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G = C2.Dic12order 96 = 25·3

1st central extension by C2 of Dic12

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.1Q16, Dic62C4, C12.44D4, C6.1SD16, C2.1Dic12, C22.7D12, C4.7(C4×S3), (C2×C8).2S3, (C2×C24).2C2, (C2×C6).12D4, (C2×C4).67D6, C12.17(C2×C4), C2.7(D6⋊C4), C32(Q8⋊C4), C4⋊Dic3.1C2, C2.1(C24⋊C2), C4.19(C3⋊D4), C6.5(C22⋊C4), (C2×Dic6).1C2, (C2×C12).80C22, SmallGroup(96,23)

Series: Derived Chief Lower central Upper central

C1C12 — C2.Dic12
C1C3C6C12C2×C12C4⋊Dic3 — C2.Dic12
C3C6C12 — C2.Dic12
C1C22C2×C4C2×C8

Generators and relations for C2.Dic12
 G = < a,b,c | a6=b8=1, c2=a3, ab=ba, cac-1=a-1, cbc-1=a3b3 >

6C4
6C4
12C4
2C8
3Q8
3Q8
6C2×C4
6Q8
6C2×C4
2Dic3
2Dic3
4Dic3
3C2×Q8
3C4⋊C4
2C2×Dic3
2C2×Dic3
2C24
2Dic6
3Q8⋊C4

Character table of C2.Dic12

 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
ρ51-11-111-11i-1-i-11-1ii-i-i1-1-11i-i-ii-i-iii    linear of order 4
ρ61-11-111-11-i-1i-11-1-i-iii1-1-11-iii-iii-i-i    linear of order 4
ρ71-11-111-1-1-i1i-11-1ii-i-i1-1-11i-i-ii-i-iii    linear of order 4
ρ81-11-111-1-1i1-i-11-1-i-iii1-1-11-iii-iii-i-i    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
ρ1022222-2-200002220000-2-2-2-200000000    orthogonal lifted from D4
ρ112222-1220000-1-1-1-2-2-2-2-1-1-1-111111111    orthogonal lifted from D6
ρ122-22-22-220000-22-20000-222-200000000    orthogonal lifted from D4
ρ132222-1-2-20000-1-1-100001111-3-3-3333-33    orthogonal lifted from D12
ρ142222-1-2-20000-1-1-100001111333-3-3-33-3    orthogonal lifted from D12
ρ152-2-222000000-2-22-22-22000022-222-2-2-2    symplectic lifted from Q16, Schur index 2
ρ162-2-222000000-2-222-22-20000-2-22-2-2222    symplectic lifted from Q16, Schur index 2
ρ172-2-22-100000011-1-22-22-3-333ζ83ζ3838ζ3ζ83ζ3838ζ3ζ83ζ328ζ328ζ87ζ385ζ385ζ87ζ385ζ385ζ87ζ328785ζ32ζ83ζ328ζ328ζ87ζ328785ζ32    symplectic lifted from Dic12, Schur index 2
ρ182-2-22-100000011-12-22-2-3-333ζ83ζ328ζ328ζ83ζ328ζ328ζ83ζ3838ζ3ζ87ζ328785ζ32ζ87ζ328785ζ32ζ87ζ385ζ385ζ83ζ3838ζ3ζ87ζ385ζ385    symplectic lifted from Dic12, Schur index 2
ρ192-2-22-100000011-12-22-233-3-3ζ87ζ328785ζ32ζ87ζ328785ζ32ζ87ζ385ζ385ζ83ζ328ζ328ζ83ζ328ζ328ζ83ζ3838ζ3ζ87ζ385ζ385ζ83ζ3838ζ3    symplectic lifted from Dic12, Schur index 2
ρ202-2-22-100000011-1-22-2233-3-3ζ87ζ385ζ385ζ87ζ385ζ385ζ87ζ328785ζ32ζ83ζ3838ζ3ζ83ζ3838ζ3ζ83ζ328ζ328ζ87ζ328785ζ32ζ83ζ328ζ328    symplectic lifted from Dic12, Schur index 2
ρ212-22-2-12-200001-112i2i-2i-2i-111-1-iii-iii-i-i    complex lifted from C4×S3
ρ222-22-2-1-2200001-1100001-1-11-3--3--3--3-3-3-3--3    complex lifted from C3⋊D4
ρ232-22-2-12-200001-11-2i-2i2i2i-111-1i-i-ii-i-iii    complex lifted from C4×S3
ρ2422-2-2-1000000-111--2-2-2--2-33-33ζ87ζ328785ζ32ζ83ζ32838ζ32ζ87ζ328785ζ32ζ87ζ38785ζ3ζ83ζ3838ζ3ζ87ζ38785ζ3ζ83ζ32838ζ32ζ83ζ3838ζ3    complex lifted from C24⋊C2
ρ2522-2-220000002-2-2-2--2--2-20000--2-2--2--2-2--2-2-2    complex lifted from SD16
ρ262-22-2-1-2200001-1100001-1-11--3-3-3-3--3--3--3-3    complex lifted from C3⋊D4
ρ2722-2-2-1000000-111-2--2--2-23-33-3ζ83ζ3838ζ3ζ87ζ38785ζ3ζ83ζ3838ζ3ζ83ζ32838ζ32ζ87ζ328785ζ32ζ83ζ32838ζ32ζ87ζ38785ζ3ζ87ζ328785ζ32    complex lifted from C24⋊C2
ρ2822-2-2-1000000-111--2-2-2--23-33-3ζ87ζ38785ζ3ζ83ζ3838ζ3ζ87ζ38785ζ3ζ87ζ328785ζ32ζ83ζ32838ζ32ζ87ζ328785ζ32ζ83ζ3838ζ3ζ83ζ32838ζ32    complex lifted from C24⋊C2
ρ2922-2-2-1000000-111-2--2--2-2-33-33ζ83ζ32838ζ32ζ87ζ328785ζ32ζ83ζ32838ζ32ζ83ζ3838ζ3ζ87ζ38785ζ3ζ83ζ3838ζ3ζ87ζ328785ζ32ζ87ζ38785ζ3    complex lifted from C24⋊C2
ρ3022-2-220000002-2-2--2-2-2--20000-2--2-2-2--2-2--2--2    complex lifted from SD16

Smallest permutation representation of C2.Dic12
Regular action on 96 points
Generators in S96
(1 53 74 87 10 35)(2 54 75 88 11 36)(3 55 76 81 12 37)(4 56 77 82 13 38)(5 49 78 83 14 39)(6 50 79 84 15 40)(7 51 80 85 16 33)(8 52 73 86 9 34)(17 62 90 66 25 43)(18 63 91 67 26 44)(19 64 92 68 27 45)(20 57 93 69 28 46)(21 58 94 70 29 47)(22 59 95 71 30 48)(23 60 96 72 31 41)(24 61 89 65 32 42)
(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 62 87 25)(2 28 88 57)(3 60 81 31)(4 26 82 63)(5 58 83 29)(6 32 84 61)(7 64 85 27)(8 30 86 59)(9 22 52 71)(10 66 53 17)(11 20 54 69)(12 72 55 23)(13 18 56 67)(14 70 49 21)(15 24 50 65)(16 68 51 19)(33 92 80 45)(34 48 73 95)(35 90 74 43)(36 46 75 93)(37 96 76 41)(38 44 77 91)(39 94 78 47)(40 42 79 89)

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

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

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

C2.Dic12 is a maximal subgroup of
C12.14Q16  C4×C24⋊C2  C42.264D6  C4×Dic12  C42.14D6  C42.16D6  C42.20D6  Dic12⋊C4  C23.39D12  C23.40D12  C23.15D12  D12.32D4  D1214D4  Dic614D4  Dic6.32D4  D4.S3⋊C4  Dic36SD16  C12⋊Q8⋊C2  (C2×C8).200D6  D4⋊(C4×S3)  D42S3⋊C4  D6⋊SD16  C3⋊C81D4  C3⋊Q16⋊C4  Dic34Q16  Dic3.1Q16  (C2×Q8).36D6  S3×Q8⋊C4  (S3×Q8)⋊C4  D61Q16  C3⋊C8.D4  Dic6.3Q8  C12⋊SD16  D12.19D4  C42.36D6  C4⋊Dic12  Dic63Q8  Dic64Q8  Dic6⋊Q8  Dic6.Q8  D6.2SD16  C6.(C4○D8)  Dic3.Q16  Dic6.2Q8  D6.2Q16  C2.D87S3  C23.28D12  C2430D4  C24.82D4  C23.51D12  C23.54D12  C242D4  C24.4D4  (C6×D8).C2  Dic6⋊D4  Dic33SD16  (C3×Q8).D4  D68SD16  Dic6.16D4  Dic33Q16  D65Q16  C36.45D4  C6.Dic12  C12.73D12  C6.4Dic12  C10.Dic12  Dic3015C4  Dic308C4  Dic30⋊C4
C2.Dic12 is a maximal quotient of
C4.8Dic12  C23.35D12  C4.Dic12  C12.47D8  C12.9C42  C36.45D4  C6.Dic12  C12.73D12  C6.4Dic12  C10.Dic12  Dic3015C4  Dic308C4  Dic30⋊C4

Matrix representation of C2.Dic12 in GL4(𝔽73) generated by

1100
72000
007272
0010
,
46000
04600
003611
006225
,
34800
473900
00345
004270
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,72,1,0,0,72,0],[46,0,0,0,0,46,0,0,0,0,36,62,0,0,11,25],[34,47,0,0,8,39,0,0,0,0,3,42,0,0,45,70] >;

C2.Dic12 in GAP, Magma, Sage, TeX

C_2.{\rm Dic}_{12}
% in TeX

G:=Group("C2.Dic12");
// GroupNames label

G:=SmallGroup(96,23);
// by ID

G=gap.SmallGroup(96,23);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,73,79,362,86,2309]);
// Polycyclic

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

Export

Subgroup lattice of C2.Dic12 in TeX
Character table of C2.Dic12 in TeX

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