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## G = C3×C2.C42order 96 = 25·3

### Direct product of C3 and C2.C42

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×C2.C42
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C22×C12 — C3×C2.C42
 Lower central C1 — C2 — C3×C2.C42
 Upper central C1 — C22×C6 — C3×C2.C42

Generators and relations for C3×C2.C42
G = < a,b,c,d | a3=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 100 in 76 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C12, C2×C6, C2×C6, C22×C4, C2×C12, C2×C12, C22×C6, C2.C42, C22×C12, C3×C2.C42
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C12, C3×D4, C3×Q8, C2.C42, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C2.C42

Smallest permutation representation of C3×C2.C42
Regular action on 96 points
Generators in S96
(1 15 11)(2 16 12)(3 13 9)(4 14 10)(5 69 65)(6 70 66)(7 71 67)(8 72 68)(17 25 21)(18 26 22)(19 27 23)(20 28 24)(29 37 33)(30 38 34)(31 39 35)(32 40 36)(41 52 45)(42 49 46)(43 50 47)(44 51 48)(53 61 57)(54 62 58)(55 63 59)(56 64 60)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(1 55)(2 56)(3 53)(4 54)(5 25)(6 26)(7 27)(8 28)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 92)(49 94)(50 95)(51 96)(52 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 87 75 19)(2 44 76 68)(3 85 73 17)(4 42 74 66)(5 61 52 37)(6 14 49 82)(7 63 50 39)(8 16 51 84)(9 89 77 21)(10 46 78 70)(11 91 79 23)(12 48 80 72)(13 93 81 25)(15 95 83 27)(18 54 86 30)(20 56 88 32)(22 58 90 34)(24 60 92 36)(26 62 94 38)(28 64 96 40)(29 65 53 41)(31 67 55 43)(33 69 57 45)(35 71 59 47)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4L 6A ··· 6N 12A ··· 12X order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C12 D4 Q8 C3×D4 C3×Q8 kernel C3×C2.C42 C22×C12 C2.C42 C2×C12 C22×C4 C2×C4 C2×C6 C2×C6 C22 C22 # reps 1 3 2 12 6 24 3 1 6 2

Matrix representation of C3×C2.C42 in GL5(𝔽13)

 3 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 6 5 0 0 0 6 7
,
 12 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 0 1 11 0 0 0 0 12

G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,6,6,0,0,0,5,7],[12,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,11,12] >;

C3×C2.C42 in GAP, Magma, Sage, TeX

C_3\times C_2.C_4^2
% in TeX

G:=Group("C3xC2.C4^2");
// GroupNames label

G:=SmallGroup(96,45);
// by ID

G=gap.SmallGroup(96,45);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,295]);
// Polycyclic

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

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