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G = C56.C4order 224 = 25·7

1st non-split extension by C56 of C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C56.1C4, C4.18D28, C28.34D4, C8.1Dic7, C22.2Dic14, (C2×C8).5D7, (C2×C56).7C2, C14.7(C4⋊C4), (C2×C14).3Q8, C71(C8.C4), C28.35(C2×C4), (C2×C4).70D14, C4.8(C2×Dic7), C2.5(C4⋊Dic7), C4.Dic7.1C2, (C2×C28).97C22, SmallGroup(224,25)

Series: Derived Chief Lower central Upper central

C1C28 — C56.C4
C1C7C14C28C2×C28C4.Dic7 — C56.C4
C7C14C28 — C56.C4
C1C4C2×C4C2×C8

Generators and relations for C56.C4
 G = < a,b,c | a8=1, b14=a4, c2=a4b7, ab=ba, cac-1=a-1, cbc-1=b13 >

2C2
2C14
14C8
14C8
7M4(2)
7M4(2)
2C7⋊C8
2C7⋊C8
7C8.C4

Smallest permutation representation of C56.C4
On 112 points
Generators in S112
(1 56 22 49 15 42 8 35)(2 29 23 50 16 43 9 36)(3 30 24 51 17 44 10 37)(4 31 25 52 18 45 11 38)(5 32 26 53 19 46 12 39)(6 33 27 54 20 47 13 40)(7 34 28 55 21 48 14 41)(57 92 64 99 71 106 78 85)(58 93 65 100 72 107 79 86)(59 94 66 101 73 108 80 87)(60 95 67 102 74 109 81 88)(61 96 68 103 75 110 82 89)(62 97 69 104 76 111 83 90)(63 98 70 105 77 112 84 91)
(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 57 22 78 15 71 8 64)(2 70 23 63 16 84 9 77)(3 83 24 76 17 69 10 62)(4 68 25 61 18 82 11 75)(5 81 26 74 19 67 12 60)(6 66 27 59 20 80 13 73)(7 79 28 72 21 65 14 58)(29 98 50 91 43 112 36 105)(30 111 51 104 44 97 37 90)(31 96 52 89 45 110 38 103)(32 109 53 102 46 95 39 88)(33 94 54 87 47 108 40 101)(34 107 55 100 48 93 41 86)(35 92 56 85 49 106 42 99)

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

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

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

C56.C4 is a maximal subgroup of
C56.Q8  C8.7Dic14  C112.C4  D56.1C4  C16⋊Dic7  C28.3D8  C28.4D8  D8.Dic7  Q16.Dic7  C28.58D8  D5611C4  D564C4  D7×C8.C4  M4(2).25D14  M4(2).Dic7  D4.3D28  D4.4D28  D4.5D28  C56.23D4  C56.44D4  C56.29D4  D85Dic7  D84Dic7
C56.C4 is a maximal quotient of
C562C8  C561C8  C28.10C42

62 conjugacy classes

class 1 2A2B4A4B4C7A7B7C8A8B8C8D8E8F8G8H14A···14I28A···28L56A···56X
order1224447778888888814···1428···2856···56
size1121122222222282828282···22···22···2

62 irreducible representations

dim1111222222222
type++++-+-++-
imageC1C2C2C4D4Q8D7Dic7D14C8.C4D28Dic14C56.C4
kernelC56.C4C4.Dic7C2×C56C56C28C2×C14C2×C8C8C2×C4C7C4C22C1
# reps12141136346624

Matrix representation of C56.C4 in GL4(𝔽113) generated by

18000
04400
0010
0001
,
98000
09800
0051
00194
,
0100
98000
008533
007928
G:=sub<GL(4,GF(113))| [18,0,0,0,0,44,0,0,0,0,1,0,0,0,0,1],[98,0,0,0,0,98,0,0,0,0,5,19,0,0,1,4],[0,98,0,0,1,0,0,0,0,0,85,79,0,0,33,28] >;

C56.C4 in GAP, Magma, Sage, TeX

C_{56}.C_4
% in TeX

G:=Group("C56.C4");
// GroupNames label

G:=SmallGroup(224,25);
// by ID

G=gap.SmallGroup(224,25);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,55,86,579,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^14=a^4,c^2=a^4*b^7,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^13>;
// generators/relations

Export

Subgroup lattice of C56.C4 in TeX

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