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## G = D7×SD16order 224 = 25·7

### Direct product of D7 and SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D7×SD16
 Chief series C1 — C7 — C14 — C28 — C4×D7 — D4×D7 — D7×SD16
 Lower central C7 — C14 — C28 — D7×SD16
 Upper central C1 — C2 — C4 — SD16

Generators and relations for D7×SD16
G = < a,b,c,d | a7=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 342 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2 [×4], C4, C4 [×3], C22 [×5], C7, C8, C8, C2×C4 [×2], D4, D4 [×2], Q8, Q8 [×2], C23, D7 [×2], D7, C14, C14, C2×C8, SD16, SD16 [×3], C2×D4, C2×Q8, Dic7, Dic7, C28, C28, D14, D14 [×3], C2×C14, C2×SD16, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, C7⋊D4, C7×D4, C7×Q8, C22×D7, C8×D7, C56⋊C2, D4.D7, Q8⋊D7, C7×SD16, D4×D7, Q8×D7, D7×SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D7, SD16 [×2], C2×D4, D14 [×3], C2×SD16, C22×D7, D4×D7, D7×SD16

Smallest permutation representation of D7×SD16
On 56 points
Generators in S56
(1 49 39 12 23 45 27)(2 50 40 13 24 46 28)(3 51 33 14 17 47 29)(4 52 34 15 18 48 30)(5 53 35 16 19 41 31)(6 54 36 9 20 42 32)(7 55 37 10 21 43 25)(8 56 38 11 22 44 26)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 13)(10 14)(11 15)(12 16)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 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 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 21)(18 24)(20 22)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(42 44)(43 47)(46 48)(50 52)(51 55)(54 56)

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

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

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

35 conjugacy classes

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

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 SD16 D14 D14 D14 D4×D7 D7×SD16 kernel D7×SD16 C8×D7 C56⋊C2 D4.D7 Q8⋊D7 C7×SD16 D4×D7 Q8×D7 Dic7 D14 SD16 D7 C8 D4 Q8 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 3 4 3 3 3 3 6

Matrix representation of D7×SD16 in GL4(𝔽113) generated by

 0 1 0 0 112 9 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 112 0 0 0 0 112
,
 112 0 0 0 0 112 0 0 0 0 26 22 0 0 36 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 81 112
G:=sub<GL(4,GF(113))| [0,112,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,26,36,0,0,22,0],[1,0,0,0,0,1,0,0,0,0,1,81,0,0,0,112] >;

D7×SD16 in GAP, Magma, Sage, TeX

D_7\times {\rm SD}_{16}
% in TeX

G:=Group("D7xSD16");
// GroupNames label

G:=SmallGroup(224,108);
// by ID

G=gap.SmallGroup(224,108);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,116,86,297,159,69,6917]);
// 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^3>;
// generators/relations

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