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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, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C2×C6, M4(2), C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C4.10D4, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C2×Dic6, C322C8, C324Q8, C2×C3⋊Dic3, C6×C12, C62.C4, C2×C324Q8, C3⋊Dic3.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, 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 47 13)(3 15 41)(5 43 9)(7 11 45)(18 32 33)(20 35 26)(22 28 37)(24 39 30)
(1 9 47 5 13 43)(2 10 48 6 14 44)(3 45 15 7 41 11)(4 46 16 8 42 12)(17 27 40 21 31 36)(18 37 32 22 33 28)(19 38 25 23 34 29)(20 30 35 24 26 39)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(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 29 7 31 5 25 3 27)(2 30 4 28 6 26 8 32)(9 19 15 21 13 23 11 17)(10 20 12 18 14 24 16 22)(33 48 39 42 37 44 35 46)(34 41 36 47 38 45 40 43)

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

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

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

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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