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## G = C6×C4.A4order 288 = 25·32

### Direct product of C6 and C4.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C6×C4.A4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C6×SL2(𝔽3) — C6×C4.A4
 Lower central Q8 — C6×C4.A4
 Upper central C1 — C2×C12

Generators and relations for C6×C4.A4
G = < a,b,c,d,e | a6=b4=e3=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b2c, ece-1=b2cd, ede-1=c >

Subgroups: 336 in 124 conjugacy classes, 46 normal (17 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, C23, C32, C12, C12, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3×C6, SL2(𝔽3), C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×C6, C2×C4○D4, C3×C12, C62, C2×SL2(𝔽3), C4.A4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, C3×SL2(𝔽3), C6×C12, C2×C4.A4, C6×C4○D4, C6×SL2(𝔽3), C3×C4.A4, C6×C4.A4
Quotients: C1, C2, C3, C22, C6, C32, A4, C2×C6, C3×C6, C2×A4, C3×A4, C62, C4.A4, C22×A4, C6×A4, C2×C4.A4, C3×C4.A4, A4×C2×C6, C6×C4.A4

Smallest permutation representation of C6×C4.A4
On 96 points
Generators in S96
(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 54 73 81)(2 49 74 82)(3 50 75 83)(4 51 76 84)(5 52 77 79)(6 53 78 80)(7 67 26 48)(8 68 27 43)(9 69 28 44)(10 70 29 45)(11 71 30 46)(12 72 25 47)(13 19 94 36)(14 20 95 31)(15 21 96 32)(16 22 91 33)(17 23 92 34)(18 24 93 35)(37 90 62 57)(38 85 63 58)(39 86 64 59)(40 87 65 60)(41 88 66 55)(42 89 61 56)
(1 19 73 36)(2 20 74 31)(3 21 75 32)(4 22 76 33)(5 23 77 34)(6 24 78 35)(7 60 26 87)(8 55 27 88)(9 56 28 89)(10 57 29 90)(11 58 30 85)(12 59 25 86)(13 54 94 81)(14 49 95 82)(15 50 96 83)(16 51 91 84)(17 52 92 79)(18 53 93 80)(37 45 62 70)(38 46 63 71)(39 47 64 72)(40 48 65 67)(41 43 66 68)(42 44 61 69)
(1 69 73 44)(2 70 74 45)(3 71 75 46)(4 72 76 47)(5 67 77 48)(6 68 78 43)(7 52 26 79)(8 53 27 80)(9 54 28 81)(10 49 29 82)(11 50 30 83)(12 51 25 84)(13 89 94 56)(14 90 95 57)(15 85 96 58)(16 86 91 59)(17 87 92 60)(18 88 93 55)(19 61 36 42)(20 62 31 37)(21 63 32 38)(22 64 33 39)(23 65 34 40)(24 66 35 41)
(1 5 3)(2 6 4)(7 58 13)(8 59 14)(9 60 15)(10 55 16)(11 56 17)(12 57 18)(19 67 38)(20 68 39)(21 69 40)(22 70 41)(23 71 42)(24 72 37)(25 90 93)(26 85 94)(27 86 95)(28 87 96)(29 88 91)(30 89 92)(31 43 64)(32 44 65)(33 45 66)(34 46 61)(35 47 62)(36 48 63)(49 53 51)(50 54 52)(73 77 75)(74 78 76)(79 83 81)(80 84 82)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C ··· 3H 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6X 6Y 6Z 6AA 6AB 12A ··· 12H 12I ··· 12AF 12AG 12AH 12AI 12AJ order 1 2 2 2 2 2 3 3 3 ··· 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 6 6 1 1 4 ··· 4 1 1 1 1 6 6 1 ··· 1 4 ··· 4 6 6 6 6 1 ··· 1 4 ··· 4 6 6 6 6

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 3 3 3 3 3 3 type + + + + + + image C1 C2 C2 C3 C3 C6 C6 C6 C6 C4.A4 C3×C4.A4 A4 C2×A4 C2×A4 C3×A4 C6×A4 C6×A4 kernel C6×C4.A4 C6×SL2(𝔽3) C3×C4.A4 C2×C4.A4 C6×C4○D4 C2×SL2(𝔽3) C4.A4 C6×Q8 C3×C4○D4 C6 C2 C2×C12 C12 C2×C6 C2×C4 C4 C22 # reps 1 1 2 6 2 6 12 2 4 12 24 1 2 1 2 4 2

Matrix representation of C6×C4.A4 in GL3(𝔽13) generated by

 12 0 0 0 3 0 0 0 3
,
 1 0 0 0 8 0 0 0 8
,
 1 0 0 0 0 1 0 12 0
,
 1 0 0 0 9 10 0 10 4
,
 9 0 0 0 3 12 0 0 9
G:=sub<GL(3,GF(13))| [12,0,0,0,3,0,0,0,3],[1,0,0,0,8,0,0,0,8],[1,0,0,0,0,12,0,1,0],[1,0,0,0,9,10,0,10,4],[9,0,0,0,3,0,0,12,9] >;

C6×C4.A4 in GAP, Magma, Sage, TeX

C_6\times C_4.A_4
% in TeX

G:=Group("C6xC4.A4");
// GroupNames label

G:=SmallGroup(288,983);
// by ID

G=gap.SmallGroup(288,983);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1016,648,172,1153,285,124]);
// Polycyclic

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

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