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## G = C2×C4×C7⋊C3order 168 = 23·3·7

### Direct product of C2×C4 and C7⋊C3

Aliases: C2×C4×C7⋊C3, C284C6, C142C12, (C2×C28)⋊C3, C73(C2×C12), (C2×C14).2C6, C14.6(C2×C6), C22.(C2×C7⋊C3), C2.1(C22×C7⋊C3), (C22×C7⋊C3).2C2, (C2×C7⋊C3).6C22, SmallGroup(168,19)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C4×C7⋊C3
G = < a,b,c,d | a2=b4=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Smallest permutation representation of C2×C4×C7⋊C3
On 56 points
Generators in S56
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)
(1 15 8 22)(2 16 9 23)(3 17 10 24)(4 18 11 25)(5 19 12 26)(6 20 13 27)(7 21 14 28)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 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 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)(37 38 40)(39 42 41)(44 45 47)(46 49 48)(51 52 54)(53 56 55)

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

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

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

C2×C4×C7⋊C3 is a maximal subgroup of   C28.C12  Dic7⋊C12  C28⋊C12  D14⋊C12  D286C6

40 conjugacy classes

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

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + + image C1 C2 C2 C3 C4 C6 C6 C12 C7⋊C3 C2×C7⋊C3 C2×C7⋊C3 C4×C7⋊C3 kernel C2×C4×C7⋊C3 C4×C7⋊C3 C22×C7⋊C3 C2×C28 C2×C7⋊C3 C28 C2×C14 C14 C2×C4 C4 C22 C2 # reps 1 2 1 2 4 4 2 8 2 4 2 8

Matrix representation of C2×C4×C7⋊C3 in GL4(𝔽337) generated by

 336 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 148 0 0 0 0 189 0 0 0 0 189 0 0 0 0 189
,
 1 0 0 0 0 124 125 1 0 1 0 0 0 0 1 0
,
 128 0 0 0 0 1 0 0 0 212 336 336 0 0 1 0
G:=sub<GL(4,GF(337))| [336,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[148,0,0,0,0,189,0,0,0,0,189,0,0,0,0,189],[1,0,0,0,0,124,1,0,0,125,0,1,0,1,0,0],[128,0,0,0,0,1,212,0,0,0,336,1,0,0,336,0] >;

C2×C4×C7⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_7\rtimes C_3
% in TeX

G:=Group("C2xC4xC7:C3");
// GroupNames label

G:=SmallGroup(168,19);
// by ID

G=gap.SmallGroup(168,19);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-7,66,314]);
// Polycyclic

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

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