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## G = D7×M4(2)  order 224 = 25·7

### Direct product of D7 and M4(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D7×M4(2)
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — D7×M4(2)
 Lower central C7 — C14 — D7×M4(2)
 Upper central C1 — C4 — M4(2)

Generators and relations for D7×M4(2)
G = < a,b,c,d | a7=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 238 in 68 conjugacy classes, 39 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, D7, C14, C14, C2×C8, M4(2), M4(2), C22×C4, Dic7, C28, D14, D14, C2×C14, C2×M4(2), C7⋊C8, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C8×D7, C8⋊D7, C4.Dic7, C7×M4(2), C2×C4×D7, D7×M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, M4(2), C22×C4, D14, C2×M4(2), C4×D7, C22×D7, C2×C4×D7, D7×M4(2)

Smallest permutation representation of D7×M4(2)
On 56 points
Generators in S56
(1 46 14 55 37 24 27)(2 47 15 56 38 17 28)(3 48 16 49 39 18 29)(4 41 9 50 40 19 30)(5 42 10 51 33 20 31)(6 43 11 52 34 21 32)(7 44 12 53 35 22 25)(8 45 13 54 36 23 26)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(49 53)(50 54)(51 55)(52 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 6)(4 8)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)

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

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

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

D7×M4(2) is a maximal subgroup of
M4(2).19D14  M4(2).21D14  C42⋊D14  M4(2).25D14  C28.70C24  C56.49C23  SD16⋊D14  D56⋊C22
D7×M4(2) is a maximal quotient of
C56⋊Q8  C42.182D14  C89D28  Dic7.C42  Dic7.5M4(2)  Dic7.M4(2)  D14⋊M4(2)  D142M4(2)  Dic7⋊M4(2)  C42.27D14  C42.200D14  C42.202D14  D143M4(2)  C28⋊M4(2)  Dic74M4(2)  D146M4(2)  C56⋊D4  C5618D4

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 14A 14B 14C 14D 14E 14F 28A ··· 28F 28G 28H 28I 56A ··· 56L order 1 2 2 2 2 2 4 4 4 4 4 4 7 7 7 8 8 8 8 8 8 8 8 14 14 14 14 14 14 28 ··· 28 28 28 28 56 ··· 56 size 1 1 2 7 7 14 1 1 2 7 7 14 2 2 2 2 2 2 2 14 14 14 14 2 2 2 4 4 4 2 ··· 2 4 4 4 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D7 M4(2) D14 D14 C4×D7 C4×D7 D7×M4(2) kernel D7×M4(2) C8×D7 C8⋊D7 C4.Dic7 C7×M4(2) C2×C4×D7 C4×D7 C2×Dic7 C22×D7 M4(2) D7 C8 C2×C4 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 3 4 6 3 6 6 6

Matrix representation of D7×M4(2) in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 112 9
,
 112 0 0 0 0 112 0 0 0 0 0 1 0 0 1 0
,
 0 1 0 0 15 0 0 0 0 0 98 0 0 0 0 98
,
 1 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,0,112,0,0,1,9],[112,0,0,0,0,112,0,0,0,0,0,1,0,0,1,0],[0,15,0,0,1,0,0,0,0,0,98,0,0,0,0,98],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1] >;

D7×M4(2) in GAP, Magma, Sage, TeX

D_7\times M_4(2)
% in TeX

G:=Group("D7xM4(2)");
// GroupNames label

G:=SmallGroup(224,101);
// by ID

G=gap.SmallGroup(224,101);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,188,50,69,6917]);
// Polycyclic

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

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