Copied to
clipboard

G = C7×SD16order 112 = 24·7

Direct product of C7 and SD16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C7×SD16, Q8⋊C14, C82C14, C566C2, D4.C14, C14.15D4, C28.18C22, (C7×Q8)⋊4C2, C2.4(C7×D4), C4.2(C2×C14), (C7×D4).2C2, SmallGroup(112,25)

Series: Derived Chief Lower central Upper central

C1C4 — C7×SD16
C1C2C4C28C7×Q8 — C7×SD16
C1C2C4 — C7×SD16
C1C14C28 — C7×SD16

Generators and relations for C7×SD16
 G = < a,b,c | a7=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

4C2
2C4
2C22
4C14
2C28
2C2×C14

Smallest permutation representation of C7×SD16
On 56 points
Generators in S56
(1 26 18 51 39 43 14)(2 27 19 52 40 44 15)(3 28 20 53 33 45 16)(4 29 21 54 34 46 9)(5 30 22 55 35 47 10)(6 31 23 56 36 48 11)(7 32 24 49 37 41 12)(8 25 17 50 38 42 13)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)(25 31)(27 29)(28 32)(33 37)(34 40)(36 38)(41 45)(42 48)(44 46)(49 53)(50 56)(52 54)

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

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

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

C7×SD16 is a maximal subgroup of   D56⋊C2  SD16⋊D7  SD163D7

49 conjugacy classes

class 1 2A2B4A4B7A···7F8A8B14A···14F14G···14L28A···28F28G···28L56A···56L
order122447···78814···1414···1428···2828···2856···56
size114241···1221···14···42···24···42···2

49 irreducible representations

dim111111112222
type+++++
imageC1C2C2C2C7C14C14C14D4SD16C7×D4C7×SD16
kernelC7×SD16C56C7×D4C7×Q8SD16C8D4Q8C14C7C2C1
# reps1111666612612

Matrix representation of C7×SD16 in GL2(𝔽43) generated by

160
016
,
82
118
,
420
01
G:=sub<GL(2,GF(43))| [16,0,0,16],[8,11,2,8],[42,0,0,1] >;

C7×SD16 in GAP, Magma, Sage, TeX

C_7\times {\rm SD}_{16}
% in TeX

G:=Group("C7xSD16");
// GroupNames label

G:=SmallGroup(112,25);
// by ID

G=gap.SmallGroup(112,25);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-2,280,301,1683,848,58]);
// Polycyclic

G:=Group<a,b,c|a^7=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations

Export

Subgroup lattice of C7×SD16 in TeX

׿
×
𝔽