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G = C2×C4×F7  order 336 = 24·3·7

Direct product of C2×C4 and F7

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C2×C4×F7, D14⋊2C12, D7⋊(C2×C12), C28⋊3(C2×C6), (C2×C28)⋊3C6, (C4×D7)⋊5C6, C14⋊1(C2×C12), C7⋊C12⋊3C22, C7⋊1(C22×C12), Dic7⋊3(C2×C6), (C2×Dic7)⋊5C6, D14.4(C2×C6), C2.1(C22×F7), C22.9(C2×F7), C14.2(C22×C6), (C22×D7).2C6, (C2×F7).4C22, (C22×F7).2C2, (C2×C4×D7)⋊C3, (C2×C7⋊C12)⋊5C2, C7⋊C3⋊1(C22×C4), (C4×C7⋊C3)⋊3C22, (C2×C14).8(C2×C6), (C2×C7⋊C3).2C23, (C22×C7⋊C3).8C22, (C2×C4×C7⋊C3)⋊3C2, (C2×C7⋊C3)⋊1(C2×C4), SmallGroup(336,122)

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 2A2B2C2D2E2F2G3A3B4A4B4C4D4E4F4G4H6A···6N 7 12A···12P14A14B14C28A28B28C28D
order1222222233444444446···6712···1214141428282828
size1111777777111177777···767···76666666

56 irreducible representations

dim1111111111116666
type++++++++
imageC1C2C2C2C2C3C4C6C6C6C6C12F7C2×F7C2×F7C4×F7
kernelC2×C4×F7C4×F7C2×C7⋊C12C2×C4×C7⋊C3C22×F7C2×C4×D7C2×F7C4×D7C2×Dic7C2×C28C22×D7D14C2×C4C4C22C2
# reps14111288222161214

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

1000000
033600000
003360000
000336000
000033600
000003360
000000336
,
148000000
018900000
001890000
000189000
000018900
000001890
000000189
,
1000000
0336336336336336336
0100000
0010000
0001000
0000100
0000010
,
128000000
033600000
000000336
000033600
003360000
0111111
000003360

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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