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## G = C2×C6×C3.A4order 432 = 24·33

### Direct product of C2×C6 and C3.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×C6×C3.A4
 Chief series C1 — C22 — C2×C6 — C62 — C3×C3.A4 — C6×C3.A4 — C2×C6×C3.A4
 Lower central C22 — C2×C6×C3.A4
 Upper central C1 — C62

Generators and relations for C2×C6×C3.A4
G = < a,b,c,d,e,f | a2=b6=c3=d2=e2=1, f3=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 502 in 214 conjugacy classes, 80 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, C22, C6, C6, C23, C23, C9, C32, C2×C6, C2×C6, C2×C6, C24, C18, C3×C6, C3×C6, C22×C6, C22×C6, C3×C9, C3.A4, C2×C18, C62, C62, C23×C6, C23×C6, C3×C18, C2×C3.A4, C2×C62, C2×C62, C3×C3.A4, C6×C18, C22×C3.A4, C22×C62, C6×C3.A4, C2×C6×C3.A4
Quotients: C1, C2, C3, C22, C6, C9, C32, A4, C2×C6, C18, C3×C6, C2×A4, C3×C9, C3.A4, C2×C18, C3×A4, C62, C22×A4, C3×C18, C2×C3.A4, C6×A4, C3×C3.A4, C6×C18, C22×C3.A4, A4×C2×C6, C6×C3.A4, C2×C6×C3.A4

Smallest permutation representation of C2×C6×C3.A4
On 108 points
Generators in S108
(1 52)(2 53)(3 54)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 19)(18 20)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)(55 68)(56 69)(57 70)(58 71)(59 72)(60 64)(61 65)(62 66)(63 67)(73 106)(74 107)(75 108)(76 100)(77 101)(78 102)(79 103)(80 104)(81 105)(82 96)(83 97)(84 98)(85 99)(86 91)(87 92)(88 93)(89 94)(90 95)
(1 60 98 103 18 28)(2 61 99 104 10 29)(3 62 91 105 11 30)(4 63 92 106 12 31)(5 55 93 107 13 32)(6 56 94 108 14 33)(7 57 95 100 15 34)(8 58 96 101 16 35)(9 59 97 102 17 36)(19 45 51 72 83 78)(20 37 52 64 84 79)(21 38 53 65 85 80)(22 39 54 66 86 81)(23 40 46 67 87 73)(24 41 47 68 88 74)(25 42 48 69 89 75)(26 43 49 70 90 76)(27 44 50 71 82 77)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 103 106)(101 104 107)(102 105 108)
(2 80)(3 81)(5 74)(6 75)(8 77)(9 78)(10 65)(11 66)(13 68)(14 69)(16 71)(17 72)(19 59)(21 61)(22 62)(24 55)(25 56)(27 58)(29 85)(30 86)(32 88)(33 89)(35 82)(36 83)(38 99)(39 91)(41 93)(42 94)(44 96)(45 97)(47 107)(48 108)(50 101)(51 102)(53 104)(54 105)
(1 79)(3 81)(4 73)(6 75)(7 76)(9 78)(11 66)(12 67)(14 69)(15 70)(17 72)(18 64)(19 59)(20 60)(22 62)(23 63)(25 56)(26 57)(28 84)(30 86)(31 87)(33 89)(34 90)(36 83)(37 98)(39 91)(40 92)(42 94)(43 95)(45 97)(46 106)(48 108)(49 100)(51 102)(52 103)(54 105)
(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 97 98 99)(100 101 102 103 104 105 106 107 108)

G:=sub<Sym(108)| (1,52)(2,53)(3,54)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,19)(18,20)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(55,68)(56,69)(57,70)(58,71)(59,72)(60,64)(61,65)(62,66)(63,67)(73,106)(74,107)(75,108)(76,100)(77,101)(78,102)(79,103)(80,104)(81,105)(82,96)(83,97)(84,98)(85,99)(86,91)(87,92)(88,93)(89,94)(90,95), (1,60,98,103,18,28)(2,61,99,104,10,29)(3,62,91,105,11,30)(4,63,92,106,12,31)(5,55,93,107,13,32)(6,56,94,108,14,33)(7,57,95,100,15,34)(8,58,96,101,16,35)(9,59,97,102,17,36)(19,45,51,72,83,78)(20,37,52,64,84,79)(21,38,53,65,85,80)(22,39,54,66,86,81)(23,40,46,67,87,73)(24,41,47,68,88,74)(25,42,48,69,89,75)(26,43,49,70,90,76)(27,44,50,71,82,77), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108), (2,80)(3,81)(5,74)(6,75)(8,77)(9,78)(10,65)(11,66)(13,68)(14,69)(16,71)(17,72)(19,59)(21,61)(22,62)(24,55)(25,56)(27,58)(29,85)(30,86)(32,88)(33,89)(35,82)(36,83)(38,99)(39,91)(41,93)(42,94)(44,96)(45,97)(47,107)(48,108)(50,101)(51,102)(53,104)(54,105), (1,79)(3,81)(4,73)(6,75)(7,76)(9,78)(11,66)(12,67)(14,69)(15,70)(17,72)(18,64)(19,59)(20,60)(22,62)(23,63)(25,56)(26,57)(28,84)(30,86)(31,87)(33,89)(34,90)(36,83)(37,98)(39,91)(40,92)(42,94)(43,95)(45,97)(46,106)(48,108)(49,100)(51,102)(52,103)(54,105), (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)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)>;

G:=Group( (1,52)(2,53)(3,54)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,19)(18,20)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(55,68)(56,69)(57,70)(58,71)(59,72)(60,64)(61,65)(62,66)(63,67)(73,106)(74,107)(75,108)(76,100)(77,101)(78,102)(79,103)(80,104)(81,105)(82,96)(83,97)(84,98)(85,99)(86,91)(87,92)(88,93)(89,94)(90,95), (1,60,98,103,18,28)(2,61,99,104,10,29)(3,62,91,105,11,30)(4,63,92,106,12,31)(5,55,93,107,13,32)(6,56,94,108,14,33)(7,57,95,100,15,34)(8,58,96,101,16,35)(9,59,97,102,17,36)(19,45,51,72,83,78)(20,37,52,64,84,79)(21,38,53,65,85,80)(22,39,54,66,86,81)(23,40,46,67,87,73)(24,41,47,68,88,74)(25,42,48,69,89,75)(26,43,49,70,90,76)(27,44,50,71,82,77), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108), (2,80)(3,81)(5,74)(6,75)(8,77)(9,78)(10,65)(11,66)(13,68)(14,69)(16,71)(17,72)(19,59)(21,61)(22,62)(24,55)(25,56)(27,58)(29,85)(30,86)(32,88)(33,89)(35,82)(36,83)(38,99)(39,91)(41,93)(42,94)(44,96)(45,97)(47,107)(48,108)(50,101)(51,102)(53,104)(54,105), (1,79)(3,81)(4,73)(6,75)(7,76)(9,78)(11,66)(12,67)(14,69)(15,70)(17,72)(18,64)(19,59)(20,60)(22,62)(23,63)(25,56)(26,57)(28,84)(30,86)(31,87)(33,89)(34,90)(36,83)(37,98)(39,91)(40,92)(42,94)(43,95)(45,97)(46,106)(48,108)(49,100)(51,102)(52,103)(54,105), (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)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108) );

G=PermutationGroup([[(1,52),(2,53),(3,54),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(17,19),(18,20),(28,37),(29,38),(30,39),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45),(55,68),(56,69),(57,70),(58,71),(59,72),(60,64),(61,65),(62,66),(63,67),(73,106),(74,107),(75,108),(76,100),(77,101),(78,102),(79,103),(80,104),(81,105),(82,96),(83,97),(84,98),(85,99),(86,91),(87,92),(88,93),(89,94),(90,95)], [(1,60,98,103,18,28),(2,61,99,104,10,29),(3,62,91,105,11,30),(4,63,92,106,12,31),(5,55,93,107,13,32),(6,56,94,108,14,33),(7,57,95,100,15,34),(8,58,96,101,16,35),(9,59,97,102,17,36),(19,45,51,72,83,78),(20,37,52,64,84,79),(21,38,53,65,85,80),(22,39,54,66,86,81),(23,40,46,67,87,73),(24,41,47,68,88,74),(25,42,48,69,89,75),(26,43,49,70,90,76),(27,44,50,71,82,77)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81),(82,85,88),(83,86,89),(84,87,90),(91,94,97),(92,95,98),(93,96,99),(100,103,106),(101,104,107),(102,105,108)], [(2,80),(3,81),(5,74),(6,75),(8,77),(9,78),(10,65),(11,66),(13,68),(14,69),(16,71),(17,72),(19,59),(21,61),(22,62),(24,55),(25,56),(27,58),(29,85),(30,86),(32,88),(33,89),(35,82),(36,83),(38,99),(39,91),(41,93),(42,94),(44,96),(45,97),(47,107),(48,108),(50,101),(51,102),(53,104),(54,105)], [(1,79),(3,81),(4,73),(6,75),(7,76),(9,78),(11,66),(12,67),(14,69),(15,70),(17,72),(18,64),(19,59),(20,60),(22,62),(23,63),(25,56),(26,57),(28,84),(30,86),(31,87),(33,89),(34,90),(36,83),(37,98),(39,91),(40,92),(42,94),(43,95),(45,97),(46,106),(48,108),(49,100),(51,102),(52,103),(54,105)], [(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),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108)]])

144 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3H 6A ··· 6X 6Y ··· 6BD 9A ··· 9R 18A ··· 18BB order 1 2 2 2 2 2 2 2 3 ··· 3 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 3 3 3 3 1 ··· 1 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C6 C6 C9 C18 A4 C2×A4 C3.A4 C3×A4 C2×C3.A4 C6×A4 kernel C2×C6×C3.A4 C6×C3.A4 C22×C3.A4 C22×C62 C2×C3.A4 C2×C62 C23×C6 C22×C6 C62 C3×C6 C2×C6 C2×C6 C6 C6 # reps 1 3 6 2 18 6 18 54 1 3 6 2 18 6

Matrix representation of C2×C6×C3.A4 in GL4(𝔽19) generated by

 18 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 11 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 1 0 0 0 0 1 0 0 0 0 18 0 0 18 0 18
,
 1 0 0 0 0 18 0 0 0 0 18 0 0 1 1 1
,
 5 0 0 0 0 0 1 0 0 18 18 17 0 14 0 1
G:=sub<GL(4,GF(19))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[11,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,1,0,18,0,0,18,0,0,0,0,18],[1,0,0,0,0,18,0,1,0,0,18,1,0,0,0,1],[5,0,0,0,0,0,18,14,0,1,18,0,0,0,17,1] >;

C2×C6×C3.A4 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_3.A_4
% in TeX

G:=Group("C2xC6xC3.A4");
// GroupNames label

G:=SmallGroup(432,548);
// by ID

G=gap.SmallGroup(432,548);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,205,2287,3989]);
// Polycyclic

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

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