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## G = C7×C7⋊C8order 392 = 23·72

### Direct product of C7 and C7⋊C8

Aliases: C7×C7⋊C8, C7⋊C56, C14.C28, C723C8, C28.8D7, C28.2C14, C14.4Dic7, C4.2(C7×D7), C2.(C7×Dic7), (C7×C28).3C2, (C7×C14).2C4, SmallGroup(392,14)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C7×C7⋊C8
G = < a,b,c | a7=b7=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C7×C7⋊C8
On 56 points
Generators in S56
(1 34 19 49 45 30 10)(2 35 20 50 46 31 11)(3 36 21 51 47 32 12)(4 37 22 52 48 25 13)(5 38 23 53 41 26 14)(6 39 24 54 42 27 15)(7 40 17 55 43 28 16)(8 33 18 56 44 29 9)
(1 45 34 30 19 10 49)(2 50 11 20 31 35 46)(3 47 36 32 21 12 51)(4 52 13 22 25 37 48)(5 41 38 26 23 14 53)(6 54 15 24 27 39 42)(7 43 40 28 17 16 55)(8 56 9 18 29 33 44)
(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)

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

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

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

140 conjugacy classes

 class 1 2 4A 4B 7A ··· 7F 7G ··· 7AA 8A 8B 8C 8D 14A ··· 14F 14G ··· 14AA 28A ··· 28L 28M ··· 28BB 56A ··· 56X order 1 2 4 4 7 ··· 7 7 ··· 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 1 ··· 1 2 ··· 2 7 7 7 7 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 7 ··· 7

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C4 C7 C8 C14 C28 C56 D7 Dic7 C7⋊C8 C7×D7 C7×Dic7 C7×C7⋊C8 kernel C7×C7⋊C8 C7×C28 C7×C14 C7⋊C8 C72 C28 C14 C7 C28 C14 C7 C4 C2 C1 # reps 1 1 2 6 4 6 12 24 3 3 6 18 18 36

Matrix representation of C7×C7⋊C8 in GL2(𝔽29) generated by

 23 0 0 23
,
 7 0 0 25
,
 0 17 1 0
G:=sub<GL(2,GF(29))| [23,0,0,23],[7,0,0,25],[0,1,17,0] >;

C7×C7⋊C8 in GAP, Magma, Sage, TeX

C_7\times C_7\rtimes C_8
% in TeX

G:=Group("C7xC7:C8");
// GroupNames label

G:=SmallGroup(392,14);
// by ID

G=gap.SmallGroup(392,14);
# by ID

G:=PCGroup([5,-2,-7,-2,-2,-7,70,42,8404]);
// Polycyclic

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

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