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## G = D7×C4.D4order 448 = 26·7

### Direct product of D7 and C4.D4

Series: Derived Chief Lower central Upper central

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

Generators and relations for D7×C4.D4
G = < a,b,c,d,e | a7=b2=c4=1, d4=c2, e2=c, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=c-1d3 >

Subgroups: 1228 in 186 conjugacy classes, 53 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, D7, D7, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C4.D4, C4.D4, C2×M4(2), C22×D4, C7⋊C8, C56, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C2×C4.D4, C8×D7, C8⋊D7, C4.Dic7, C7×M4(2), C2×C4×D7, C2×D28, D4×D7, C2×C7⋊D4, D4×C14, C23×D7, C28.46D4, C28.D4, C7×C4.D4, D7×M4(2), C2×D4×D7, D7×C4.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C4.D4, C2×C22⋊C4, C4×D7, C22×D7, C2×C4.D4, C2×C4×D7, D4×D7, D7×C22⋊C4, D7×C4.D4

Smallest permutation representation of D7×C4.D4
On 56 points
Generators in S56
(1 37 12 42 32 24 56)(2 38 13 43 25 17 49)(3 39 14 44 26 18 50)(4 40 15 45 27 19 51)(5 33 16 46 28 20 52)(6 34 9 47 29 21 53)(7 35 10 48 30 22 54)(8 36 11 41 31 23 55)
(1 56)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 38)(18 39)(19 40)(20 33)(21 34)(22 35)(23 36)(24 37)
(1 7 5 3)(2 4 6 8)(9 11 13 15)(10 16 14 12)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(33 39 37 35)(34 36 38 40)(41 43 45 47)(42 48 46 44)(49 51 53 55)(50 56 54 52)
(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 8 7 2 5 4 3 6)(9 12 11 10 13 16 15 14)(17 20 19 18 21 24 23 22)(25 28 27 26 29 32 31 30)(33 40 39 34 37 36 35 38)(41 48 43 46 45 44 47 42)(49 52 51 50 53 56 55 54)

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

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

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

55 conjugacy classes

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

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 8 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 D4 D7 D14 D14 C4×D7 C4.D4 D4×D7 D7×C4.D4 kernel D7×C4.D4 C28.46D4 C28.D4 C7×C4.D4 D7×M4(2) C2×D4×D7 C2×C7⋊D4 C23×D7 C4×D7 C4.D4 M4(2) C2×D4 C23 D7 C4 C1 # reps 1 2 1 1 2 1 4 4 4 3 6 3 12 2 6 3

Matrix representation of D7×C4.D4 in GL6(𝔽113)

 0 1 0 0 0 0 112 24 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 111 0 0 0 0 1 1 0 0 0 0 0 18 112 69 0 0 98 0 36 1
,
 98 0 0 0 0 0 0 98 0 0 0 0 0 0 98 0 36 2 0 0 0 0 0 112 0 0 0 0 15 95 0 0 1 1 0 0
,
 98 0 0 0 0 0 0 98 0 0 0 0 0 0 98 0 36 0 0 0 0 0 0 1 0 0 0 69 15 0 0 0 1 1 0 0

G:=sub<GL(6,GF(113))| [0,112,0,0,0,0,1,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,1,0,98,0,0,111,1,18,0,0,0,0,0,112,36,0,0,0,0,69,1],[98,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,1,0,0,0,0,0,1,0,0,36,0,15,0,0,0,2,112,95,0],[98,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,1,0,0,0,0,69,1,0,0,36,0,15,0,0,0,0,1,0,0] >;

D7×C4.D4 in GAP, Magma, Sage, TeX

D_7\times C_4.D_4
% in TeX

G:=Group("D7xC4.D4");
// GroupNames label

G:=SmallGroup(448,278);
// by ID

G=gap.SmallGroup(448,278);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,58,570,136,438,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^2=c^4=1,d^4=c^2,e^2=c,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=c^-1*d^3>;
// generators/relations

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