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G = C28.C8order 224 = 25·7

1st non-split extension by C28 of C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C28.1C8, C56.4C4, C72M5(2), C8.22D14, C8.2Dic7, C56.22C22, C4.(C7⋊C8), C7⋊C165C2, C22.(C7⋊C8), (C2×C8).7D7, (C2×C14).3C8, C14.9(C2×C8), (C2×C28).8C4, C28.39(C2×C4), (C2×C56).10C2, (C2×C4).5Dic7, C4.11(C2×Dic7), C2.4(C2×C7⋊C8), SmallGroup(224,18)

Series: Derived Chief Lower central Upper central

C1C14 — C28.C8
C1C7C14C28C56C7⋊C16 — C28.C8
C7C14 — C28.C8
C1C8C2×C8

Generators and relations for C28.C8
 G = < a,b | a56=1, b4=a42, bab-1=a13 >

2C2
2C14
7C16
7C16
7M5(2)

Smallest permutation representation of C28.C8
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 64 50 85 43 106 36 71 29 92 22 57 15 78 8 99)(2 77 51 98 44 63 37 84 30 105 23 70 16 91 9 112)(3 90 52 111 45 76 38 97 31 62 24 83 17 104 10 69)(4 103 53 68 46 89 39 110 32 75 25 96 18 61 11 82)(5 60 54 81 47 102 40 67 33 88 26 109 19 74 12 95)(6 73 55 94 48 59 41 80 34 101 27 66 20 87 13 108)(7 86 56 107 49 72 42 93 35 58 28 79 21 100 14 65)

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

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

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

C28.C8 is a maximal subgroup of
C56.16Q8  C28.15C42  C56.Q8  D568C4  C8.Dic14  D56.C4  C56.8D4  D28.C8  C56.9Q8  C112⋊C4  M5(2)⋊D7  C56.D4  C56.92D4  D8.Dic7  Q16.Dic7  D82Dic7  D28.4C8  D7×M5(2)  C56.70C23  D8.D14  Q16.D14  Q16⋊D14  C56.31C23
C28.C8 is a maximal quotient of
C56.C8  C28⋊C16  C56.91D4

68 conjugacy classes

class 1 2A2B4A4B4C7A7B7C8A8B8C8D8E8F14A···14I16A···16H28A···28L56A···56X
order12244477788888814···1416···1628···2856···56
size1121122221111222···214···142···22···2

68 irreducible representations

dim111111122222222
type++++-+-
imageC1C2C2C4C4C8C8D7Dic7D14Dic7M5(2)C7⋊C8C7⋊C8C28.C8
kernelC28.C8C7⋊C16C2×C56C56C2×C28C28C2×C14C2×C8C8C8C2×C4C7C4C22C1
# reps1212244333346624

Matrix representation of C28.C8 in GL2(𝔽113) generated by

520
5172
,
74111
1739
G:=sub<GL(2,GF(113))| [52,51,0,72],[74,17,111,39] >;

C28.C8 in GAP, Magma, Sage, TeX

C_{28}.C_8
% in TeX

G:=Group("C28.C8");
// GroupNames label

G:=SmallGroup(224,18);
// by ID

G=gap.SmallGroup(224,18);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,217,50,69,6917]);
// Polycyclic

G:=Group<a,b|a^56=1,b^4=a^42,b*a*b^-1=a^13>;
// generators/relations

Export

Subgroup lattice of C28.C8 in TeX

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