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G = C4.Dic7order 112 = 24·7

The non-split extension by C4 of Dic7 acting via Dic7/C14=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C4.Dic7, C28.1C4, C72M4(2), C4.15D14, C22.Dic7, C28.15C22, C7⋊C85C2, (C2×C4).2D7, C14.7(C2×C4), (C2×C28).5C2, (C2×C14).3C4, C2.3(C2×Dic7), SmallGroup(112,9)

Series: Derived Chief Lower central Upper central

C1C14 — C4.Dic7
C1C7C14C28C7⋊C8 — C4.Dic7
C7C14 — C4.Dic7
C1C4C2×C4

Generators and relations for C4.Dic7
 G = < a,b,c | a4=1, b14=a2, c2=a2b7, ab=ba, cac-1=a-1, cbc-1=b13 >

2C2
2C14
7C8
7C8
7M4(2)

Smallest permutation representation of C4.Dic7
On 56 points
Generators in S56
(1 22 15 8)(2 23 16 9)(3 24 17 10)(4 25 18 11)(5 26 19 12)(6 27 20 13)(7 28 21 14)(29 36 43 50)(30 37 44 51)(31 38 45 52)(32 39 46 53)(33 40 47 54)(34 41 48 55)(35 42 49 56)
(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)
(1 43 22 36 15 29 8 50)(2 56 23 49 16 42 9 35)(3 41 24 34 17 55 10 48)(4 54 25 47 18 40 11 33)(5 39 26 32 19 53 12 46)(6 52 27 45 20 38 13 31)(7 37 28 30 21 51 14 44)

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

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

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

C4.Dic7 is a maximal subgroup of
Dic14⋊C4  C56.C4  C28.53D4  C28.46D4  C4.12D28  C28.D4  C28.10D4  D42Dic7  D28.2C4  D7×M4(2)  D4.D14  C28.C23  Q8.Dic7  D4⋊D14  D4.9D14  C28.C12  D6.Dic7  C84.C4
C4.Dic7 is a maximal quotient of
C42.D7  C28⋊C8  C28.55D4  D6.Dic7  C84.C4

34 conjugacy classes

class 1 2A2B4A4B4C7A7B7C8A8B8C8D14A···14I28A···28L
order122444777888814···1428···28
size112112222141414142···22···2

34 irreducible representations

dim11111222222
type++++-+-
imageC1C2C2C4C4D7M4(2)Dic7D14Dic7C4.Dic7
kernelC4.Dic7C7⋊C8C2×C28C28C2×C14C2×C4C7C4C4C22C1
# reps121223233312

Matrix representation of C4.Dic7 in GL2(𝔽29) generated by

170
012
,
260
010
,
012
10
G:=sub<GL(2,GF(29))| [17,0,0,12],[26,0,0,10],[0,1,12,0] >;

C4.Dic7 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_7
% in TeX

G:=Group("C4.Dic7");
// GroupNames label

G:=SmallGroup(112,9);
// by ID

G=gap.SmallGroup(112,9);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,20,101,42,2404]);
// Polycyclic

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

Export

Subgroup lattice of C4.Dic7 in TeX

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