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## G = C8×D7order 112 = 24·7

### Direct product of C8 and D7

Aliases: C8×D7, C563C2, C8Dic7, D14.2C4, C4.12D14, Dic7.2C4, C28.12C22, C8(C7⋊C8), C7⋊C86C2, C71(C2×C8), C2.1(C4×D7), C14.1(C2×C4), (C4×D7).3C2, SmallGroup(112,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C8×D7
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C8×D7
 Lower central C7 — C8×D7
 Upper central C1 — C8

Generators and relations for C8×D7
G = < a,b,c | a8=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C8×D7
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 49 46 19 16 34 31)(2 50 47 20 9 35 32)(3 51 48 21 10 36 25)(4 52 41 22 11 37 26)(5 53 42 23 12 38 27)(6 54 43 24 13 39 28)(7 55 44 17 14 40 29)(8 56 45 18 15 33 30)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 41)(16 42)(17 21)(18 22)(19 23)(20 24)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)

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

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

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

C8×D7 is a maximal subgroup of   C16⋊D7  D28.2C4  D28.C4  D83D7  SD163D7  Q8.D14  D21⋊C8
C8×D7 is a maximal quotient of   C16⋊D7  Dic7⋊C8  D14⋊C8  D21⋊C8

40 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 14A 14B 14C 28A ··· 28F 56A ··· 56L order 1 2 2 2 4 4 4 4 7 7 7 8 8 8 8 8 8 8 8 14 14 14 28 ··· 28 56 ··· 56 size 1 1 7 7 1 1 7 7 2 2 2 1 1 1 1 7 7 7 7 2 2 2 2 ··· 2 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C8 D7 D14 C4×D7 C8×D7 kernel C8×D7 C7⋊C8 C56 C4×D7 Dic7 D14 D7 C8 C4 C2 C1 # reps 1 1 1 1 2 2 8 3 3 6 12

Matrix representation of C8×D7 in GL2(𝔽41) generated by

 14 0 0 14
,
 0 9 9 37
,
 37 29 32 4
G:=sub<GL(2,GF(41))| [14,0,0,14],[0,9,9,37],[37,32,29,4] >;

C8×D7 in GAP, Magma, Sage, TeX

C_8\times D_7
% in TeX

G:=Group("C8xD7");
// GroupNames label

G:=SmallGroup(112,3);
// by ID

G=gap.SmallGroup(112,3);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,26,42,2404]);
// Polycyclic

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

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