Copied to
clipboard

G = C8×Dic3order 96 = 25·3

Direct product of C8 and Dic3

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C8×Dic3, C244C4, C6.3C42, C3⋊C85C4, C32(C4×C8), C82(C3⋊C8), C2.2(S3×C8), C6.3(C2×C8), C4.20(C4×S3), (C2×C8).10S3, (C2×C4).90D6, C12.40(C2×C4), (C2×C24).14C2, C22.8(C4×S3), C2.2(C4×Dic3), (C2×Dic3).7C4, C4.12(C2×Dic3), (C4×Dic3).10C2, (C2×C12).104C22, C8(C2×C3⋊C8), (C2×C8)(C3⋊C8), (C2×C3⋊C8).12C2, (C2×C6).9(C2×C4), (C2×C8)(C4×Dic3), SmallGroup(96,20)

Series: Derived Chief Lower central Upper central

C1C3 — C8×Dic3
C1C3C6C2×C6C2×C12C4×Dic3 — C8×Dic3
C3 — C8×Dic3
C1C2×C8

Generators and relations for C8×Dic3
 G = < a,b,c | a8=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C4
3C4
3C4
3C8
3C2×C4
3C2×C4
3C8
3C42
3C2×C8
3C4×C8

Smallest permutation representation of C8×Dic3
Regular action 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 75 25 24 86 71)(2 76 26 17 87 72)(3 77 27 18 88 65)(4 78 28 19 81 66)(5 79 29 20 82 67)(6 80 30 21 83 68)(7 73 31 22 84 69)(8 74 32 23 85 70)(9 41 60 49 90 38)(10 42 61 50 91 39)(11 43 62 51 92 40)(12 44 63 52 93 33)(13 45 64 53 94 34)(14 46 57 54 95 35)(15 47 58 55 96 36)(16 48 59 56 89 37)
(1 64 24 34)(2 57 17 35)(3 58 18 36)(4 59 19 37)(5 60 20 38)(6 61 21 39)(7 62 22 40)(8 63 23 33)(9 67 49 29)(10 68 50 30)(11 69 51 31)(12 70 52 32)(13 71 53 25)(14 72 54 26)(15 65 55 27)(16 66 56 28)(41 82 90 79)(42 83 91 80)(43 84 92 73)(44 85 93 74)(45 86 94 75)(46 87 95 76)(47 88 96 77)(48 81 89 78)

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

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

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

C8×Dic3 is a maximal subgroup of
Dic3⋊C16  C4810C4  C24.97D4  S3×C4×C8  D6.C42  C24⋊Q8  Dic35M4(2)  D6.4C42  Dic3.5M4(2)  C24⋊C4⋊C2  C3⋊D4⋊C8  Dic3⋊M4(2)  Dic34D8  Dic36SD16  Dic3.SD16  (C2×C8).200D6  Dic37SD16  Dic34Q16  Dic3.1Q16  Q8⋊C4⋊S3  C42.27D6  Dic6⋊C8  C42.200D6  C42.31D6  Dic38SD16  C245Q8  C8.8Dic6  Dic35D8  Dic35Q16  C242Q8  C8.6Dic6  D247C4  C12.12C42  C12.7C42  C2421D4  C24.100D4  C245D4  C24.22D4  C24.43D4  C2415D4  C24.26D4  C24.28D4  D85Dic3  C6.(S3×C8)  Dic154C8  C30.C42
C8×Dic3 is a maximal quotient of
C42.279D6  C4810C4  (C2×C24)⋊5C4  C6.(S3×C8)  Dic154C8  C30.C42

48 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E···4L6A6B6C8A···8H8I···8P12A12B12C12D24A···24H
order1222344444···46668···88···81212121224···24
size1111211113···32221···13···322222···2

48 irreducible representations

dim11111111222222
type+++++-+
imageC1C2C2C2C4C4C4C8S3Dic3D6C4×S3C4×S3S3×C8
kernelC8×Dic3C2×C3⋊C8C4×Dic3C2×C24C3⋊C8C24C2×Dic3Dic3C2×C8C8C2×C4C4C22C2
# reps111144416121228

Matrix representation of C8×Dic3 in GL3(𝔽73) generated by

2700
0510
0051
,
7200
0721
0720
,
2700
0260
06271
G:=sub<GL(3,GF(73))| [27,0,0,0,51,0,0,0,51],[72,0,0,0,72,72,0,1,0],[27,0,0,0,2,62,0,60,71] >;

C8×Dic3 in GAP, Magma, Sage, TeX

C_8\times {\rm Dic}_3
% in TeX

G:=Group("C8xDic3");
// GroupNames label

G:=SmallGroup(96,20);
// by ID

G=gap.SmallGroup(96,20);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,55,69,2309]);
// Polycyclic

G:=Group<a,b,c|a^8=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C8×Dic3 in TeX

׿
×
𝔽