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

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

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

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

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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