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## G = D4×C14order 112 = 24·7

### Direct product of C14 and D4

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

Aliases: D4×C14, C232C14, C284C22, C14.11C23, C4⋊(C2×C14), (C2×C4)⋊2C14, (C2×C28)⋊6C2, C22⋊(C2×C14), (C22×C14)⋊1C2, (C2×C14)⋊2C22, C2.1(C22×C14), SmallGroup(112,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C14
 Chief series C1 — C2 — C14 — C2×C14 — C7×D4 — D4×C14
 Lower central C1 — C2 — D4×C14
 Upper central C1 — C2×C14 — D4×C14

Generators and relations for D4×C14
G = < a,b,c | a14=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 70 in 54 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, D4, C23, C14, C14, C14, C2×D4, C28, C2×C14, C2×C14, C2×C14, C2×C28, C7×D4, C22×C14, D4×C14
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C7×D4, C22×C14, D4×C14

Smallest permutation representation of D4×C14
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 28 36 46)(2 15 37 47)(3 16 38 48)(4 17 39 49)(5 18 40 50)(6 19 41 51)(7 20 42 52)(8 21 29 53)(9 22 30 54)(10 23 31 55)(11 24 32 56)(12 25 33 43)(13 26 34 44)(14 27 35 45)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 43)(41 44)(42 45)

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

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

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

D4×C14 is a maximal subgroup of
D4⋊Dic7  C28.D4  C23⋊Dic7  D4.D14  C23.18D14  C28.17D4  C23⋊D14  C282D4  Dic7⋊D4  C28⋊D4  D46D14

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 7A ··· 7F 14A ··· 14R 14S ··· 14AP 28A ··· 28L order 1 2 2 2 2 2 2 2 4 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C7 C14 C14 C14 D4 C7×D4 kernel D4×C14 C2×C28 C7×D4 C22×C14 C2×D4 C2×C4 D4 C23 C14 C2 # reps 1 1 4 2 6 6 24 12 2 12

Matrix representation of D4×C14 in GL3(𝔽29) generated by

 28 0 0 0 4 0 0 0 4
,
 1 0 0 0 0 1 0 28 0
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(29))| [28,0,0,0,4,0,0,0,4],[1,0,0,0,0,28,0,1,0],[1,0,0,0,0,1,0,1,0] >;

D4×C14 in GAP, Magma, Sage, TeX

D_4\times C_{14}
% in TeX

G:=Group("D4xC14");
// GroupNames label

G:=SmallGroup(112,38);
// by ID

G=gap.SmallGroup(112,38);
# by ID

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

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

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