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G = C2×C4⋊F7  order 336 = 24·3·7

Direct product of C2 and C4⋊F7

direct product, metabelian, supersoluble, monomial

Aliases: C2×C4⋊F7, D28⋊5C6, (C2×D28)⋊C3, C7⋊1(C6×D4), C4⋊2(C2×F7), (C2×C4)⋊2F7, (C2×C28)⋊2C6, C28⋊2(C2×C6), C14⋊1(C3×D4), D14⋊1(C2×C6), (C22×D7)⋊1C6, (C22×F7)⋊1C2, (C2×F7)⋊1C22, C2.4(C22×F7), C14.3(C22×C6), C22.10(C2×F7), C7⋊C3⋊1(C2×D4), (C2×C7⋊C3)⋊1D4, (C4×C7⋊C3)⋊2C22, (C2×C14).9(C2×C6), (C2×C7⋊C3).3C23, (C22×C7⋊C3).9C22, (C2×C4×C7⋊C3)⋊2C2, SmallGroup(336,123)

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 2A2B2C2D2E2F2G3A3B4A4B6A···6F6G···6N 7 12A12B12C12D14A14B14C28A28B28C28D
order1222222233446···66···671212121214141428282828
size11111414141477227···714···146141414146666666

38 irreducible representations

dim11111111226666
type+++++++++
imageC1C2C2C2C3C6C6C6D4C3×D4F7C2×F7C2×F7C4⋊F7
kernelC2×C4⋊F7C4⋊F7C2×C4×C7⋊C3C22×F7C2×D28D28C2×C28C22×D7C2×C7⋊C3C14C2×C4C4C22C2
# reps14122824241214

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

1000000000
0100000000
003360000000
000336000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
336700000000
96100000000
003367000000
00961000000
000033600000
000003360000
000000336000
000000033600
000000003360
000000000336
,
1000000000
0100000000
0010000000
0001000000
0000010000
0000001000
0000000100
0000000010
0000000001
0000336336336336336336
,
20922200000000
012800000000
00209222000000
000128000000
000033600000
000000000336
000000033600
000003360000
0000111111
000000003360

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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