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

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

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

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

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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