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## G = C7×C32⋊C4order 252 = 22·32·7

### Direct product of C7 and C32⋊C4

Aliases: C7×C32⋊C4, C32⋊C28, C3⋊S3.C14, (C3×C21)⋊1C4, (C7×C3⋊S3).1C2, SmallGroup(252,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C7×C32⋊C4
 Chief series C1 — C32 — C3⋊S3 — C7×C3⋊S3 — C7×C32⋊C4
 Lower central C32 — C7×C32⋊C4
 Upper central C1 — C7

Generators and relations for C7×C32⋊C4
G = < a,b,c,d | a7=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Smallest permutation representation of C7×C32⋊C4
On 42 points
Generators in S42
(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)
(8 42 16)(9 36 17)(10 37 18)(11 38 19)(12 39 20)(13 40 21)(14 41 15)
(1 30 23)(2 31 24)(3 32 25)(4 33 26)(5 34 27)(6 35 28)(7 29 22)(8 42 16)(9 36 17)(10 37 18)(11 38 19)(12 39 20)(13 40 21)(14 41 15)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 36)(8 35 16 28)(9 29 17 22)(10 30 18 23)(11 31 19 24)(12 32 20 25)(13 33 21 26)(14 34 15 27)

G:=sub<Sym(42)| (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), (8,42,16)(9,36,17)(10,37,18)(11,38,19)(12,39,20)(13,40,21)(14,41,15), (1,30,23)(2,31,24)(3,32,25)(4,33,26)(5,34,27)(6,35,28)(7,29,22)(8,42,16)(9,36,17)(10,37,18)(11,38,19)(12,39,20)(13,40,21)(14,41,15), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,36)(8,35,16,28)(9,29,17,22)(10,30,18,23)(11,31,19,24)(12,32,20,25)(13,33,21,26)(14,34,15,27)>;

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), (8,42,16)(9,36,17)(10,37,18)(11,38,19)(12,39,20)(13,40,21)(14,41,15), (1,30,23)(2,31,24)(3,32,25)(4,33,26)(5,34,27)(6,35,28)(7,29,22)(8,42,16)(9,36,17)(10,37,18)(11,38,19)(12,39,20)(13,40,21)(14,41,15), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,36)(8,35,16,28)(9,29,17,22)(10,30,18,23)(11,31,19,24)(12,32,20,25)(13,33,21,26)(14,34,15,27) );

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)], [(8,42,16),(9,36,17),(10,37,18),(11,38,19),(12,39,20),(13,40,21),(14,41,15)], [(1,30,23),(2,31,24),(3,32,25),(4,33,26),(5,34,27),(6,35,28),(7,29,22),(8,42,16),(9,36,17),(10,37,18),(11,38,19),(12,39,20),(13,40,21),(14,41,15)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,36),(8,35,16,28),(9,29,17,22),(10,30,18,23),(11,31,19,24),(12,32,20,25),(13,33,21,26),(14,34,15,27)]])

42 conjugacy classes

 class 1 2 3A 3B 4A 4B 7A ··· 7F 14A ··· 14F 21A ··· 21L 28A ··· 28L order 1 2 3 3 4 4 7 ··· 7 14 ··· 14 21 ··· 21 28 ··· 28 size 1 9 4 4 9 9 1 ··· 1 9 ··· 9 4 ··· 4 9 ··· 9

42 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + image C1 C2 C4 C7 C14 C28 C32⋊C4 C7×C32⋊C4 kernel C7×C32⋊C4 C7×C3⋊S3 C3×C21 C32⋊C4 C3⋊S3 C32 C7 C1 # reps 1 1 2 6 6 12 2 12

Matrix representation of C7×C32⋊C4 in GL4(𝔽337) generated by

 52 0 0 0 0 52 0 0 0 0 52 0 0 0 0 52
,
 1 0 0 0 0 1 0 0 0 0 0 1 336 336 336 336
,
 0 336 0 0 1 336 0 0 0 1 0 1 336 0 336 336
,
 0 0 336 1 336 336 335 336 0 0 1 0 0 1 1 0
G:=sub<GL(4,GF(337))| [52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[1,0,0,336,0,1,0,336,0,0,0,336,0,0,1,336],[0,1,0,336,336,336,1,0,0,0,0,336,0,0,1,336],[0,336,0,0,0,336,0,1,336,335,1,1,1,336,0,0] >;

C7×C32⋊C4 in GAP, Magma, Sage, TeX

C_7\times C_3^2\rtimes C_4
% in TeX

G:=Group("C7xC3^2:C4");
// GroupNames label

G:=SmallGroup(252,31);
// by ID

G=gap.SmallGroup(252,31);
# by ID

G:=PCGroup([5,-2,-7,-2,-3,3,70,3923,93,5604,314]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations

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