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## G = C3×C4.S4order 288 = 25·32

### Direct product of C3 and C4.S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C3×C4.S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C3×CSU2(𝔽3) — C3×C4.S4
 Lower central SL2(𝔽3) — C3×C4.S4
 Upper central C1 — C6 — C12

Generators and relations for C3×C4.S4
G = < a,b,c,d,e,f | a3=b4=e3=1, c2=d2=f2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=b-1, dcd-1=b2c, ece-1=b2cd, fcf-1=cd, ede-1=c, fdf-1=b2d, fef-1=e-1 >

Subgroups: 246 in 77 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, C12, C12, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×C6, C24, SL2(𝔽3), SL2(𝔽3), Dic6, C2×C12, C3×D4, C3×Q8, C3×Q8, C8.C22, C3×Dic3, C3×C12, C3×M4(2), C3×SD16, C3×Q16, CSU2(𝔽3), C4.A4, C4.A4, C6×Q8, C3×C4○D4, C3×SL2(𝔽3), C3×Dic6, C3×C8.C22, C4.S4, C3×CSU2(𝔽3), C3×C4.A4, C3×C4.S4
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S4, S3×C6, C2×S4, C3×S4, C4.S4, C6×S4, C3×C4.S4

Smallest permutation representation of C3×C4.S4
On 96 points
Generators in S96
(1 33 29)(2 34 30)(3 35 31)(4 36 32)(5 38 22)(6 39 23)(7 40 24)(8 37 21)(9 41 52)(10 42 49)(11 43 50)(12 44 51)(13 45 17)(14 46 18)(15 47 19)(16 48 20)(25 77 91)(26 78 92)(27 79 89)(28 80 90)(53 57 61)(54 58 62)(55 59 63)(56 60 64)(65 84 94)(66 81 95)(67 82 96)(68 83 93)(69 73 86)(70 74 87)(71 75 88)(72 76 85)
(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)(73 74 75 76)(77 78 79 80)(81 82 83 84)(85 86 87 88)(89 90 91 92)(93 94 95 96)
(1 37 3 39)(2 38 4 40)(5 32 7 30)(6 29 8 31)(9 19 11 17)(10 20 12 18)(13 41 15 43)(14 42 16 44)(21 35 23 33)(22 36 24 34)(25 64 27 62)(26 61 28 63)(45 52 47 50)(46 49 48 51)(53 80 55 78)(54 77 56 79)(57 90 59 92)(58 91 60 89)(65 86 67 88)(66 87 68 85)(69 82 71 84)(70 83 72 81)(73 96 75 94)(74 93 76 95)
(1 17 3 19)(2 18 4 20)(5 51 7 49)(6 52 8 50)(9 37 11 39)(10 38 12 40)(13 35 15 33)(14 36 16 34)(21 43 23 41)(22 44 24 42)(25 76 27 74)(26 73 28 75)(29 45 31 47)(30 46 32 48)(53 65 55 67)(54 66 56 68)(57 84 59 82)(58 81 60 83)(61 94 63 96)(62 95 64 93)(69 90 71 92)(70 91 72 89)(77 85 79 87)(78 86 80 88)
(1 29 33)(2 30 34)(3 31 35)(4 32 36)(5 14 10)(6 15 11)(7 16 12)(8 13 9)(17 52 21)(18 49 22)(19 50 23)(20 51 24)(25 66 70)(26 67 71)(27 68 72)(28 65 69)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(53 57 61)(54 58 62)(55 59 63)(56 60 64)(73 80 84)(74 77 81)(75 78 82)(76 79 83)(85 89 93)(86 90 94)(87 91 95)(88 92 96)
(1 54 3 56)(2 53 4 55)(5 75 7 73)(6 74 8 76)(9 79 11 77)(10 78 12 80)(13 83 15 81)(14 82 16 84)(17 68 19 66)(18 67 20 65)(21 72 23 70)(22 71 24 69)(25 52 27 50)(26 51 28 49)(29 62 31 64)(30 61 32 63)(33 58 35 60)(34 57 36 59)(37 85 39 87)(38 88 40 86)(41 89 43 91)(42 92 44 90)(45 93 47 95)(46 96 48 94)

G:=sub<Sym(96)| (1,33,29)(2,34,30)(3,35,31)(4,36,32)(5,38,22)(6,39,23)(7,40,24)(8,37,21)(9,41,52)(10,42,49)(11,43,50)(12,44,51)(13,45,17)(14,46,18)(15,47,19)(16,48,20)(25,77,91)(26,78,92)(27,79,89)(28,80,90)(53,57,61)(54,58,62)(55,59,63)(56,60,64)(65,84,94)(66,81,95)(67,82,96)(68,83,93)(69,73,86)(70,74,87)(71,75,88)(72,76,85), (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)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96), (1,37,3,39)(2,38,4,40)(5,32,7,30)(6,29,8,31)(9,19,11,17)(10,20,12,18)(13,41,15,43)(14,42,16,44)(21,35,23,33)(22,36,24,34)(25,64,27,62)(26,61,28,63)(45,52,47,50)(46,49,48,51)(53,80,55,78)(54,77,56,79)(57,90,59,92)(58,91,60,89)(65,86,67,88)(66,87,68,85)(69,82,71,84)(70,83,72,81)(73,96,75,94)(74,93,76,95), (1,17,3,19)(2,18,4,20)(5,51,7,49)(6,52,8,50)(9,37,11,39)(10,38,12,40)(13,35,15,33)(14,36,16,34)(21,43,23,41)(22,44,24,42)(25,76,27,74)(26,73,28,75)(29,45,31,47)(30,46,32,48)(53,65,55,67)(54,66,56,68)(57,84,59,82)(58,81,60,83)(61,94,63,96)(62,95,64,93)(69,90,71,92)(70,91,72,89)(77,85,79,87)(78,86,80,88), (1,29,33)(2,30,34)(3,31,35)(4,32,36)(5,14,10)(6,15,11)(7,16,12)(8,13,9)(17,52,21)(18,49,22)(19,50,23)(20,51,24)(25,66,70)(26,67,71)(27,68,72)(28,65,69)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(53,57,61)(54,58,62)(55,59,63)(56,60,64)(73,80,84)(74,77,81)(75,78,82)(76,79,83)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (1,54,3,56)(2,53,4,55)(5,75,7,73)(6,74,8,76)(9,79,11,77)(10,78,12,80)(13,83,15,81)(14,82,16,84)(17,68,19,66)(18,67,20,65)(21,72,23,70)(22,71,24,69)(25,52,27,50)(26,51,28,49)(29,62,31,64)(30,61,32,63)(33,58,35,60)(34,57,36,59)(37,85,39,87)(38,88,40,86)(41,89,43,91)(42,92,44,90)(45,93,47,95)(46,96,48,94)>;

G:=Group( (1,33,29)(2,34,30)(3,35,31)(4,36,32)(5,38,22)(6,39,23)(7,40,24)(8,37,21)(9,41,52)(10,42,49)(11,43,50)(12,44,51)(13,45,17)(14,46,18)(15,47,19)(16,48,20)(25,77,91)(26,78,92)(27,79,89)(28,80,90)(53,57,61)(54,58,62)(55,59,63)(56,60,64)(65,84,94)(66,81,95)(67,82,96)(68,83,93)(69,73,86)(70,74,87)(71,75,88)(72,76,85), (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)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96), (1,37,3,39)(2,38,4,40)(5,32,7,30)(6,29,8,31)(9,19,11,17)(10,20,12,18)(13,41,15,43)(14,42,16,44)(21,35,23,33)(22,36,24,34)(25,64,27,62)(26,61,28,63)(45,52,47,50)(46,49,48,51)(53,80,55,78)(54,77,56,79)(57,90,59,92)(58,91,60,89)(65,86,67,88)(66,87,68,85)(69,82,71,84)(70,83,72,81)(73,96,75,94)(74,93,76,95), (1,17,3,19)(2,18,4,20)(5,51,7,49)(6,52,8,50)(9,37,11,39)(10,38,12,40)(13,35,15,33)(14,36,16,34)(21,43,23,41)(22,44,24,42)(25,76,27,74)(26,73,28,75)(29,45,31,47)(30,46,32,48)(53,65,55,67)(54,66,56,68)(57,84,59,82)(58,81,60,83)(61,94,63,96)(62,95,64,93)(69,90,71,92)(70,91,72,89)(77,85,79,87)(78,86,80,88), (1,29,33)(2,30,34)(3,31,35)(4,32,36)(5,14,10)(6,15,11)(7,16,12)(8,13,9)(17,52,21)(18,49,22)(19,50,23)(20,51,24)(25,66,70)(26,67,71)(27,68,72)(28,65,69)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(53,57,61)(54,58,62)(55,59,63)(56,60,64)(73,80,84)(74,77,81)(75,78,82)(76,79,83)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (1,54,3,56)(2,53,4,55)(5,75,7,73)(6,74,8,76)(9,79,11,77)(10,78,12,80)(13,83,15,81)(14,82,16,84)(17,68,19,66)(18,67,20,65)(21,72,23,70)(22,71,24,69)(25,52,27,50)(26,51,28,49)(29,62,31,64)(30,61,32,63)(33,58,35,60)(34,57,36,59)(37,85,39,87)(38,88,40,86)(41,89,43,91)(42,92,44,90)(45,93,47,95)(46,96,48,94) );

G=PermutationGroup([[(1,33,29),(2,34,30),(3,35,31),(4,36,32),(5,38,22),(6,39,23),(7,40,24),(8,37,21),(9,41,52),(10,42,49),(11,43,50),(12,44,51),(13,45,17),(14,46,18),(15,47,19),(16,48,20),(25,77,91),(26,78,92),(27,79,89),(28,80,90),(53,57,61),(54,58,62),(55,59,63),(56,60,64),(65,84,94),(66,81,95),(67,82,96),(68,83,93),(69,73,86),(70,74,87),(71,75,88),(72,76,85)], [(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),(73,74,75,76),(77,78,79,80),(81,82,83,84),(85,86,87,88),(89,90,91,92),(93,94,95,96)], [(1,37,3,39),(2,38,4,40),(5,32,7,30),(6,29,8,31),(9,19,11,17),(10,20,12,18),(13,41,15,43),(14,42,16,44),(21,35,23,33),(22,36,24,34),(25,64,27,62),(26,61,28,63),(45,52,47,50),(46,49,48,51),(53,80,55,78),(54,77,56,79),(57,90,59,92),(58,91,60,89),(65,86,67,88),(66,87,68,85),(69,82,71,84),(70,83,72,81),(73,96,75,94),(74,93,76,95)], [(1,17,3,19),(2,18,4,20),(5,51,7,49),(6,52,8,50),(9,37,11,39),(10,38,12,40),(13,35,15,33),(14,36,16,34),(21,43,23,41),(22,44,24,42),(25,76,27,74),(26,73,28,75),(29,45,31,47),(30,46,32,48),(53,65,55,67),(54,66,56,68),(57,84,59,82),(58,81,60,83),(61,94,63,96),(62,95,64,93),(69,90,71,92),(70,91,72,89),(77,85,79,87),(78,86,80,88)], [(1,29,33),(2,30,34),(3,31,35),(4,32,36),(5,14,10),(6,15,11),(7,16,12),(8,13,9),(17,52,21),(18,49,22),(19,50,23),(20,51,24),(25,66,70),(26,67,71),(27,68,72),(28,65,69),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(53,57,61),(54,58,62),(55,59,63),(56,60,64),(73,80,84),(74,77,81),(75,78,82),(76,79,83),(85,89,93),(86,90,94),(87,91,95),(88,92,96)], [(1,54,3,56),(2,53,4,55),(5,75,7,73),(6,74,8,76),(9,79,11,77),(10,78,12,80),(13,83,15,81),(14,82,16,84),(17,68,19,66),(18,67,20,65),(21,72,23,70),(22,71,24,69),(25,52,27,50),(26,51,28,49),(29,62,31,64),(30,61,32,63),(33,58,35,60),(34,57,36,59),(37,85,39,87),(38,88,40,86),(41,89,43,91),(42,92,44,90),(45,93,47,95),(46,96,48,94)]])

39 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 8A 8B 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 24A 24B 24C 24D order 1 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 8 8 12 12 12 12 12 ··· 12 12 12 12 12 24 24 24 24 size 1 1 6 1 1 8 8 8 2 6 12 12 1 1 6 6 8 8 8 12 12 2 2 6 6 8 ··· 8 12 12 12 12 12 12 12 12

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 3 4 4 type + + + + + + + - image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 S4 C2×S4 C3×S4 C6×S4 C4.S4 C3×C4.S4 kernel C3×C4.S4 C3×CSU2(𝔽3) C3×C4.A4 C4.S4 CSU2(𝔽3) C4.A4 C3×C4○D4 C3×Q8 C4○D4 Q8 C12 C6 C4 C2 C3 C1 # reps 1 2 1 2 4 2 1 1 2 2 2 2 4 4 3 6

Matrix representation of C3×C4.S4 in GL4(𝔽7) generated by

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

C3×C4.S4 in GAP, Magma, Sage, TeX

C_3\times C_4.S_4
% in TeX

G:=Group("C3xC4.S4");
// GroupNames label

G:=SmallGroup(288,902);
// by ID

G=gap.SmallGroup(288,902);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,2045,1016,675,2524,655,172,1517,404,285,124]);
// Polycyclic

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

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