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G = Dic14⋊C4order 224 = 25·7

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D281C4, C423D7, C4.17D28, C28.33D4, Dic141C4, C71C4≀C2, (C4×C28)⋊6C2, C4.6(C4×D7), C28.16(C2×C4), C4○D28.1C2, (C2×C14).26D4, (C2×C4).66D14, C4.Dic71C2, C2.3(D14⋊C4), C14.1(C22⋊C4), (C2×C28).96C22, C22.7(C7⋊D4), SmallGroup(224,11)

Series: Derived Chief Lower central Upper central

C1C28 — Dic14⋊C4
C1C7C14C2×C14C2×C28C4○D28 — Dic14⋊C4
C7C14C28 — Dic14⋊C4
C1C4C2×C4C42

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

2C2
28C2
2C4
2C4
14C22
14C4
2C14
4D7
2C2×C4
7D4
7Q8
14C8
14D4
14C2×C4
2C28
2C28
2D14
2Dic7
7M4(2)
7C4○D4
2C7⋊D4
2C4×D7
2C7⋊C8
2C2×C28
7C4≀C2

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 36 15 50)(2 35 16 49)(3 34 17 48)(4 33 18 47)(5 32 19 46)(6 31 20 45)(7 30 21 44)(8 29 22 43)(9 56 23 42)(10 55 24 41)(11 54 25 40)(12 53 26 39)(13 52 27 38)(14 51 28 37)
(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), (1,36,15,50)(2,35,16,49)(3,34,17,48)(4,33,18,47)(5,32,19,46)(6,31,20,45)(7,30,21,44)(8,29,22,43)(9,56,23,42)(10,55,24,41)(11,54,25,40)(12,53,26,39)(13,52,27,38)(14,51,28,37), (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), (1,36,15,50)(2,35,16,49)(3,34,17,48)(4,33,18,47)(5,32,19,46)(6,31,20,45)(7,30,21,44)(8,29,22,43)(9,56,23,42)(10,55,24,41)(11,54,25,40)(12,53,26,39)(13,52,27,38)(14,51,28,37), (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)], [(1,36,15,50),(2,35,16,49),(3,34,17,48),(4,33,18,47),(5,32,19,46),(6,31,20,45),(7,30,21,44),(8,29,22,43),(9,56,23,42),(10,55,24,41),(11,54,25,40),(12,53,26,39),(13,52,27,38),(14,51,28,37)], [(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 2A2B2C4A4B4C···4G4H7A7B7C8A8B14A···14I28A···28AJ
order1222444···447778814···1428···28
size11228112···22822228282···22···2

62 irreducible representations

dim111111222222222
type+++++++++
imageC1C2C2C2C4C4D4D4D7D14C4≀C2C4×D7D28C7⋊D4Dic14⋊C4
kernelDic14⋊C4C4.Dic7C4×C28C4○D28Dic14D28C28C2×C14C42C2×C4C7C4C4C22C1
# reps1111221133466624

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

227
510
,
152
314
,
185
227
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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Subgroup lattice of Dic14⋊C4 in TeX

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