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G = D28⋊6C6  order 336 = 24·3·7

2nd semidirect product of D28 and C6 acting via C6/C2=C3

metabelian, supersoluble, monomial

Aliases: D28⋊6C6, Dic14⋊6C6, C4○D28⋊C3, C4⋊F7⋊5C2, (C2×C4)⋊3F7, (C2×C28)⋊1C6, (C4×D7)⋊4C6, C7⋊D4⋊3C6, (C4×F7)⋊4C2, C4.F7⋊5C2, C4.16(C2×F7), C28.12(C2×C6), Dic7⋊C6⋊3C2, D14.1(C2×C6), C7⋊C12.2C22, C22.2(C2×F7), C2.5(C22×F7), C14.4(C22×C6), Dic7.2(C2×C6), (C2×F7).1C22, C7⋊1(C3×C4○D4), C7⋊C3⋊1(C4○D4), (C2×C7⋊C3).4C23, (C2×C14).10(C2×C6), (C4×C7⋊C3).12C22, (C22×C7⋊C3).10C22, (C2×C4×C7⋊C3)⋊1C2, SmallGroup(336,124)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — D28⋊6C6
Chief series
C1 — C7 — C14 — C2×C7⋊C3 — C2×F7 — C4×F7 — D28⋊6C6
Lower central
C7 — C14 — D28⋊6C6
Upper central
C1 — C4 — C2×C4

Generators and relations for D28⋊6C6
 G = < a,b,c | a28=b2=c6=1, bab=a-1, cac-1=a25, cbc-1=a10b >

Subgroups: 320 in 80 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C7, C2×C4, C2×C4, D4, Q8, C12, C2×C6, D7, C14, C14, C4○D4, C7⋊C3, C2×C12, C3×D4, C3×Q8, Dic7, C28, D14, C2×C14, F7, C2×C7⋊C3, C2×C7⋊C3, C3×C4○D4, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C7⋊C12, C4×C7⋊C3, C2×F7, C22×C7⋊C3, C4○D28, C4.F7, C4×F7, C4⋊F7, Dic7⋊C6, C2×C4×C7⋊C3, D28⋊6C6
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C4○D4, C22×C6, F7, C3×C4○D4, C2×F7, C22×F7, D28⋊6C6

Smallest permutation representation of D28⋊6C6
►On 56 points
Generators in S56
(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 36)(30 35)(31 34)(32 33)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)

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

38 conjugacy classes

class 1 2A2B2C2D3A3B4A4B4C4D4E6A6B6C···6H 7 12A12B12C12D12E···12J14A14B14C28A28B28C28D
order122223344444666···671212121212···1214141428282828
size11214147711214147714···146777714···146666666

38 irreducible representations

dim111111111111226666
type+++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6C4○D4C3×C4○D4F7C2×F7C2×F7D28⋊6C6
kernelD28⋊6C6C4.F7C4×F7C4⋊F7Dic7⋊C6C2×C4×C7⋊C3C4○D28Dic14C4×D7D28C7⋊D4C2×C28C7⋊C3C7C2×C4C4C22C1
# reps112121224242241214

Matrix representation of D28⋊6C6 ►in GL8(𝔽337)

148296000000
0189000000
0000003361
0000003360
0010003360
0001003360
0000103360
0000013360
,
148296000000
189189000000
0000013360
0000103360
0001003360
0010003360
0000003360
0000003361
,
20981000000
0128000000
0000033600
0033600000
0000003360
0003360000
0000000336
0000336000

G:=sub<GL(8,GF(337))| [148,0,0,0,0,0,0,0,296,189,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,336,336,336,336,336,336,0,0,1,0,0,0,0,0],[148,189,0,0,0,0,0,0,296,189,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,336,336,336,336,336,336,0,0,0,0,0,0,0,1],[209,0,0,0,0,0,0,0,81,128,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,336,0,0,336,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,336,0] >;
 

D28⋊6C6 in GAP, Magma, Sage, TeX 

D_{28}\rtimes_6C_6
 
% in TeX
 
G:=Group("D28:6C6");
 
// GroupNames label
 
G:=SmallGroup(336,124);
 
// by ID
 
G=gap.SmallGroup(336,124);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-2,-7,151,506,10373,887]);
 
// Polycyclic
 
G:=Group<a,b,c|a^28=b^2=c^6=1,b*a*b=a^-1,c*a*c^-1=a^25,c*b*c^-1=a^10*b>;
 
// generators/relations
 

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