GroupNames label

G:=SmallGroup(224,106);
// by ID

G=gap.SmallGroup(224,106);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,362,116,297,159,69,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^7=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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