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G = C7xSD16order 112 = 24·7

Direct product of C7 and SD16

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

Aliases: C7xSD16, Q8:C14, C8:2C14, C56:6C2, D4.C14, C14.15D4, C28.18C22, (C7xQ8):4C2, C2.4(C7xD4), C4.2(C2xC14), (C7xD4).2C2, SmallGroup(112,25)

Series: Derived Chief Lower central Upper central

C1C4 — C7xSD16
C1C2C4C28C7xQ8 — C7xSD16
C1C2C4 — C7xSD16
C1C14C28 — C7xSD16

Generators and relations for C7xSD16
 G = < a,b,c | a7=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Subgroups: 30 in 20 conjugacy classes, 14 normal (all characteristic)
Quotients: C1, C2, C22, C7, D4, C14, SD16, C2xC14, C7xD4, C7xSD16
4C2
2C4
2C22
4C14
2C28
2C2xC14

Smallest permutation representation of C7xSD16
On 56 points
Generators in S56
(1 26 38 51 22 43 10)(2 27 39 52 23 44 11)(3 28 40 53 24 45 12)(4 29 33 54 17 46 13)(5 30 34 55 18 47 14)(6 31 35 56 19 48 15)(7 32 36 49 20 41 16)(8 25 37 50 21 42 9)
(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 39)(35 37)(36 40)(41 45)(42 48)(44 46)(49 53)(50 56)(52 54)

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

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

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

C7xSD16 is a maximal subgroup of   D56:C2  SD16:D7  SD16:3D7

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+++++
imageC1C2C2C2C7C14C14C14D4SD16C7xD4C7xSD16
kernelC7xSD16C56C7xD4C7xQ8SD16C8D4Q8C14C7C2C1
# reps1111666612612

Matrix representation of C7xSD16 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] >;

C7xSD16 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 C7xSD16 in TeX

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