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## G = C2×C4×F7order 336 = 24·3·7

### Direct product of C2×C4 and F7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C2×C4×F7
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C2×F7 — C22×F7 — C2×C4×F7
 Lower central C7 — C2×C4×F7
 Upper central C1 — C2×C4

Generators and relations for C2×C4×F7
G = < a,b,c,d | a2=b4=c7=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 384 in 108 conjugacy classes, 62 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C7, C2×C4, C2×C4, C23, C12, C2×C6, D7, C14, C14, C22×C4, C7⋊C3, C2×C12, C22×C6, Dic7, C28, D14, C2×C14, F7, C2×C7⋊C3, C2×C7⋊C3, C22×C12, C4×D7, C2×Dic7, C2×C28, C22×D7, C7⋊C12, C4×C7⋊C3, C2×F7, C22×C7⋊C3, C2×C4×D7, C4×F7, C2×C7⋊C12, C2×C4×C7⋊C3, C22×F7, C2×C4×F7
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C2×C12, C22×C6, F7, C22×C12, C2×F7, C4×F7, C22×F7, C2×C4×F7

Smallest permutation representation of C2×C4×F7
On 56 points
Generators in S56
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 50 36 43)(30 51 37 44)(31 52 38 45)(32 53 39 46)(33 54 40 47)(34 55 41 48)(35 56 42 49)
(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)(2 11 3 14 5 13)(4 10 7 12 6 9)(15 22)(16 25 17 28 19 27)(18 24 21 26 20 23)(29 36)(30 39 31 42 33 41)(32 38 35 40 34 37)(43 50)(44 53 45 56 47 55)(46 52 49 54 48 51)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6N 7 12A ··· 12P 14A 14B 14C 28A 28B 28C 28D order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 6 ··· 6 7 12 ··· 12 14 14 14 28 28 28 28 size 1 1 1 1 7 7 7 7 7 7 1 1 1 1 7 7 7 7 7 ··· 7 6 7 ··· 7 6 6 6 6 6 6 6

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 F7 C2×F7 C2×F7 C4×F7 kernel C2×C4×F7 C4×F7 C2×C7⋊C12 C2×C4×C7⋊C3 C22×F7 C2×C4×D7 C2×F7 C4×D7 C2×Dic7 C2×C28 C22×D7 D14 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 2 8 8 2 2 2 16 1 2 1 4

Matrix representation of C2×C4×F7 in GL7(𝔽337)

 1 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 336
,
 148 0 0 0 0 0 0 0 189 0 0 0 0 0 0 0 189 0 0 0 0 0 0 0 189 0 0 0 0 0 0 0 189 0 0 0 0 0 0 0 189 0 0 0 0 0 0 0 189
,
 1 0 0 0 0 0 0 0 336 336 336 336 336 336 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0
,
 128 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 0 0 0 336 0 0 0 0 336 0 0 0 0 336 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 336 0

G:=sub<GL(7,GF(337))| [1,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336],[148,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,189],[1,0,0,0,0,0,0,0,336,1,0,0,0,0,0,336,0,1,0,0,0,0,336,0,0,1,0,0,0,336,0,0,0,1,0,0,336,0,0,0,0,1,0,336,0,0,0,0,0],[128,0,0,0,0,0,0,0,336,0,0,0,1,0,0,0,0,0,336,1,0,0,0,0,0,0,1,0,0,0,0,336,0,1,0,0,0,0,0,0,1,336,0,0,336,0,0,1,0] >;

C2×C4×F7 in GAP, Magma, Sage, TeX

C_2\times C_4\times F_7
% in TeX

G:=Group("C2xC4xF7");
// GroupNames label

G:=SmallGroup(336,122);
// by ID

G=gap.SmallGroup(336,122);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,122,10373,887]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^7=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations

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