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G = C8⋊Dic3order 96 = 25·3

2nd semidirect product of C8 and Dic3 acting via Dic3/C6=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C242C4, C82Dic3, C12.4Q8, C4.4Dic6, C6.2SD16, C22.8D12, (C2×C8).6S3, C32(C4.Q8), C6.5(C4⋊C4), (C2×C24).8C2, (C2×C6).13D4, (C2×C4).68D6, C12.33(C2×C4), C4⋊Dic3.2C2, C4.6(C2×Dic3), C2.2(C24⋊C2), C2.3(C4⋊Dic3), (C2×C12).81C22, SmallGroup(96,24)

Series: Derived Chief Lower central Upper central

C1C12 — C8⋊Dic3
C1C3C6C2×C6C2×C12C4⋊Dic3 — C8⋊Dic3
C3C6C12 — C8⋊Dic3
C1C22C2×C4C2×C8

Generators and relations for C8⋊Dic3
 G = < a,b,c | a8=b6=1, c2=b3, ab=ba, cac-1=a3, cbc-1=b-1 >

12C4
12C4
6C2×C4
6C2×C4
4Dic3
4Dic3
3C4⋊C4
3C4⋊C4
2C2×Dic3
2C2×Dic3
3C4.Q8

Character table of C8⋊Dic3

 class 12A2B2C34A4B4C4D4E4F6A6B6C8A8B8C8D12A12B12C12D24A24B24C24D24E24F24G24H
 size 1111222121212122222222222222222222
ρ1111111111111111111111111111111    trivial
ρ211111111-11-1111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111111-11-11111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111111-1-1-1-11111111111111111111    linear of order 2
ρ511-1-11-11ii-i-i-11-1-111-1-11-11-111-111-1-1    linear of order 4
ρ611-1-11-11i-i-ii-11-11-1-11-11-111-1-11-1-111    linear of order 4
ρ711-1-11-11-iii-i-11-11-1-11-11-111-1-11-1-111    linear of order 4
ρ811-1-11-11-i-iii-11-1-111-1-11-11-111-111-1-1    linear of order 4
ρ92222-1220000-1-1-12222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ102222-1-2-20000-1-1-1000011113-33-33-33-3    orthogonal lifted from D12
ρ1122222-2-200002220000-2-2-2-200000000    orthogonal lifted from D4
ρ122222-1220000-1-1-1-2-2-2-2-1-1-1-111111111    orthogonal lifted from D6
ρ132222-1-2-20000-1-1-100001111-33-33-33-33    orthogonal lifted from D12
ρ1422-2-2-1-2200001-112-2-221-11-1-111-111-1-1    symplectic lifted from Dic3, Schur index 2
ρ1522-2-2-1-2200001-11-222-21-11-11-1-11-1-111    symplectic lifted from Dic3, Schur index 2
ρ1622-2-222-20000-22-200002-22-200000000    symplectic lifted from Q8, Schur index 2
ρ1722-2-2-12-200001-110000-11-1133-3-3-333-3    symplectic lifted from Dic6, Schur index 2
ρ1822-2-2-12-200001-110000-11-11-3-3333-3-33    symplectic lifted from Dic6, Schur index 2
ρ192-2-2220000002-2-2-2--2-2--20000--2-2-2--2--2--2-2-2    complex lifted from SD16
ρ202-2-2220000002-2-2--2-2--2-20000-2--2--2-2-2-2--2--2    complex lifted from SD16
ρ212-22-22000000-2-22--2--2-2-20000-2-2-2-2--2--2--2--2    complex lifted from SD16
ρ222-22-22000000-2-22-2-2--2--20000--2--2--2--2-2-2-2-2    complex lifted from SD16
ρ232-22-2-100000011-1--2--2-2-233-3-387ζ3285ζ328587ζ385ζ38587ζ3285ζ328587ζ385ζ38583ζ328ζ32883ζ38ζ3883ζ328ζ32883ζ38ζ38    complex lifted from C24⋊C2
ρ242-2-22-1000000-111--2-2--2-2-333-387ζ3285ζ328583ζ38ζ3883ζ328ζ32887ζ385ζ38587ζ3285ζ328587ζ385ζ38583ζ328ζ32883ζ38ζ38    complex lifted from C24⋊C2
ρ252-22-2-100000011-1--2--2-2-2-3-33387ζ385ζ38587ζ3285ζ328587ζ385ζ38587ζ3285ζ328583ζ38ζ3883ζ328ζ32883ζ38ζ3883ζ328ζ328    complex lifted from C24⋊C2
ρ262-2-22-1000000-111--2-2--2-23-3-3387ζ385ζ38583ζ328ζ32883ζ38ζ3887ζ3285ζ328587ζ385ζ38587ζ3285ζ328583ζ38ζ3883ζ328ζ328    complex lifted from C24⋊C2
ρ272-22-2-100000011-1-2-2--2--233-3-383ζ328ζ32883ζ38ζ3883ζ328ζ32883ζ38ζ3887ζ3285ζ328587ζ385ζ38587ζ3285ζ328587ζ385ζ385    complex lifted from C24⋊C2
ρ282-2-22-1000000-111-2--2-2--23-3-3383ζ38ζ3887ζ3285ζ328587ζ385ζ38583ζ328ζ32883ζ38ζ3883ζ328ζ32887ζ385ζ38587ζ3285ζ3285    complex lifted from C24⋊C2
ρ292-22-2-100000011-1-2-2--2--2-3-33383ζ38ζ3883ζ328ζ32883ζ38ζ3883ζ328ζ32887ζ385ζ38587ζ3285ζ328587ζ385ζ38587ζ3285ζ3285    complex lifted from C24⋊C2
ρ302-2-22-1000000-111-2--2-2--2-333-383ζ328ζ32887ζ385ζ38587ζ3285ζ328583ζ38ζ3883ζ328ζ32883ζ38ζ3887ζ3285ζ328587ζ385ζ385    complex lifted from C24⋊C2

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

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

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

C8⋊Dic3 is a maximal subgroup of
C24.6Q8  C24.Q8  D242C4  D82Dic3  C249Q8  C24.13Q8  C4×C24⋊C2  C8⋊Dic6  D24⋊C4  Dic12⋊C4  C23.39D12  C23.15D12  C23.43D12  C23.18D12  D4⋊Dic6  D4.Dic6  D6.SD16  D6⋊C811C2  Q82Dic6  Q8.4Dic6  D6.1SD16  C8⋊Dic3⋊C2  Dic6.3Q8  D123Q8  D12.3Q8  Dic64Q8  C245Q8  C8.8Dic6  S3×C4.Q8  (S3×C8)⋊C4  C244Q8  C8⋊S3⋊C4  C23.27D12  C2430D4  C23.52D12  C243D4  C24.4D4  D8⋊Dic3  C2412D4  Dic3×SD16  C2414D4  Q16⋊Dic3  C24.36D4  C8⋊Dic9  C12.Dic6  C242Dic3  C60.7Q8  C12010C4  C120⋊C4
C8⋊Dic3 is a maximal quotient of
C242C8  C24.Q8  C12.9C42  C8⋊Dic9  C12.Dic6  C242Dic3  C60.7Q8  C12010C4  C120⋊C4

Matrix representation of C8⋊Dic3 in GL4(𝔽73) generated by

1000
0100
00220
006463
,
07200
1100
0010
0001
,
394700
83400
00673
00376
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,22,64,0,0,0,63],[0,1,0,0,72,1,0,0,0,0,1,0,0,0,0,1],[39,8,0,0,47,34,0,0,0,0,67,37,0,0,3,6] >;

C8⋊Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C8:Dic3");
// GroupNames label

G:=SmallGroup(96,24);
// by ID

G=gap.SmallGroup(96,24);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,55,579,69,2309]);
// Polycyclic

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

Export

Subgroup lattice of C8⋊Dic3 in TeX
Character table of C8⋊Dic3 in TeX

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