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## G = C2×C4⋊F7order 336 = 24·3·7

### Direct product of C2 and C4⋊F7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×C4⋊F7
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C2×F7 — C22×F7 — C2×C4⋊F7
 Lower central C7 — C14 — C2×C4⋊F7
 Upper central C1 — C22 — 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, dbd-1=b-1, dcd-1=c5 >

Subgroups: 512 in 108 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C7, C2×C4, D4, C23, C12, C2×C6, D7, C14, C14, C2×D4, C7⋊C3, C2×C12, C3×D4, C22×C6, C28, D14, D14, C2×C14, F7, C2×C7⋊C3, C2×C7⋊C3, C6×D4, D28, C2×C28, C22×D7, C4×C7⋊C3, C2×F7, C2×F7, C22×C7⋊C3, C2×D28, C4⋊F7, C2×C4×C7⋊C3, C22×F7, C2×C4⋊F7
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, F7, C6×D4, C2×F7, C4⋊F7, C22×F7, C2×C4⋊F7

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 6 6 6 6 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 C3×D4 F7 C2×F7 C2×F7 C4⋊F7 kernel C2×C4⋊F7 C4⋊F7 C2×C4×C7⋊C3 C22×F7 C2×D28 D28 C2×C28 C22×D7 C2×C7⋊C3 C14 C2×C4 C4 C22 C2 # reps 1 4 1 2 2 8 2 4 2 4 1 2 1 4

Matrix representation of C2×C4⋊F7 in GL10(𝔽337)

 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 336 7 0 0 0 0 0 0 0 0 96 1 0 0 0 0 0 0 0 0 0 0 336 7 0 0 0 0 0 0 0 0 96 1 0 0 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 0 0 336
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 336 336 336 336 336 336
,
 209 222 0 0 0 0 0 0 0 0 0 128 0 0 0 0 0 0 0 0 0 0 209 222 0 0 0 0 0 0 0 0 0 128 0 0 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 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 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 336 0

G:=sub<GL(10,GF(337))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[336,96,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,0,336,96,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,0,336],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,336,0,0,0,0,1,0,0,0,0,336,0,0,0,0,0,1,0,0,0,336,0,0,0,0,0,0,1,0,0,336,0,0,0,0,0,0,0,1,0,336,0,0,0,0,0,0,0,0,1,336],[209,0,0,0,0,0,0,0,0,0,222,128,0,0,0,0,0,0,0,0,0,0,209,0,0,0,0,0,0,0,0,0,222,128,0,0,0,0,0,0,0,0,0,0,336,0,0,0,1,0,0,0,0,0,0,0,0,336,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,336,0,1,0,0,0,0,0,0,0,0,0,1,336,0,0,0,0,0,336,0,0,1,0] >;

C2×C4⋊F7 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes F_7
% in TeX

G:=Group("C2xC4:F7");
// GroupNames label

G:=SmallGroup(336,123);
// by ID

G=gap.SmallGroup(336,123);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,506,122,10373,887]);
// 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,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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