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## G = Dic14⋊C4order 224 = 25·7

### 1st semidirect product of Dic14 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — Dic14⋊C4
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C4○D28 — Dic14⋊C4
 Lower central C7 — C14 — C28 — Dic14⋊C4
 Upper central C1 — C4 — C2×C4 — C42

Generators and relations for Dic14⋊C4
G = < a,b,c | a28=c4=1, b2=a14, bab-1=a-1, ac=ca, cbc-1=a7b >

Smallest permutation representation of Dic14⋊C4
On 56 points
Generators in S56
(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 50 15 36)(2 49 16 35)(3 48 17 34)(4 47 18 33)(5 46 19 32)(6 45 20 31)(7 44 21 30)(8 43 22 29)(9 42 23 56)(10 41 24 55)(11 40 25 54)(12 39 26 53)(13 38 27 52)(14 37 28 51)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 50 43 36)(30 51 44 37)(31 52 45 38)(32 53 46 39)(33 54 47 40)(34 55 48 41)(35 56 49 42)

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

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

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

Dic14⋊C4 is a maximal subgroup of
D5611C4  D564C4  D7×C4≀C2  C42⋊D14  D44D28  M4(2).22D14  C42.196D14  M4(2)⋊D14  D4.9D28  D4.10D28  C424D14  C425D14  D28.14D4  D285D4  D28.15D4
Dic14⋊C4 is a maximal quotient of
C14.C4≀C2  C4⋊Dic7⋊C4  C4.8Dic28  C4.17D56  C42.D14  C42.2D14  C28.8C42

62 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C ··· 4G 4H 7A 7B 7C 8A 8B 14A ··· 14I 28A ··· 28AJ order 1 2 2 2 4 4 4 ··· 4 4 7 7 7 8 8 14 ··· 14 28 ··· 28 size 1 1 2 28 1 1 2 ··· 2 28 2 2 2 28 28 2 ··· 2 2 ··· 2

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D4 D7 D14 C4≀C2 C4×D7 D28 C7⋊D4 Dic14⋊C4 kernel Dic14⋊C4 C4.Dic7 C4×C28 C4○D28 Dic14 D28 C28 C2×C14 C42 C2×C4 C7 C4 C4 C22 C1 # reps 1 1 1 1 2 2 1 1 3 3 4 6 6 6 24

Matrix representation of Dic14⋊C4 in GL2(𝔽29) generated by

 2 27 5 10
,
 15 2 3 14
,
 18 5 2 27
G:=sub<GL(2,GF(29))| [2,5,27,10],[15,3,2,14],[18,2,5,27] >;

Dic14⋊C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{14}\rtimes C_4
% in TeX

G:=Group("Dic14:C4");
// GroupNames label

G:=SmallGroup(224,11);
// by ID

G=gap.SmallGroup(224,11);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,121,31,362,579,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^28=c^4=1,b^2=a^14,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;
// generators/relations

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