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G = C28⋊C12order 336 = 24·3·7

1st semidirect product of C28 and C12 acting via C12/C2=C6

metacyclic, supersoluble, monomial

Aliases: C281C12, C4⋊(C7⋊C12), C4⋊Dic7⋊C3, (C2×C4).3F7, (C2×C28).2C6, C14.4(C3×D4), C2.1(C4⋊F7), C14.2(C3×Q8), C14.8(C2×C12), C2.2(C4.F7), C22.5(C2×F7), (C2×Dic7).2C6, C72(C3×C4⋊C4), C7⋊C32(C4⋊C4), (C4×C7⋊C3)⋊1C4, C2.4(C2×C7⋊C12), (C2×C7⋊C3).4D4, (C2×C7⋊C3).2Q8, (C2×C7⋊C12).2C2, (C2×C14).4(C2×C6), (C22×C7⋊C3).4C22, (C2×C4×C7⋊C3).2C2, (C2×C7⋊C3).8(C2×C4), SmallGroup(336,16)

Series: Derived Chief Lower central Upper central

C1C14 — C28⋊C12
C1C7C14C2×C14C22×C7⋊C3C2×C7⋊C12 — C28⋊C12
C7C14 — C28⋊C12
C1C22C2×C4

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

7C3
14C4
14C4
7C6
7C6
7C6
7C2×C4
7C2×C4
7C12
7C12
7C2×C6
14C12
14C12
2Dic7
2Dic7
7C4⋊C4
7C2×C12
7C2×C12
7C2×C12
2C7⋊C12
2C7⋊C12
7C3×C4⋊C4

Smallest permutation representation of C28⋊C12
On 112 points
Generators in S112
(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)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 106 46 74)(2 109 55 73 26 97 47 77 10 105 43 65)(3 112 36 72 23 88 48 80 19 104 40 84)(4 87 45 71 20 107 49 83 28 103 37 75)(5 90 54 70 17 98 50 58 9 102 34 66)(6 93 35 69 14 89 51 61 18 101 31 57)(7 96 44 68 11 108 52 64 27 100 56 76)(8 99 53 67)(12 111 33 63 24 91 29 79 16 95 41 59)(13 86 42 62 21 110 30 82 25 94 38 78)(15 92 32 60)(22 85 39 81)

G:=sub<Sym(112)| (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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,106,46,74)(2,109,55,73,26,97,47,77,10,105,43,65)(3,112,36,72,23,88,48,80,19,104,40,84)(4,87,45,71,20,107,49,83,28,103,37,75)(5,90,54,70,17,98,50,58,9,102,34,66)(6,93,35,69,14,89,51,61,18,101,31,57)(7,96,44,68,11,108,52,64,27,100,56,76)(8,99,53,67)(12,111,33,63,24,91,29,79,16,95,41,59)(13,86,42,62,21,110,30,82,25,94,38,78)(15,92,32,60)(22,85,39,81)>;

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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,106,46,74)(2,109,55,73,26,97,47,77,10,105,43,65)(3,112,36,72,23,88,48,80,19,104,40,84)(4,87,45,71,20,107,49,83,28,103,37,75)(5,90,54,70,17,98,50,58,9,102,34,66)(6,93,35,69,14,89,51,61,18,101,31,57)(7,96,44,68,11,108,52,64,27,100,56,76)(8,99,53,67)(12,111,33,63,24,91,29,79,16,95,41,59)(13,86,42,62,21,110,30,82,25,94,38,78)(15,92,32,60)(22,85,39,81) );

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),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,106,46,74),(2,109,55,73,26,97,47,77,10,105,43,65),(3,112,36,72,23,88,48,80,19,104,40,84),(4,87,45,71,20,107,49,83,28,103,37,75),(5,90,54,70,17,98,50,58,9,102,34,66),(6,93,35,69,14,89,51,61,18,101,31,57),(7,96,44,68,11,108,52,64,27,100,56,76),(8,99,53,67),(12,111,33,63,24,91,29,79,16,95,41,59),(13,86,42,62,21,110,30,82,25,94,38,78),(15,92,32,60),(22,85,39,81)]])

38 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F6A···6F 7 12A···12L14A14B14C28A28B28C28D
order1222334444446···6712···1214141428282828
size11117722141414147···7614···146666666

38 irreducible representations

dim11111111222266666
type++++-+-+-+
imageC1C2C2C3C4C6C6C12D4Q8C3×D4C3×Q8F7C7⋊C12C2×F7C4.F7C4⋊F7
kernelC28⋊C12C2×C7⋊C12C2×C4×C7⋊C3C4⋊Dic7C4×C7⋊C3C2×Dic7C2×C28C28C2×C7⋊C3C2×C7⋊C3C14C14C2×C4C4C22C2C2
# reps12124428112212122

Matrix representation of C28⋊C12 in GL8(𝔽337)

0336000000
10000000
00001000
00000100
00000010
00000001
00336336336336336336
00100000
,
15232000000
232322000000
006610810801080
0002290229229295
001080066108108
000661081080108
00424227142271271
0022902292292950

G:=sub<GL(8,GF(337))| [0,1,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,0,0,0,0,336,1,0,0,0,0,0,0,336,0,0,0,1,0,0,0,336,0,0,0,0,1,0,0,336,0,0,0,0,0,1,0,336,0,0,0,0,0,0,1,336,0],[15,232,0,0,0,0,0,0,232,322,0,0,0,0,0,0,0,0,66,0,108,0,42,229,0,0,108,229,0,66,42,0,0,0,108,0,0,108,271,229,0,0,0,229,66,108,42,229,0,0,108,229,108,0,271,295,0,0,0,295,108,108,271,0] >;

C28⋊C12 in GAP, Magma, Sage, TeX

C_{28}\rtimes C_{12}
% in TeX

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

G:=SmallGroup(336,16);
// by ID

G=gap.SmallGroup(336,16);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,72,313,151,10373,1745]);
// Polycyclic

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

Export

Subgroup lattice of C28⋊C12 in TeX

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