Copied to
clipboard

## G = C12×D8order 192 = 26·3

### Direct product of C12 and D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C12×D8
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C3×C4⋊C4 — C3×D4⋊C4 — C12×D8
 Lower central C1 — C2 — C4 — C12×D8
 Upper central C1 — C2×C12 — C4×C12 — C12×D8

Generators and relations for C12×D8
G = < a,b,c | a12=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 250 in 134 conjugacy classes, 74 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, C23, C12, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C4×C8, D4⋊C4, C2.D8, C4×D4, C2×D8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C3×D8, C22×C12, C6×D4, C4×D8, C4×C24, C3×D4⋊C4, C3×C2.D8, D4×C12, C6×D8, C12×D8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, D8, C22×C4, C2×D4, C4○D4, C2×C12, C3×D4, C22×C6, C4×D4, C2×D8, C4○D8, C3×D8, C22×C12, C6×D4, C3×C4○D4, C4×D8, D4×C12, C6×D8, C3×C4○D8, C12×D8

Smallest permutation representation of C12×D8
On 96 points
Generators in S96
(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 73 90 71 22 54 45 27)(2 74 91 72 23 55 46 28)(3 75 92 61 24 56 47 29)(4 76 93 62 13 57 48 30)(5 77 94 63 14 58 37 31)(6 78 95 64 15 59 38 32)(7 79 96 65 16 60 39 33)(8 80 85 66 17 49 40 34)(9 81 86 67 18 50 41 35)(10 82 87 68 19 51 42 36)(11 83 88 69 20 52 43 25)(12 84 89 70 21 53 44 26)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 25)(12 26)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 61)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 73)(46 74)(47 75)(48 76)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A ··· 6F 6G ··· 6N 8A ··· 8H 12A ··· 12H 12I ··· 12P 12Q ··· 12X 24A ··· 24P order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 4 4 4 4 1 1 1 1 1 1 2 2 2 2 4 4 4 4 1 ··· 1 4 ··· 4 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C12 D4 D8 C4○D4 C3×D4 C4○D8 C3×D8 C3×C4○D4 C3×C4○D8 kernel C12×D8 C4×C24 C3×D4⋊C4 C3×C2.D8 D4×C12 C6×D8 C4×D8 C3×D8 C4×C8 D4⋊C4 C2.D8 C4×D4 C2×D8 D8 C2×C12 C12 C12 C2×C4 C6 C4 C4 C2 # reps 1 1 2 1 2 1 2 8 2 4 2 4 2 16 2 4 2 4 4 8 4 8

Matrix representation of C12×D8 in GL3(𝔽73) generated by

 46 0 0 0 64 0 0 0 64
,
 72 0 0 0 16 57 0 16 16
,
 1 0 0 0 16 57 0 57 57
G:=sub<GL(3,GF(73))| [46,0,0,0,64,0,0,0,64],[72,0,0,0,16,16,0,57,16],[1,0,0,0,16,57,0,57,57] >;

C12×D8 in GAP, Magma, Sage, TeX

C_{12}\times D_8
% in TeX

G:=Group("C12xD8");
// GroupNames label

G:=SmallGroup(192,870);
// by ID

G=gap.SmallGroup(192,870);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,268,4204,2111,172]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

׿
×
𝔽