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## G = C2×Dic3.A4order 288 = 25·32

### Direct product of C2 and Dic3.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — C2×Dic3.A4
 Chief series C1 — C2 — C6 — C3×Q8 — C3×SL2(𝔽3) — Dic3.A4 — C2×Dic3.A4
 Lower central C3×Q8 — C2×Dic3.A4
 Upper central C1 — C22

Generators and relations for C2×Dic3.A4
G = < a,b,c,d,e,f | a2=b6=f3=1, c2=d2=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=b3d, fdf-1=b3de, fef-1=d >

Subgroups: 486 in 105 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, Q8, Q8, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×C6, SL2(𝔽3), SL2(𝔽3), C4×S3, D12, C2×Dic3, C2×C12, C3×Q8, C3×Q8, C22×S3, C2×C4○D4, C3×Dic3, C62, C2×SL2(𝔽3), C2×SL2(𝔽3), C4.A4, S3×C2×C4, C2×D12, Q83S3, Q83S3, C6×Q8, C3×SL2(𝔽3), C6×Dic3, C2×C4.A4, C2×Q83S3, Dic3.A4, C6×SL2(𝔽3), C2×Dic3.A4
Quotients: C1, C2, C3, C22, S3, C6, A4, D6, C2×C6, C3×S3, C2×A4, S3×C6, C4.A4, C22×A4, S3×A4, C2×C4.A4, Dic3.A4, C2×S3×A4, C2×Dic3.A4

Smallest permutation representation of C2×Dic3.A4
On 96 points
Generators in S96
(1 28)(2 29)(3 30)(4 25)(5 26)(6 27)(7 34)(8 35)(9 36)(10 31)(11 32)(12 33)(13 40)(14 41)(15 42)(16 37)(17 38)(18 39)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)(49 76)(50 77)(51 78)(52 73)(53 74)(54 75)(55 82)(56 83)(57 84)(58 79)(59 80)(60 81)(61 88)(62 89)(63 90)(64 85)(65 86)(66 87)(67 94)(68 95)(69 96)(70 91)(71 92)(72 93)
(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 50 4 53)(2 49 5 52)(3 54 6 51)(7 56 10 59)(8 55 11 58)(9 60 12 57)(13 62 16 65)(14 61 17 64)(15 66 18 63)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 74 28 77)(26 73 29 76)(27 78 30 75)(31 80 34 83)(32 79 35 82)(33 84 36 81)(37 86 40 89)(38 85 41 88)(39 90 42 87)(43 92 46 95)(44 91 47 94)(45 96 48 93)
(1 13 4 16)(2 14 5 17)(3 15 6 18)(7 19 10 22)(8 20 11 23)(9 21 12 24)(25 37 28 40)(26 38 29 41)(27 39 30 42)(31 43 34 46)(32 44 35 47)(33 45 36 48)(49 61 52 64)(50 62 53 65)(51 63 54 66)(55 67 58 70)(56 68 59 71)(57 69 60 72)(73 85 76 88)(74 86 77 89)(75 87 78 90)(79 91 82 94)(80 92 83 95)(81 93 84 96)
(1 7 4 10)(2 8 5 11)(3 9 6 12)(13 22 16 19)(14 23 17 20)(15 24 18 21)(25 31 28 34)(26 32 29 35)(27 33 30 36)(37 46 40 43)(38 47 41 44)(39 48 42 45)(49 55 52 58)(50 56 53 59)(51 57 54 60)(61 70 64 67)(62 71 65 68)(63 72 66 69)(73 79 76 82)(74 80 77 83)(75 81 78 84)(85 94 88 91)(86 95 89 92)(87 96 90 93)
(1 3 5)(2 4 6)(7 21 17)(8 22 18)(9 23 13)(10 24 14)(11 19 15)(12 20 16)(25 27 29)(26 28 30)(31 45 41)(32 46 42)(33 47 37)(34 48 38)(35 43 39)(36 44 40)(49 53 51)(50 54 52)(55 71 63)(56 72 64)(57 67 65)(58 68 66)(59 69 61)(60 70 62)(73 77 75)(74 78 76)(79 95 87)(80 96 88)(81 91 89)(82 92 90)(83 93 85)(84 94 86)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D ··· 6I 6J ··· 6O 12A ··· 12J order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 6 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 18 18 2 4 4 8 8 3 3 3 3 6 6 2 2 2 4 ··· 4 8 ··· 8 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 3 3 3 4 4 6 6 type + + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 C4.A4 A4 C2×A4 C2×A4 Dic3.A4 Dic3.A4 S3×A4 C2×S3×A4 kernel C2×Dic3.A4 Dic3.A4 C6×SL2(𝔽3) C2×Q8⋊3S3 Q8⋊3S3 C6×Q8 C2×SL2(𝔽3) SL2(𝔽3) C2×Q8 Q8 C6 C2×Dic3 Dic3 C2×C6 C2 C2 C22 C2 # reps 1 2 1 2 4 2 1 1 2 2 12 1 2 1 2 4 1 1

Matrix representation of C2×Dic3.A4 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 9 0 0 0 0 3
,
 5 0 0 0 0 5 0 0 0 0 0 9 0 0 3 0
,
 6 6 0 0 9 7 0 0 0 0 1 0 0 0 0 1
,
 12 2 0 0 12 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 10 3 0 0 0 0 3 0 0 0 0 3
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,9,0,0,0,0,3],[5,0,0,0,0,5,0,0,0,0,0,3,0,0,9,0],[6,9,0,0,6,7,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,2,1,0,0,0,0,1,0,0,0,0,1],[1,10,0,0,0,3,0,0,0,0,3,0,0,0,0,3] >;

C2×Dic3.A4 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3.A_4
% in TeX

G:=Group("C2xDic3.A4");
// GroupNames label

G:=SmallGroup(288,921);
// by ID

G=gap.SmallGroup(288,921);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-3,-2,1008,269,360,123,515,242,4037]);
// Polycyclic

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

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