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## G = C7×C4.D4order 224 = 25·7

### Direct product of C7 and C4.D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C4.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C7×M4(2) — C7×C4.D4
 Lower central C1 — C2 — C22 — C7×C4.D4
 Upper central C1 — C14 — C2×C28 — C7×C4.D4

Generators and relations for C7×C4.D4
G = < a,b,c,d | a7=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Smallest permutation representation of C7×C4.D4
On 56 points
Generators in S56
(1 27 17 53 16 45 35)(2 28 18 54 9 46 36)(3 29 19 55 10 47 37)(4 30 20 56 11 48 38)(5 31 21 49 12 41 39)(6 32 22 50 13 42 40)(7 25 23 51 14 43 33)(8 26 24 52 15 44 34)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)(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 4 3 2 5 8 7 6)(9 12 15 14 13 16 11 10)(17 20 19 18 21 24 23 22)(25 32 27 30 29 28 31 26)(33 40 35 38 37 36 39 34)(41 44 43 42 45 48 47 46)(49 52 51 50 53 56 55 54)

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

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

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

C7×C4.D4 is a maximal subgroup of   C23.3D28  C23.4D28  M4(2).19D14  D28.1D4  D281D4  D28.2D4  D28.3D4

77 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 7A ··· 7F 8A 8B 8C 8D 14A ··· 14F 14G ··· 14L 14M ··· 14X 28A ··· 28L 56A ··· 56X order 1 2 2 2 2 4 4 7 ··· 7 8 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 2 4 4 2 2 1 ··· 1 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

77 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + image C1 C2 C2 C4 C7 C14 C14 C28 D4 C7×D4 C4.D4 C7×C4.D4 kernel C7×C4.D4 C7×M4(2) D4×C14 C22×C14 C4.D4 M4(2) C2×D4 C23 C28 C4 C7 C1 # reps 1 2 1 4 6 12 6 24 2 12 1 6

Matrix representation of C7×C4.D4 in GL6(𝔽113)

 106 0 0 0 0 0 0 106 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
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 0 1 0 0 0 0 112 0 0 0 0 0 0 0 0 1 0 0 0 0 112 0
,
 1 0 0 0 0 0 112 112 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 112 0 0 0 0 0 0 1 0 0
,
 1 2 0 0 0 0 112 112 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 112 0 0 0

G:=sub<GL(6,GF(113))| [106,0,0,0,0,0,0,106,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0],[1,112,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0],[1,112,0,0,0,0,2,112,0,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C7×C4.D4 in GAP, Magma, Sage, TeX

C_7\times C_4.D_4
% in TeX

G:=Group("C7xC4.D4");
// GroupNames label

G:=SmallGroup(224,49);
// by ID

G=gap.SmallGroup(224,49);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,3363,2530,88]);
// Polycyclic

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

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