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G = C28.C12order 336 = 24·3·7

1st non-split extension by C28 of C12 acting via C12/C2=C6

metacyclic, supersoluble, monomial

Aliases: C28.1C12, C7⋊C85C6, C7⋊C245C2, C4.(C7⋊C12), C4.Dic7⋊C3, C7⋊C32M4(2), (C2×C4).2F7, (C2×C28).1C6, C22.(C7⋊C12), C72(C3×M4(2)), C4.15(C2×F7), (C2×C14).2C12, C14.7(C2×C12), C28.16(C2×C6), (C4×C7⋊C3).1C4, C2.3(C2×C7⋊C12), (C22×C7⋊C3).2C4, (C4×C7⋊C3).16C22, (C2×C4×C7⋊C3).1C2, (C2×C7⋊C3).6(C2×C4), SmallGroup(336,13)

Series: Derived Chief Lower central Upper central

C1C14 — C28.C12
C1C7C14C28C4×C7⋊C3C7⋊C24 — C28.C12
C7C14 — C28.C12
C1C4C2×C4

Generators and relations for C28.C12
 G = < a,b | a28=1, b12=a14, bab-1=a19 >

2C2
7C3
7C6
14C6
2C14
7C8
7C8
7C12
7C12
7C2×C6
2C2×C7⋊C3
7M4(2)
7C24
7C24
7C2×C12
7C3×M4(2)

Smallest permutation representation of C28.C12
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 51 22 30 15 37 8 44)(2 54 3 29 12 56 9 47 10 50 19 49 16 40 17 43 26 42 23 33 24 36 5 35)(4 32 21 55 6 38 11 53 28 48 13 31 18 46 7 41 20 52 25 39 14 34 27 45)

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

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

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

38 conjugacy classes

class 1 2A2B3A3B4A4B4C6A6B6C6D 7 8A8B8C8D12A12B12C12D12E12F14A14B14C24A···24H28A28B28C28D
order1223344466667888812121212121214141424···2428282828
size112771127714146141414147777141466614···146666

38 irreducible representations

dim11111111112266666
type++++-+-
imageC1C2C2C3C4C4C6C6C12C12M4(2)C3×M4(2)F7C7⋊C12C2×F7C7⋊C12C28.C12
kernelC28.C12C7⋊C24C2×C4×C7⋊C3C4.Dic7C4×C7⋊C3C22×C7⋊C3C7⋊C8C2×C28C28C2×C14C7⋊C3C7C2×C4C4C4C22C1
# reps12122242442411114

Matrix representation of C28.C12 in GL6(𝔽337)

0148189000
154035000
1890183000
000015435
000189189183
0000189183
,
000010
0002131125
0001125336
01890000
15418935000
18935148000

G:=sub<GL(6,GF(337))| [0,154,189,0,0,0,148,0,0,0,0,0,189,35,183,0,0,0,0,0,0,0,189,0,0,0,0,154,189,189,0,0,0,35,183,183],[0,0,0,0,154,189,0,0,0,189,189,35,0,0,0,0,35,148,0,213,1,0,0,0,1,1,125,0,0,0,0,125,336,0,0,0] >;

C28.C12 in GAP, Magma, Sage, TeX

C_{28}.C_{12}
% in TeX

G:=Group("C28.C12");
// GroupNames label

G:=SmallGroup(336,13);
// by ID

G=gap.SmallGroup(336,13);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,72,313,69,10373,1745]);
// Polycyclic

G:=Group<a,b|a^28=1,b^12=a^14,b*a*b^-1=a^19>;
// generators/relations

Export

Subgroup lattice of C28.C12 in TeX

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