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## G = M4(2)×C7⋊C3order 336 = 24·3·7

### Direct product of M4(2) and C7⋊C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — M4(2)×C7⋊C3
 Chief series C1 — C7 — C14 — C28 — C4×C7⋊C3 — C2×C4×C7⋊C3 — M4(2)×C7⋊C3
 Lower central C7 — C14 — M4(2)×C7⋊C3
 Upper central C1 — C4 — M4(2)

Generators and relations for M4(2)×C7⋊C3
G = < a,b,c,d | a8=b2=c7=d3=1, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Smallest permutation representation of M4(2)×C7⋊C3
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 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)
(1 56 38 30 12 21 46)(2 49 39 31 13 22 47)(3 50 40 32 14 23 48)(4 51 33 25 15 24 41)(5 52 34 26 16 17 42)(6 53 35 27 9 18 43)(7 54 36 28 10 19 44)(8 55 37 29 11 20 45)
(9 53 35)(10 54 36)(11 55 37)(12 56 38)(13 49 39)(14 50 40)(15 51 33)(16 52 34)(17 26 42)(18 27 43)(19 28 44)(20 29 45)(21 30 46)(22 31 47)(23 32 48)(24 25 41)

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

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

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

50 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 6C 6D 7A 7B 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 14A 14B 14C 14D 24A ··· 24H 28A 28B 28C 28D 28E 28F 56A ··· 56H order 1 2 2 3 3 4 4 4 6 6 6 6 7 7 8 8 8 8 12 12 12 12 12 12 14 14 14 14 24 ··· 24 28 28 28 28 28 28 56 ··· 56 size 1 1 2 7 7 1 1 2 7 7 14 14 3 3 2 2 2 2 7 7 7 7 14 14 3 3 6 6 14 ··· 14 3 3 3 3 6 6 6 ··· 6

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 3 3 3 3 3 6 type + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 M4(2) C3×M4(2) C7⋊C3 C2×C7⋊C3 C2×C7⋊C3 C4×C7⋊C3 C4×C7⋊C3 M4(2)×C7⋊C3 kernel M4(2)×C7⋊C3 C8×C7⋊C3 C2×C4×C7⋊C3 C7×M4(2) C4×C7⋊C3 C22×C7⋊C3 C56 C2×C28 C28 C2×C14 C7⋊C3 C7 M4(2) C8 C2×C4 C4 C22 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 4 2 4 2 4 4 4

Matrix representation of M4(2)×C7⋊C3 in GL5(𝔽337)

 149 335 0 0 0 222 188 0 0 0 0 0 189 0 0 0 0 0 189 0 0 0 0 0 189
,
 1 0 0 0 0 149 336 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 213 212 212 0 0 336 0 336
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 124 1 0 0 0 1 0 0 0 336 0 336

G:=sub<GL(5,GF(337))| [149,222,0,0,0,335,188,0,0,0,0,0,189,0,0,0,0,0,189,0,0,0,0,0,189],[1,149,0,0,0,0,336,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,213,336,0,0,1,212,0,0,0,1,212,336],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,336,0,0,124,1,0,0,0,1,0,336] >;

M4(2)×C7⋊C3 in GAP, Magma, Sage, TeX

M_4(2)\times C_7\rtimes C_3
% in TeX

G:=Group("M4(2)xC7:C3");
// GroupNames label

G:=SmallGroup(336,52);
// by ID

G=gap.SmallGroup(336,52);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,313,79,69,881]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^7=d^3=1,b*a*b=a^5,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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