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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

C1C2 — D4×C14
C1C2C14C2×C14C7×D4 — D4×C14
C1C2 — D4×C14
C1C2×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 [×2], C2 [×4], C4 [×2], C22, C22 [×4], C22 [×4], C7, C2×C4, D4 [×4], C23 [×2], C14, C14 [×2], C14 [×4], C2×D4, C28 [×2], C2×C14, C2×C14 [×4], C2×C14 [×4], C2×C28, C7×D4 [×4], C22×C14 [×2], D4×C14
Quotients: C1, C2 [×7], C22 [×7], C7, D4 [×2], C23, C14 [×7], C2×D4, C2×C14 [×7], C7×D4 [×2], 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 24 33 52)(2 25 34 53)(3 26 35 54)(4 27 36 55)(5 28 37 56)(6 15 38 43)(7 16 39 44)(8 17 40 45)(9 18 41 46)(10 19 42 47)(11 20 29 48)(12 21 30 49)(13 22 31 50)(14 23 32 51)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 15)(14 16)(29 55)(30 56)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)

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

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

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

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 2A2B2C2D2E2F2G4A4B7A···7F14A···14R14S···14AP28A···28L
order12222222447···714···1414···1428···28
size11112222221···11···12···22···2

70 irreducible representations

dim1111111122
type+++++
imageC1C2C2C2C7C14C14C14D4C7×D4
kernelD4×C14C2×C28C7×D4C22×C14C2×D4C2×C4D4C23C14C2
# reps1142662412212

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

2800
040
004
,
100
001
0280
,
100
001
010
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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