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## G = C32⋊Dic11order 396 = 22·32·11

### The semidirect product of C32 and Dic11 acting via Dic11/C11=C4

Aliases: C32⋊Dic11, (C3×C33)⋊2C4, C11⋊(C32⋊C4), C3⋊S3.D11, (C11×C3⋊S3).2C2, SmallGroup(396,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C33 — C32⋊Dic11
 Chief series C1 — C11 — C3×C33 — C11×C3⋊S3 — C32⋊Dic11
 Lower central C3×C33 — C32⋊Dic11
 Upper central C1

Generators and relations for C32⋊Dic11
G = < a,b,c,d | a3=b3=c22=1, d2=c11, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a-1b-1, dcd-1=c-1 >

Smallest permutation representation of C32⋊Dic11
On 66 points
Generators in S66
(1 25 36)(2 37 26)(3 27 38)(4 39 28)(5 29 40)(6 41 30)(7 31 42)(8 43 32)(9 33 44)(10 23 34)(11 35 24)(12 62 51)(13 52 63)(14 64 53)(15 54 65)(16 66 55)(17 56 45)(18 46 57)(19 58 47)(20 48 59)(21 60 49)(22 50 61)
(12 51 62)(13 63 52)(14 53 64)(15 65 54)(16 55 66)(17 45 56)(18 57 46)(19 47 58)(20 59 48)(21 49 60)(22 61 50)
(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 57 58 59 60 61 62 63 64 65 66)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 22)(8 21)(9 20)(10 19)(11 18)(23 58 34 47)(24 57 35 46)(25 56 36 45)(26 55 37 66)(27 54 38 65)(28 53 39 64)(29 52 40 63)(30 51 41 62)(31 50 42 61)(32 49 43 60)(33 48 44 59)

G:=sub<Sym(66)| (1,25,36)(2,37,26)(3,27,38)(4,39,28)(5,29,40)(6,41,30)(7,31,42)(8,43,32)(9,33,44)(10,23,34)(11,35,24)(12,62,51)(13,52,63)(14,64,53)(15,54,65)(16,66,55)(17,56,45)(18,46,57)(19,58,47)(20,48,59)(21,60,49)(22,50,61), (12,51,62)(13,63,52)(14,53,64)(15,65,54)(16,55,66)(17,45,56)(18,57,46)(19,47,58)(20,59,48)(21,49,60)(22,61,50), (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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,22)(8,21)(9,20)(10,19)(11,18)(23,58,34,47)(24,57,35,46)(25,56,36,45)(26,55,37,66)(27,54,38,65)(28,53,39,64)(29,52,40,63)(30,51,41,62)(31,50,42,61)(32,49,43,60)(33,48,44,59)>;

G:=Group( (1,25,36)(2,37,26)(3,27,38)(4,39,28)(5,29,40)(6,41,30)(7,31,42)(8,43,32)(9,33,44)(10,23,34)(11,35,24)(12,62,51)(13,52,63)(14,64,53)(15,54,65)(16,66,55)(17,56,45)(18,46,57)(19,58,47)(20,48,59)(21,60,49)(22,50,61), (12,51,62)(13,63,52)(14,53,64)(15,65,54)(16,55,66)(17,45,56)(18,57,46)(19,47,58)(20,59,48)(21,49,60)(22,61,50), (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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,22)(8,21)(9,20)(10,19)(11,18)(23,58,34,47)(24,57,35,46)(25,56,36,45)(26,55,37,66)(27,54,38,65)(28,53,39,64)(29,52,40,63)(30,51,41,62)(31,50,42,61)(32,49,43,60)(33,48,44,59) );

G=PermutationGroup([[(1,25,36),(2,37,26),(3,27,38),(4,39,28),(5,29,40),(6,41,30),(7,31,42),(8,43,32),(9,33,44),(10,23,34),(11,35,24),(12,62,51),(13,52,63),(14,64,53),(15,54,65),(16,66,55),(17,56,45),(18,46,57),(19,58,47),(20,48,59),(21,60,49),(22,50,61)], [(12,51,62),(13,63,52),(14,53,64),(15,65,54),(16,55,66),(17,45,56),(18,57,46),(19,47,58),(20,59,48),(21,49,60),(22,61,50)], [(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),(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,22),(8,21),(9,20),(10,19),(11,18),(23,58,34,47),(24,57,35,46),(25,56,36,45),(26,55,37,66),(27,54,38,65),(28,53,39,64),(29,52,40,63),(30,51,41,62),(31,50,42,61),(32,49,43,60),(33,48,44,59)]])

36 conjugacy classes

 class 1 2 3A 3B 4A 4B 11A ··· 11E 22A ··· 22E 33A ··· 33T order 1 2 3 3 4 4 11 ··· 11 22 ··· 22 33 ··· 33 size 1 9 4 4 99 99 2 ··· 2 18 ··· 18 4 ··· 4

36 irreducible representations

 dim 1 1 1 2 2 4 4 type + + + - + image C1 C2 C4 D11 Dic11 C32⋊C4 C32⋊Dic11 kernel C32⋊Dic11 C11×C3⋊S3 C3×C33 C3⋊S3 C32 C11 C1 # reps 1 1 2 5 5 2 20

Matrix representation of C32⋊Dic11 in GL4(𝔽397) generated by

 0 1 0 0 396 396 0 0 0 0 396 396 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 396 396
,
 393 0 0 0 4 4 0 0 0 0 99 0 0 0 298 298
,
 0 0 1 0 0 0 0 1 1 0 0 0 396 396 0 0
G:=sub<GL(4,GF(397))| [0,396,0,0,1,396,0,0,0,0,396,1,0,0,396,0],[1,0,0,0,0,1,0,0,0,0,0,396,0,0,1,396],[393,4,0,0,0,4,0,0,0,0,99,298,0,0,0,298],[0,0,1,396,0,0,0,396,1,0,0,0,0,1,0,0] >;

C32⋊Dic11 in GAP, Magma, Sage, TeX

C_3^2\rtimes {\rm Dic}_{11}
% in TeX

G:=Group("C3^2:Dic11");
// GroupNames label

G:=SmallGroup(396,18);
// by ID

G=gap.SmallGroup(396,18);
# by ID

G:=PCGroup([5,-2,-2,-3,3,-11,10,302,67,323,248,9004]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^22=1,d^2=c^11,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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