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G = C2×C4×C12order 96 = 25·3

Abelian group of type [2,4,12]

direct product, abelian, monomial, 2-elementary

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

Series: Derived Chief Lower central Upper central

C1 — C2×C4×C12
C1C2C22C2×C6C2×C12C4×C12 — C2×C4×C12
C1 — C2×C4×C12
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 [×7], C3, C4 [×12], C22, C22 [×6], C6 [×7], C2×C4 [×18], C23, C12 [×12], C2×C6, C2×C6 [×6], C42 [×4], C22×C4 [×3], C2×C12 [×18], C22×C6, C2×C42, C4×C12 [×4], C22×C12 [×3], C2×C4×C12
Quotients: C1, C2 [×7], C3, C4 [×12], C22 [×7], C6 [×7], C2×C4 [×18], C23, C12 [×12], C2×C6 [×7], C42 [×4], C22×C4 [×3], C2×C12 [×18], C22×C6, C2×C42, C4×C12 [×4], C22×C12 [×3], C2×C4×C12

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

C2×C4×C12 is a maximal subgroup of
C12.8C42  (C2×C12)⋊3C8  C127M4(2)  C42.285D6  C42.270D6  C124(C4⋊C4)  (C2×Dic6)⋊7C4  C426Dic3  (C2×C42).6S3  C4210Dic3  C4211Dic3  C427Dic3  (C2×C4)⋊6D12  (C2×C42)⋊3S3  C42.274D6  C42.276D6  C42.277D6

96 conjugacy classes

class 1 2A···2G3A3B4A···4X6A···6N12A···12AV
order12···2334···46···612···12
size11···1111···11···11···1

96 irreducible representations

dim11111111
type+++
imageC1C2C2C3C4C6C6C12
kernelC2×C4×C12C4×C12C22×C12C2×C42C2×C12C42C22×C4C2×C4
# reps1432248648

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

1200
010
001
,
100
050
001
,
500
030
005
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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