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## G = Dic3.S4order 288 = 25·32

### 3rd non-split extension by Dic3 of S4 acting via S4/A4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C6×A4 — Dic3.S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4 — Dic3×A4 — Dic3.S4
 Lower central C3×A4 — C6×A4 — Dic3.S4
 Upper central C1 — C2

Generators and relations for Dic3.S4
G = < a,b,c,d,e,f | a6=c2=d2=e3=1, b2=f2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=a3b, ece-1=fcf-1=cd=dc, ede-1=c, df=fd, fef-1=e-1 >

Subgroups: 450 in 90 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, Q8, C23, C32, Dic3, Dic3, C12, A4, A4, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×C12, C2×A4, C2×A4, C22×C6, C22⋊Q8, C3×Dic3, C3⋊Dic3, C3×A4, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, A4⋊C4, A4⋊C4, C4×A4, C2×Dic6, C22×Dic3, C322Q8, C6×A4, Dic3.D4, A4⋊Q8, C3×A4⋊C4, C6.7S4, Dic3×A4, Dic3.S4
Quotients: C1, C2, C22, S3, Q8, D6, Dic6, S4, S32, C2×S4, C322Q8, A4⋊Q8, S3×S4, Dic3.S4

Character table of Dic3.S4

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E 12F size 1 1 3 3 2 8 16 6 12 12 18 36 36 2 6 6 8 16 12 12 12 12 24 24 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 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 linear of order 2 ρ5 2 2 2 2 -1 2 -1 0 2 2 0 0 0 -1 -1 -1 2 -1 -1 -1 -1 -1 0 0 orthogonal lifted from S3 ρ6 2 2 2 2 2 -1 -1 -2 0 0 -2 0 0 2 2 2 -1 -1 0 0 0 0 1 1 orthogonal lifted from D6 ρ7 2 2 2 2 -1 2 -1 0 -2 -2 0 0 0 -1 -1 -1 2 -1 1 1 1 1 0 0 orthogonal lifted from D6 ρ8 2 2 2 2 2 -1 -1 2 0 0 2 0 0 2 2 2 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ9 2 -2 -2 2 2 2 2 0 0 0 0 0 0 -2 2 -2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ10 2 -2 -2 2 2 -1 -1 0 0 0 0 0 0 -2 2 -2 1 1 0 0 0 0 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ11 2 -2 -2 2 -1 2 -1 0 0 0 0 0 0 1 -1 1 -2 1 √3 -√3 -√3 √3 0 0 symplectic lifted from Dic6, Schur index 2 ρ12 2 -2 -2 2 2 -1 -1 0 0 0 0 0 0 -2 2 -2 1 1 0 0 0 0 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ13 2 -2 -2 2 -1 2 -1 0 0 0 0 0 0 1 -1 1 -2 1 -√3 √3 √3 -√3 0 0 symplectic lifted from Dic6, Schur index 2 ρ14 3 3 -1 -1 3 0 0 3 1 -1 -1 -1 1 3 -1 -1 0 0 -1 1 -1 1 0 0 orthogonal lifted from S4 ρ15 3 3 -1 -1 3 0 0 -3 1 -1 1 1 -1 3 -1 -1 0 0 -1 1 -1 1 0 0 orthogonal lifted from C2×S4 ρ16 3 3 -1 -1 3 0 0 3 -1 1 -1 1 -1 3 -1 -1 0 0 1 -1 1 -1 0 0 orthogonal lifted from S4 ρ17 3 3 -1 -1 3 0 0 -3 -1 1 1 -1 1 3 -1 -1 0 0 1 -1 1 -1 0 0 orthogonal lifted from C2×S4 ρ18 4 4 4 4 -2 -2 1 0 0 0 0 0 0 -2 -2 -2 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ19 4 -4 -4 4 -2 -2 1 0 0 0 0 0 0 2 -2 2 2 -1 0 0 0 0 0 0 symplectic lifted from C32⋊2Q8, Schur index 2 ρ20 6 6 -2 -2 -3 0 0 0 -2 2 0 0 0 -3 1 1 0 0 -1 1 -1 1 0 0 orthogonal lifted from S3×S4 ρ21 6 6 -2 -2 -3 0 0 0 2 -2 0 0 0 -3 1 1 0 0 1 -1 1 -1 0 0 orthogonal lifted from S3×S4 ρ22 6 -6 2 -2 6 0 0 0 0 0 0 0 0 -6 -2 2 0 0 0 0 0 0 0 0 symplectic lifted from A4⋊Q8, Schur index 2 ρ23 6 -6 2 -2 -3 0 0 0 0 0 0 0 0 3 1 -1 0 0 √3 √3 -√3 -√3 0 0 symplectic faithful, Schur index 2 ρ24 6 -6 2 -2 -3 0 0 0 0 0 0 0 0 3 1 -1 0 0 -√3 -√3 √3 √3 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of Dic3.S4
On 72 points
Generators in S72
(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)(67 68 69 70 71 72)
(1 43 4 46)(2 48 5 45)(3 47 6 44)(7 23 10 20)(8 22 11 19)(9 21 12 24)(13 51 16 54)(14 50 17 53)(15 49 18 52)(25 70 28 67)(26 69 29 72)(27 68 30 71)(31 61 34 64)(32 66 35 63)(33 65 36 62)(37 58 40 55)(38 57 41 60)(39 56 42 59)
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 69)(8 70)(9 71)(10 72)(11 67)(12 68)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(55 58)(56 59)(57 60)(61 64)(62 65)(63 66)
(1 4)(2 5)(3 6)(7 72)(8 67)(9 68)(10 69)(11 70)(12 71)(13 16)(14 17)(15 18)(19 28)(20 29)(21 30)(22 25)(23 26)(24 27)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)(43 46)(44 47)(45 48)(49 52)(50 53)(51 54)(55 64)(56 65)(57 66)(58 61)(59 62)(60 63)
(1 28 40)(2 29 41)(3 30 42)(4 25 37)(5 26 38)(6 27 39)(7 66 54)(8 61 49)(9 62 50)(10 63 51)(11 64 52)(12 65 53)(13 23 35)(14 24 36)(15 19 31)(16 20 32)(17 21 33)(18 22 34)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)
(1 49 4 52)(2 50 5 53)(3 51 6 54)(7 42 10 39)(8 37 11 40)(9 38 12 41)(13 44 16 47)(14 45 17 48)(15 46 18 43)(19 58 22 55)(20 59 23 56)(21 60 24 57)(25 64 28 61)(26 65 29 62)(27 66 30 63)(31 70 34 67)(32 71 35 68)(33 72 36 69)

G:=sub<Sym(72)| (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)(67,68,69,70,71,72), (1,43,4,46)(2,48,5,45)(3,47,6,44)(7,23,10,20)(8,22,11,19)(9,21,12,24)(13,51,16,54)(14,50,17,53)(15,49,18,52)(25,70,28,67)(26,69,29,72)(27,68,30,71)(31,61,34,64)(32,66,35,63)(33,65,36,62)(37,58,40,55)(38,57,41,60)(39,56,42,59), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,69)(8,70)(9,71)(10,72)(11,67)(12,68)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(55,58)(56,59)(57,60)(61,64)(62,65)(63,66), (1,4)(2,5)(3,6)(7,72)(8,67)(9,68)(10,69)(11,70)(12,71)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)(43,46)(44,47)(45,48)(49,52)(50,53)(51,54)(55,64)(56,65)(57,66)(58,61)(59,62)(60,63), (1,28,40)(2,29,41)(3,30,42)(4,25,37)(5,26,38)(6,27,39)(7,66,54)(8,61,49)(9,62,50)(10,63,51)(11,64,52)(12,65,53)(13,23,35)(14,24,36)(15,19,31)(16,20,32)(17,21,33)(18,22,34)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60), (1,49,4,52)(2,50,5,53)(3,51,6,54)(7,42,10,39)(8,37,11,40)(9,38,12,41)(13,44,16,47)(14,45,17,48)(15,46,18,43)(19,58,22,55)(20,59,23,56)(21,60,24,57)(25,64,28,61)(26,65,29,62)(27,66,30,63)(31,70,34,67)(32,71,35,68)(33,72,36,69)>;

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)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72), (1,43,4,46)(2,48,5,45)(3,47,6,44)(7,23,10,20)(8,22,11,19)(9,21,12,24)(13,51,16,54)(14,50,17,53)(15,49,18,52)(25,70,28,67)(26,69,29,72)(27,68,30,71)(31,61,34,64)(32,66,35,63)(33,65,36,62)(37,58,40,55)(38,57,41,60)(39,56,42,59), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,69)(8,70)(9,71)(10,72)(11,67)(12,68)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(55,58)(56,59)(57,60)(61,64)(62,65)(63,66), (1,4)(2,5)(3,6)(7,72)(8,67)(9,68)(10,69)(11,70)(12,71)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)(43,46)(44,47)(45,48)(49,52)(50,53)(51,54)(55,64)(56,65)(57,66)(58,61)(59,62)(60,63), (1,28,40)(2,29,41)(3,30,42)(4,25,37)(5,26,38)(6,27,39)(7,66,54)(8,61,49)(9,62,50)(10,63,51)(11,64,52)(12,65,53)(13,23,35)(14,24,36)(15,19,31)(16,20,32)(17,21,33)(18,22,34)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60), (1,49,4,52)(2,50,5,53)(3,51,6,54)(7,42,10,39)(8,37,11,40)(9,38,12,41)(13,44,16,47)(14,45,17,48)(15,46,18,43)(19,58,22,55)(20,59,23,56)(21,60,24,57)(25,64,28,61)(26,65,29,62)(27,66,30,63)(31,70,34,67)(32,71,35,68)(33,72,36,69) );

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),(43,44,45,46,47,48),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72)], [(1,43,4,46),(2,48,5,45),(3,47,6,44),(7,23,10,20),(8,22,11,19),(9,21,12,24),(13,51,16,54),(14,50,17,53),(15,49,18,52),(25,70,28,67),(26,69,29,72),(27,68,30,71),(31,61,34,64),(32,66,35,63),(33,65,36,62),(37,58,40,55),(38,57,41,60),(39,56,42,59)], [(1,15),(2,16),(3,17),(4,18),(5,13),(6,14),(7,69),(8,70),(9,71),(10,72),(11,67),(12,68),(19,25),(20,26),(21,27),(22,28),(23,29),(24,30),(31,34),(32,35),(33,36),(37,40),(38,41),(39,42),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54),(55,58),(56,59),(57,60),(61,64),(62,65),(63,66)], [(1,4),(2,5),(3,6),(7,72),(8,67),(9,68),(10,69),(11,70),(12,71),(13,16),(14,17),(15,18),(19,28),(20,29),(21,30),(22,25),(23,26),(24,27),(31,37),(32,38),(33,39),(34,40),(35,41),(36,42),(43,46),(44,47),(45,48),(49,52),(50,53),(51,54),(55,64),(56,65),(57,66),(58,61),(59,62),(60,63)], [(1,28,40),(2,29,41),(3,30,42),(4,25,37),(5,26,38),(6,27,39),(7,66,54),(8,61,49),(9,62,50),(10,63,51),(11,64,52),(12,65,53),(13,23,35),(14,24,36),(15,19,31),(16,20,32),(17,21,33),(18,22,34),(43,67,55),(44,68,56),(45,69,57),(46,70,58),(47,71,59),(48,72,60)], [(1,49,4,52),(2,50,5,53),(3,51,6,54),(7,42,10,39),(8,37,11,40),(9,38,12,41),(13,44,16,47),(14,45,17,48),(15,46,18,43),(19,58,22,55),(20,59,23,56),(21,60,24,57),(25,64,28,61),(26,65,29,62),(27,66,30,63),(31,70,34,67),(32,71,35,68),(33,72,36,69)]])

Matrix representation of Dic3.S4 in GL5(𝔽13)

 12 5 0 0 0 2 2 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 6 3 0 0 0 5 7 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 12 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 12 0 0 0 1 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 11 0 0 0 0 12 0 0 0 1 12
,
 9 9 0 0 0 1 4 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(13))| [12,2,0,0,0,5,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[6,5,0,0,0,3,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,12,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,11,12,12],[9,1,0,0,0,9,4,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

Dic3.S4 in GAP, Magma, Sage, TeX

{\rm Dic}_3.S_4
% in TeX

G:=Group("Dic3.S4");
// GroupNames label

G:=SmallGroup(288,852);
// by ID

G=gap.SmallGroup(288,852);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,28,85,36,234,1684,3036,782,1777,1350]);
// Polycyclic

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

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