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## G = C2×C4×C12order 96 = 25·3

### Abelian group of type [2,4,12]

Aliases: C2×C4×C12, SmallGroup(96,161)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4×C12
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C4×C12 — C2×C4×C12
 Lower central C1 — C2×C4×C12
 Upper central C1 — C2×C4×C12

Generators and relations for C2×C4×C12
G = < a,b,c | a2=b4=c12=1, ab=ba, ac=ca, bc=cb >

Subgroups: 108, all normal (8 characteristic)
C1, C2, C3, C4, C22, C22, C6, C2×C4, C23, C12, C2×C6, C2×C6, C42, C22×C4, C2×C12, C22×C6, C2×C42, C4×C12, C22×C12, C2×C4×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C42, C22×C4, C2×C12, C22×C6, C2×C42, C4×C12, C22×C12, C2×C4×C12

Smallest permutation representation of C2×C4×C12
Regular action on 96 points
Generators in S96
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 73)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(61 94)(62 95)(63 96)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)
(1 72 44 17)(2 61 45 18)(3 62 46 19)(4 63 47 20)(5 64 48 21)(6 65 37 22)(7 66 38 23)(8 67 39 24)(9 68 40 13)(10 69 41 14)(11 70 42 15)(12 71 43 16)(25 60 79 86)(26 49 80 87)(27 50 81 88)(28 51 82 89)(29 52 83 90)(30 53 84 91)(31 54 73 92)(32 55 74 93)(33 56 75 94)(34 57 76 95)(35 58 77 96)(36 59 78 85)
(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)

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

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

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

96 conjugacy classes

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

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C6 C6 C12 kernel C2×C4×C12 C4×C12 C22×C12 C2×C42 C2×C12 C42 C22×C4 C2×C4 # reps 1 4 3 2 24 8 6 48

Matrix representation of C2×C4×C12 in GL3(𝔽13) generated by

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

C2×C4×C12 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_{12}
% in TeX

G:=Group("C2xC4xC12");
// GroupNames label

G:=SmallGroup(96,161);
// by ID

G=gap.SmallGroup(96,161);
# by ID

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

G:=Group<a,b,c|a^2=b^4=c^12=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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