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## 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

 Derived series C1 — C4 — C7×SD16
 Chief series C1 — C2 — C4 — C28 — C7×Q8 — C7×SD16
 Lower central C1 — C2 — C4 — C7×SD16
 Upper central C1 — C14 — C28 — C7×SD16

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

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 2A 2B 4A 4B 7A ··· 7F 8A 8B 14A ··· 14F 14G ··· 14L 28A ··· 28F 28G ··· 28L 56A ··· 56L order 1 2 2 4 4 7 ··· 7 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 4 2 4 1 ··· 1 2 2 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2

49 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C2 C7 C14 C14 C14 D4 SD16 C7×D4 C7×SD16 kernel C7×SD16 C56 C7×D4 C7×Q8 SD16 C8 D4 Q8 C14 C7 C2 C1 # reps 1 1 1 1 6 6 6 6 1 2 6 12

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

 16 0 0 16
,
 8 2 11 8
,
 42 0 0 1
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

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