Copied to
clipboard

G = C2×D12.C4order 192 = 26·3

Direct product of C2 and D12.C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×D12.C4
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×C4○D12 — C2×D12.C4
 Lower central C3 — C6 — C2×D12.C4
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for C2×D12.C4
G = < a,b,c,d | a2=b12=c2=1, d4=b6, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b7, dcd-1=b6c >

Subgroups: 504 in 266 conjugacy classes, 151 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, S3×C8, C8⋊S3, C2×C3⋊C8, C2×C3⋊C8, C2×C24, C3×M4(2), C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C2×C8○D4, S3×C2×C8, C2×C8⋊S3, D12.C4, C22×C3⋊C8, C6×M4(2), C2×C4○D12, C2×D12.C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C8○D4, C23×C4, S3×C2×C4, S3×C23, C2×C8○D4, D12.C4, S3×C22×C4, C2×D12.C4

Smallest permutation representation of C2×D12.C4
On 96 points
Generators in S96
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 37)(12 38)(13 82)(14 83)(15 84)(16 73)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 81)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 85)(34 86)(35 87)(36 88)(49 68)(50 69)(51 70)(52 71)(53 72)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)
(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 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 48)(13 76)(14 75)(15 74)(16 73)(17 84)(18 83)(19 82)(20 81)(21 80)(22 79)(23 78)(24 77)(25 89)(26 88)(27 87)(28 86)(29 85)(30 96)(31 95)(32 94)(33 93)(34 92)(35 91)(36 90)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 72)(58 71)(59 70)(60 69)
(1 54 85 15 7 60 91 21)(2 49 86 22 8 55 92 16)(3 56 87 17 9 50 93 23)(4 51 88 24 10 57 94 18)(5 58 89 19 11 52 95 13)(6 53 90 14 12 59 96 20)(25 76 37 71 31 82 43 65)(26 83 38 66 32 77 44 72)(27 78 39 61 33 84 45 67)(28 73 40 68 34 79 46 62)(29 80 41 63 35 74 47 69)(30 75 42 70 36 81 48 64)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 6E 8A ··· 8H 8I ··· 8P 8Q 8R 8S 8T 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 ··· 8 8 ··· 8 8 8 8 8 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 6 6 6 6 2 1 1 1 1 2 2 6 6 6 6 2 2 2 4 4 2 ··· 2 3 ··· 3 6 6 6 6 2 2 2 2 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D6 D6 D6 C4×S3 C4×S3 C8○D4 D12.C4 kernel C2×D12.C4 S3×C2×C8 C2×C8⋊S3 D12.C4 C22×C3⋊C8 C6×M4(2) C2×C4○D12 C2×Dic6 C2×D12 C4○D12 C2×C3⋊D4 C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C23 C6 C2 # reps 1 2 2 8 1 1 1 2 2 8 4 1 2 4 1 6 2 8 4

Matrix representation of C2×D12.C4 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 46 55 0 0 0 27 0 0 0 0 1 72 0 0 1 0
,
 72 0 0 0 3 1 0 0 0 0 1 0 0 0 1 72
,
 51 34 0 0 66 22 0 0 0 0 72 0 0 0 0 72
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[46,0,0,0,55,27,0,0,0,0,1,1,0,0,72,0],[72,3,0,0,0,1,0,0,0,0,1,1,0,0,0,72],[51,66,0,0,34,22,0,0,0,0,72,0,0,0,0,72] >;

C2×D12.C4 in GAP, Magma, Sage, TeX

C_2\times D_{12}.C_4
% in TeX

G:=Group("C2xD12.C4");
// GroupNames label

G:=SmallGroup(192,1303);
// by ID

G=gap.SmallGroup(192,1303);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,297,80,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^2=1,d^4=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^7,d*c*d^-1=b^6*c>;
// generators/relations

׿
×
𝔽