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G = C7×M4(2)  order 112 = 24·7

Direct product of C7 and M4(2)

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C7×M4(2), C4.C28, C567C2, C83C14, C28.4C4, C22.C28, C28.22C22, C2.3(C2×C28), (C2×C28).8C2, (C2×C14).1C4, (C2×C4).2C14, C4.6(C2×C14), C14.12(C2×C4), SmallGroup(112,23)

Series: Derived Chief Lower central Upper central

C1C2 — C7×M4(2)
C1C2C4C28C56 — C7×M4(2)
C1C2 — C7×M4(2)
C1C28 — C7×M4(2)

Generators and relations for C7×M4(2)
 G = < a,b,c | a7=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

2C2
2C14

Smallest permutation representation of C7×M4(2)
On 56 points
Generators in S56
(1 26 17 51 37 43 11)(2 27 18 52 38 44 12)(3 28 19 53 39 45 13)(4 29 20 54 40 46 14)(5 30 21 55 33 47 15)(6 31 22 56 34 48 16)(7 32 23 49 35 41 9)(8 25 24 50 36 42 10)
(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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)

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

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

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

C7×M4(2) is a maximal subgroup of   C28.53D4  C28.46D4  C4.12D28  D284C4  D28.C4  C8⋊D14  C8.D14

70 conjugacy classes

class 1 2A2B4A4B4C7A···7F8A8B8C8D14A···14F14G···14L28A···28L28M···28R56A···56X
order1224447···7888814···1414···1428···2828···2856···56
size1121121···122221···12···21···12···22···2

70 irreducible representations

dim111111111122
type+++
imageC1C2C2C4C4C7C14C14C28C28M4(2)C7×M4(2)
kernelC7×M4(2)C56C2×C28C28C2×C14M4(2)C8C2×C4C4C22C7C1
# reps1212261261212212

Matrix representation of C7×M4(2) in GL2(𝔽29) generated by

230
023
,
010
220
,
280
01
G:=sub<GL(2,GF(29))| [23,0,0,23],[0,22,10,0],[28,0,0,1] >;

C7×M4(2) in GAP, Magma, Sage, TeX

C_7\times M_4(2)
% in TeX

G:=Group("C7xM4(2)");
// GroupNames label

G:=SmallGroup(112,23);
// by ID

G=gap.SmallGroup(112,23);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-2,140,581,58]);
// Polycyclic

G:=Group<a,b,c|a^7=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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Subgroup lattice of C7×M4(2) in TeX

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