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## G = C3⋊Dic3.D4order 288 = 25·32

### 9th non-split extension by C3⋊Dic3 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C3⋊Dic3.D4
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C62.C4 — C3⋊Dic3.D4
 Lower central C32 — C3×C6 — C62 — C3⋊Dic3.D4
 Upper central C1 — C2 — C22 — C2×C4

Generators and relations for C3⋊Dic3.D4
G = < a,b,c,d,e | a3=b6=1, c2=d4=b3, e2=dcd-1=b3c, ab=ba, cac-1=a-1, dad-1=eae-1=ab4, cbc-1=b-1, dbd-1=ebe-1=ab-1, ce=ec, ede-1=b3cd3 >

Subgroups: 360 in 76 conjugacy classes, 16 normal (12 characteristic)
C1, C2, C2, C3 [×2], C4 [×4], C22, C6 [×6], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C32, Dic3 [×8], C12 [×4], C2×C6 [×2], M4(2) [×2], C2×Q8, C3×C6, C3×C6, Dic6 [×8], C2×Dic3 [×4], C2×C12 [×2], C4.10D4, C3⋊Dic3 [×2], C3⋊Dic3, C3×C12, C62, C2×Dic6 [×2], C322C8 [×2], C324Q8 [×2], C2×C3⋊Dic3 [×2], C6×C12, C62.C4 [×2], C2×C324Q8, C3⋊Dic3.D4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, C4.10D4, C32⋊C4, C2×C32⋊C4, C62⋊C4, C3⋊Dic3.D4

Character table of C3⋊Dic3.D4

 class 1 2A 2B 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 2 4 4 4 18 18 36 4 4 4 4 4 4 36 36 36 36 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 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 -1 -1 -1 linear of order 2 ρ3 1 1 1 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 ρ4 1 1 1 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 ρ5 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 -i -i i i 1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 i i -i -i 1 1 1 1 1 1 1 1 linear of order 4 ρ7 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 -i i i -i -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ8 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 i -i -i i -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ9 2 2 -2 2 2 0 -2 2 0 -2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 2 2 0 2 -2 0 -2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 -4 1 -2 0 0 0 0 -1 1 -1 2 -2 2 0 0 0 0 0 -3 0 3 3 0 0 -3 orthogonal lifted from C62⋊C4 ρ12 4 4 -4 1 -2 0 0 0 0 -1 1 -1 2 -2 2 0 0 0 0 0 3 0 -3 -3 0 0 3 orthogonal lifted from C62⋊C4 ρ13 4 4 -4 -2 1 0 0 0 0 2 -2 2 -1 1 -1 0 0 0 0 3 0 -3 0 0 3 -3 0 orthogonal lifted from C62⋊C4 ρ14 4 4 4 1 -2 4 0 0 0 1 1 1 -2 -2 -2 0 0 0 0 -2 1 -2 1 1 -2 -2 1 orthogonal lifted from C32⋊C4 ρ15 4 4 4 -2 1 -4 0 0 0 -2 -2 -2 1 1 1 0 0 0 0 -1 2 -1 2 2 -1 -1 2 orthogonal lifted from C2×C32⋊C4 ρ16 4 4 4 -2 1 4 0 0 0 -2 -2 -2 1 1 1 0 0 0 0 1 -2 1 -2 -2 1 1 -2 orthogonal lifted from C32⋊C4 ρ17 4 4 -4 -2 1 0 0 0 0 2 -2 2 -1 1 -1 0 0 0 0 -3 0 3 0 0 -3 3 0 orthogonal lifted from C62⋊C4 ρ18 4 4 4 1 -2 -4 0 0 0 1 1 1 -2 -2 -2 0 0 0 0 2 -1 2 -1 -1 2 2 -1 orthogonal lifted from C2×C32⋊C4 ρ19 4 -4 0 4 4 0 0 0 0 0 -4 0 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2 ρ20 4 -4 0 -2 1 0 0 0 0 0 2 0 3 -1 -3 0 0 0 0 √3 0 -√3 2√3 -2√3 -√3 √3 0 symplectic faithful, Schur index 2 ρ21 4 -4 0 1 -2 0 0 0 0 -3 -1 3 0 2 0 0 0 0 0 0 -√3 -2√3 -√3 √3 0 2√3 √3 symplectic faithful, Schur index 2 ρ22 4 -4 0 1 -2 0 0 0 0 -3 -1 3 0 2 0 0 0 0 0 0 √3 2√3 √3 -√3 0 -2√3 -√3 symplectic faithful, Schur index 2 ρ23 4 -4 0 1 -2 0 0 0 0 3 -1 -3 0 2 0 0 0 0 0 -2√3 -√3 0 √3 -√3 2√3 0 √3 symplectic faithful, Schur index 2 ρ24 4 -4 0 1 -2 0 0 0 0 3 -1 -3 0 2 0 0 0 0 0 2√3 √3 0 -√3 √3 -2√3 0 -√3 symplectic faithful, Schur index 2 ρ25 4 -4 0 -2 1 0 0 0 0 0 2 0 -3 -1 3 0 0 0 0 √3 -2√3 √3 0 0 -√3 -√3 2√3 symplectic faithful, Schur index 2 ρ26 4 -4 0 -2 1 0 0 0 0 0 2 0 -3 -1 3 0 0 0 0 -√3 2√3 -√3 0 0 √3 √3 -2√3 symplectic faithful, Schur index 2 ρ27 4 -4 0 -2 1 0 0 0 0 0 2 0 3 -1 -3 0 0 0 0 -√3 0 √3 -2√3 2√3 √3 -√3 0 symplectic faithful, Schur index 2

Smallest permutation representation of C3⋊Dic3.D4
On 48 points
Generators in S48
(1 45 26)(3 28 47)(5 41 30)(7 32 43)(9 17 39)(11 33 19)(13 21 35)(15 37 23)
(1 30 45 5 26 41)(2 31 46 6 27 42)(3 43 28 7 47 32)(4 44 29 8 48 25)(9 35 17 13 39 21)(10 36 18 14 40 22)(11 23 33 15 19 37)(12 24 34 16 20 38)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 19 21 23)(18 24 22 20)(25 31 29 27)(26 28 30 32)(33 35 37 39)(34 40 38 36)(41 43 45 47)(42 48 46 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)
(1 16 7 10 5 12 3 14)(2 9 4 15 6 13 8 11)(17 29 23 31 21 25 19 27)(18 30 20 28 22 26 24 32)(33 46 39 48 37 42 35 44)(34 47 36 45 38 43 40 41)

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

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

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

Matrix representation of C3⋊Dic3.D4 in GL4(𝔽73) generated by

 72 1 0 0 72 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 72 1 0 0 0 0 1 72 0 0 1 0
,
 61 63 0 0 51 12 0 0 0 0 12 10 0 0 22 61
,
 0 0 1 0 0 0 0 1 61 63 0 0 51 12 0 0
,
 0 0 66 14 0 0 59 7 68 19 0 0 14 5 0 0
G:=sub<GL(4,GF(73))| [72,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,1,1,0,0,0,0,1,1,0,0,72,0],[61,51,0,0,63,12,0,0,0,0,12,22,0,0,10,61],[0,0,61,51,0,0,63,12,1,0,0,0,0,1,0,0],[0,0,68,14,0,0,19,5,66,59,0,0,14,7,0,0] >;

C3⋊Dic3.D4 in GAP, Magma, Sage, TeX

C_3\rtimes {\rm Dic}_3.D_4
% in TeX

G:=Group("C3:Dic3.D4");
// GroupNames label

G:=SmallGroup(288,428);
// by ID

G=gap.SmallGroup(288,428);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,120,219,100,675,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^6=1,c^2=d^4=b^3,e^2=d*c*d^-1=b^3*c,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=e*a*e^-1=a*b^4,c*b*c^-1=b^-1,d*b*d^-1=e*b*e^-1=a*b^-1,c*e=e*c,e*d*e^-1=b^3*c*d^3>;
// generators/relations

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