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## G = C12×C3.A4order 432 = 24·33

### Direct product of C12 and C3.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C12×C3.A4
 Chief series C1 — C22 — C2×C6 — C22×C6 — C2×C62 — C6×C3.A4 — C12×C3.A4
 Lower central C22 — C12×C3.A4
 Upper central C1 — C3×C12

Generators and relations for C12×C3.A4
G = < a,b,c,d,e | a12=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 222 in 102 conjugacy classes, 48 normal (21 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4, C4, C22, C22 [×2], C6, C6 [×3], C6 [×8], C2×C4 [×2], C23, C9 [×3], C32, C12, C12 [×3], C12 [×4], C2×C6, C2×C6 [×3], C2×C6 [×8], C22×C4, C18 [×3], C3×C6, C3×C6 [×2], C2×C12 [×8], C22×C6, C22×C6 [×3], C3×C9, C36 [×3], C3.A4 [×3], C3×C12, C3×C12, C62, C62 [×2], C22×C12, C22×C12 [×3], C3×C18, C2×C3.A4 [×3], C6×C12 [×2], C2×C62, C3×C36, C3×C3.A4, C4×C3.A4 [×3], C2×C6×C12, C6×C3.A4, C12×C3.A4
Quotients: C1, C2, C3 [×4], C4, C6 [×4], C9 [×3], C32, C12 [×4], A4, C18 [×3], C3×C6, C2×A4, C3×C9, C36 [×3], C3.A4 [×3], C3×C12, C3×A4, C4×A4, C3×C18, C2×C3.A4 [×3], C6×A4, C3×C36, C3×C3.A4, C4×C3.A4 [×3], C12×A4, C6×C3.A4, C12×C3.A4

Smallest permutation representation of C12×C3.A4
On 108 points
Generators in S108
(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)
(1 57 78)(2 58 79)(3 59 80)(4 60 81)(5 49 82)(6 50 83)(7 51 84)(8 52 73)(9 53 74)(10 54 75)(11 55 76)(12 56 77)(13 102 69)(14 103 70)(15 104 71)(16 105 72)(17 106 61)(18 107 62)(19 108 63)(20 97 64)(21 98 65)(22 99 66)(23 100 67)(24 101 68)(25 48 95)(26 37 96)(27 38 85)(28 39 86)(29 40 87)(30 41 88)(31 42 89)(32 43 90)(33 44 91)(34 45 92)(35 46 93)(36 47 94)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 24 94 57 101 36 78 68 47)(2 13 95 58 102 25 79 69 48)(3 14 96 59 103 26 80 70 37)(4 15 85 60 104 27 81 71 38)(5 16 86 49 105 28 82 72 39)(6 17 87 50 106 29 83 61 40)(7 18 88 51 107 30 84 62 41)(8 19 89 52 108 31 73 63 42)(9 20 90 53 97 32 74 64 43)(10 21 91 54 98 33 75 65 44)(11 22 92 55 99 34 76 66 45)(12 23 93 56 100 35 77 67 46)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A 4B 4C 4D 6A ··· 6H 6I ··· 6X 9A ··· 9R 12A ··· 12P 12Q ··· 12AF 18A ··· 18R 36A ··· 36AJ order 1 2 2 2 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 3 3 1 ··· 1 1 1 3 3 1 ··· 1 3 ··· 3 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C4 C6 C6 C9 C12 C12 C18 C36 A4 C2×A4 C3.A4 C3×A4 C4×A4 C2×C3.A4 C6×A4 C4×C3.A4 C12×A4 kernel C12×C3.A4 C6×C3.A4 C4×C3.A4 C2×C6×C12 C3×C3.A4 C2×C3.A4 C2×C62 C22×C12 C3.A4 C62 C22×C6 C2×C6 C3×C12 C3×C6 C12 C12 C32 C6 C6 C3 C3 # reps 1 1 6 2 2 6 2 18 12 4 18 36 1 1 6 2 2 6 2 12 4

Matrix representation of C12×C3.A4 in GL4(𝔽37) generated by

 23 0 0 0 0 23 0 0 0 0 23 0 0 0 0 23
,
 1 0 0 0 0 26 0 0 0 0 26 0 0 0 0 26
,
 1 0 0 0 0 1 0 0 0 0 36 0 0 0 0 36
,
 1 0 0 0 0 36 0 0 0 0 36 0 0 0 0 1
,
 10 0 0 0 0 0 1 0 0 0 0 1 0 26 0 0
G:=sub<GL(4,GF(37))| [23,0,0,0,0,23,0,0,0,0,23,0,0,0,0,23],[1,0,0,0,0,26,0,0,0,0,26,0,0,0,0,26],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,36,0,0,0,0,36,0,0,0,0,1],[10,0,0,0,0,0,0,26,0,1,0,0,0,0,1,0] >;

C12×C3.A4 in GAP, Magma, Sage, TeX

C_{12}\times C_3.A_4
% in TeX

G:=Group("C12xC3.A4");
// GroupNames label

G:=SmallGroup(432,331);
// by ID

G=gap.SmallGroup(432,331);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,-3,-2,2,126,260,4548,7951]);
// Polycyclic

G:=Group<a,b,c,d,e|a^12=b^3=c^2=d^2=1,e^3=b,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,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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