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## G = C32⋊Dic7order 252 = 22·32·7

### The semidirect product of C32 and Dic7 acting via Dic7/C7=C4

Aliases: C32⋊Dic7, C7⋊(C32⋊C4), C3⋊S3.D7, (C3×C21)⋊2C4, (C7×C3⋊S3).2C2, SmallGroup(252,32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C21 — C32⋊Dic7
 Chief series C1 — C7 — C3×C21 — C7×C3⋊S3 — C32⋊Dic7
 Lower central C3×C21 — C32⋊Dic7
 Upper central C1

Generators and relations for C32⋊Dic7
G = < a,b,c,d | a3=b3=c14=1, d2=c7, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a-1b-1, dcd-1=c-1 >

Character table of C32⋊Dic7

 class 1 2 3A 3B 4A 4B 7A 7B 7C 14A 14B 14C 21A 21B 21C 21D 21E 21F 21G 21H 21I 21J 21K 21L size 1 9 4 4 63 63 2 2 2 18 18 18 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 1 1 -i i 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ4 1 -1 1 1 i -i 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ5 2 2 2 2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 orthogonal lifted from D7 ρ6 2 2 2 2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 orthogonal lifted from D7 ρ7 2 2 2 2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 orthogonal lifted from D7 ρ8 2 -2 2 2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 symplectic lifted from Dic7, Schur index 2 ρ9 2 -2 2 2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 symplectic lifted from Dic7, Schur index 2 ρ10 2 -2 2 2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 symplectic lifted from Dic7, Schur index 2 ρ11 4 0 -2 1 0 0 4 4 4 0 0 0 1 -2 -2 -2 -2 -2 1 1 1 1 -2 1 orthogonal lifted from C32⋊C4 ρ12 4 0 1 -2 0 0 4 4 4 0 0 0 -2 1 1 1 1 1 -2 -2 -2 -2 1 -2 orthogonal lifted from C32⋊C4 ρ13 4 0 -2 1 0 0 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 0 0 0 2ζ76-ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 2ζ75-ζ72 -ζ76+2ζ7 2ζ74-ζ73 -ζ75+2ζ72 -ζ74-ζ73 -ζ74+2ζ73 complex faithful ρ14 4 0 1 -2 0 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 0 0 0 -ζ75-ζ72 2ζ75-ζ72 2ζ74-ζ73 -ζ74+2ζ73 -ζ75+2ζ72 -ζ76+2ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 2ζ76-ζ7 -ζ76-ζ7 complex faithful ρ15 4 0 -2 1 0 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 0 0 0 2ζ74-ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ76+2ζ7 -ζ74+2ζ73 2ζ75-ζ72 2ζ76-ζ7 -ζ75-ζ72 -ζ75+2ζ72 complex faithful ρ16 4 0 -2 1 0 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 0 0 0 -ζ75+2ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 2ζ74-ζ73 2ζ75-ζ72 2ζ76-ζ7 -ζ74+2ζ73 -ζ76-ζ7 -ζ76+2ζ7 complex faithful ρ17 4 0 1 -2 0 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 0 0 0 -ζ74-ζ73 -ζ74+2ζ73 -ζ76+2ζ7 2ζ76-ζ7 2ζ74-ζ73 -ζ75+2ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 2ζ75-ζ72 -ζ75-ζ72 complex faithful ρ18 4 0 1 -2 0 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 0 0 0 -ζ74-ζ73 2ζ74-ζ73 2ζ76-ζ7 -ζ76+2ζ7 -ζ74+2ζ73 2ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ75+2ζ72 -ζ75-ζ72 complex faithful ρ19 4 0 1 -2 0 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 0 0 0 -ζ75-ζ72 -ζ75+2ζ72 -ζ74+2ζ73 2ζ74-ζ73 2ζ75-ζ72 2ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ76+2ζ7 -ζ76-ζ7 complex faithful ρ20 4 0 1 -2 0 0 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 0 0 0 -ζ76-ζ7 2ζ76-ζ7 -ζ75+2ζ72 2ζ75-ζ72 -ζ76+2ζ7 2ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ74+2ζ73 -ζ74-ζ73 complex faithful ρ21 4 0 -2 1 0 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 0 0 0 2ζ75-ζ72 -ζ75-ζ72 -ζ74-ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ74+2ζ73 -ζ75+2ζ72 -ζ76+2ζ7 2ζ74-ζ73 -ζ76-ζ7 2ζ76-ζ7 complex faithful ρ22 4 0 1 -2 0 0 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 0 0 0 -ζ76-ζ7 -ζ76+2ζ7 2ζ75-ζ72 -ζ75+2ζ72 2ζ76-ζ7 -ζ74+2ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 2ζ74-ζ73 -ζ74-ζ73 complex faithful ρ23 4 0 -2 1 0 0 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 0 0 0 -ζ76+2ζ7 -ζ76-ζ7 -ζ75-ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ75+2ζ72 2ζ76-ζ7 -ζ74+2ζ73 2ζ75-ζ72 -ζ74-ζ73 2ζ74-ζ73 complex faithful ρ24 4 0 -2 1 0 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 0 0 0 -ζ74+2ζ73 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 2ζ76-ζ7 2ζ74-ζ73 -ζ75+2ζ72 -ζ76+2ζ7 -ζ75-ζ72 2ζ75-ζ72 complex faithful

Smallest permutation representation of C32⋊Dic7
On 42 points
Generators in S42
(8 42 35)(9 36 29)(10 30 37)(11 38 31)(12 32 39)(13 40 33)(14 34 41)
(1 16 23)(2 24 17)(3 18 25)(4 26 19)(5 20 27)(6 28 21)(7 22 15)(8 42 35)(9 36 29)(10 30 37)(11 38 31)(12 32 39)(13 40 33)(14 34 41)
(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)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 40 22 33)(16 39 23 32)(17 38 24 31)(18 37 25 30)(19 36 26 29)(20 35 27 42)(21 34 28 41)

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

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

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

Matrix representation of C32⋊Dic7 in GL4(𝔽337) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 297 0 336 336
,
 1 269 0 0 5 335 0 0 0 8 0 1 297 8 336 336
,
 8 0 0 0 40 329 0 0 232 0 295 0 227 0 42 42
,
 329 0 269 0 0 0 336 1 105 0 8 0 110 336 8 0
G:=sub<GL(4,GF(337))| [1,0,0,297,0,1,0,0,0,0,0,336,0,0,1,336],[1,5,0,297,269,335,8,8,0,0,0,336,0,0,1,336],[8,40,232,227,0,329,0,0,0,0,295,42,0,0,0,42],[329,0,105,110,0,0,0,336,269,336,8,8,0,1,0,0] >;

C32⋊Dic7 in GAP, Magma, Sage, TeX

C_3^2\rtimes {\rm Dic}_7
% in TeX

G:=Group("C3^2:Dic7");
// GroupNames label

G:=SmallGroup(252,32);
// by ID

G=gap.SmallGroup(252,32);
# by ID

G:=PCGroup([5,-2,-2,-3,3,-7,10,302,67,323,248,5404]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^14=1,d^2=c^7,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations

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