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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, C56⋊7C2, C8⋊3C14, 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

Derived series
C1 — C2 — C7×M4(2)
Chief series
C1 — C2 — C4 — C28 — C56 — C7×M4(2)
Lower central
C1 — C2 — C7×M4(2)
Upper central
C1 — C28 — C7×M4(2)

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

C1
C2
2C2
C7
C4
C22
C4
C14
2C14
C8
C2×C4
C8
C2×C14
C28
C28
M4(2)
C56
C56
C2×C28
C7×M4(2)

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

(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)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)
 
G:=sub<Sym(56)| (1,26,19,51,14,43,39)(2,27,20,52,15,44,40)(3,28,21,53,16,45,33)(4,29,22,54,9,46,34)(5,30,23,55,10,47,35)(6,31,24,56,11,48,36)(7,32,17,49,12,41,37)(8,25,18,50,13,42,38), (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)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)>;
 
G:=Group( (1,26,19,51,14,43,39)(2,27,20,52,15,44,40)(3,28,21,53,16,45,33)(4,29,22,54,9,46,34)(5,30,23,55,10,47,35)(6,31,24,56,11,48,36)(7,32,17,49,12,41,37)(8,25,18,50,13,42,38), (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)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56) );
 
G=PermutationGroup([[(1,26,19,51,14,43,39),(2,27,20,52,15,44,40),(3,28,21,53,16,45,33),(4,29,22,54,9,46,34),(5,30,23,55,10,47,35),(6,31,24,56,11,48,36),(7,32,17,49,12,41,37),(8,25,18,50,13,42,38)], [(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),(9,13),(11,15),(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  D28⋊4C4  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
 

Export

Subgroup lattice of C7×M4(2) in TeX

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