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G = C122Q8order 96 = 25·3

1st semidirect product of C12 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C122Q8, C42Dic6, C4.4D12, C12.27D4, C42.4S3, C31(C4⋊Q8), C6.1(C2×D4), C6.2(C2×Q8), (C4×C12).2C2, C2.4(C2×D12), (C2×C4).73D6, C4⋊Dic3.4C2, C2.4(C2×Dic6), (C2×C6).10C23, (C2×Dic6).2C2, (C2×C12).85C22, C22.34(C22×S3), (C2×Dic3).1C22, SmallGroup(96,76)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C122Q8
C1C3C6C2×C6C2×Dic3C2×Dic6 — C122Q8
C3C2×C6 — C122Q8
C1C22C42

Generators and relations for C122Q8
 G = < a,b,c | a12=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 138 in 68 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2 [×2], C3, C4 [×6], C4 [×4], C22, C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], Dic3 [×4], C12 [×6], C2×C6, C42, C4⋊C4 [×4], C2×Q8 [×2], Dic6 [×4], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C4⋊Q8, C4⋊Dic3 [×4], C4×C12, C2×Dic6 [×2], C122Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D6 [×3], C2×D4, C2×Q8 [×2], Dic6 [×4], D12 [×2], C22×S3, C4⋊Q8, C2×Dic6 [×2], C2×D12, C122Q8

Character table of C122Q8

 class 12A2B2C34A4B4C4D4E4F4G4H4I4J6A6B6C12A12B12C12D12E12F12G12H12I12J12K12L
 size 1111222222212121212222222222222222
ρ1111111111111111111111111111111    trivial
ρ211111-1-111-1-1-1-111111-11-1-11-1-1-11-1-11    linear of order 2
ρ3111111-1-1-11-1-111-11111-1-11-1-111-1-1-1-1    linear of order 2
ρ411111-11-1-1-111-11-1111-1-11-1-11-1-1-111-1    linear of order 2
ρ5111111-1-1-11-11-1-111111-1-11-1-111-1-1-1-1    linear of order 2
ρ611111-11-1-1-11-11-11111-1-11-1-11-1-1-111-1    linear of order 2
ρ711111111111-1-1-1-1111111111111111    linear of order 2
ρ811111-1-111-1-111-1-1111-11-1-11-1-1-11-1-11    linear of order 2
ρ922-2-2200-22000000-2-220200-2000-2002    orthogonal lifted from D4
ρ102222-12-2-2-22-20000-1-1-1-111-111-1-11111    orthogonal lifted from D6
ρ112222-12222220000-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1222-2-22002-2000000-2-220-2002000200-2    orthogonal lifted from D4
ρ1322-2-2-1002-200000011-131-3-3-13-33-1-331    orthogonal lifted from D12
ρ142222-1-2-222-2-20000-1-1-11-111-1111-111-1    orthogonal lifted from D6
ρ1522-2-2-100-2200000011-13-13-31-3-3313-3-1    orthogonal lifted from D12
ρ1622-2-2-100-2200000011-1-3-1-33133-31-33-1    orthogonal lifted from D12
ρ172222-1-22-2-2-220000-1-1-111-111-1111-1-11    orthogonal lifted from D6
ρ1822-2-2-1002-200000011-1-3133-1-33-3-13-31    orthogonal lifted from D12
ρ192-22-220-200020000-22-2002002000-2-20    symplectic lifted from Q8, Schur index 2
ρ202-2-2222000-2000002-2-2-200-200220000    symplectic lifted from Q8, Schur index 2
ρ212-2-222-20002000002-2-2200200-2-20000    symplectic lifted from Q8, Schur index 2
ρ222-2-22-12000-200000-11113313-3-1-1-3-33-3    symplectic lifted from Dic6, Schur index 2
ρ232-22-2202000-20000-22-200-200-2000220    symplectic lifted from Q8, Schur index 2
ρ242-22-2-102000-200001-113-31-3313-3-3-1-13    symplectic lifted from Dic6, Schur index 2
ρ252-2-22-1-2000200000-111-13-3-13311-33-3-3    symplectic lifted from Dic6, Schur index 2
ρ262-22-2-102000-200001-11-3313-31-333-1-1-3    symplectic lifted from Dic6, Schur index 2
ρ272-22-2-10-2000200001-1133-1-3-3-13-3311-3    symplectic lifted from Dic6, Schur index 2
ρ282-2-22-1-2000200000-111-1-33-1-3-3113-333    symplectic lifted from Dic6, Schur index 2
ρ292-2-22-12000-200000-1111-3-31-33-1-133-33    symplectic lifted from Dic6, Schur index 2
ρ302-22-2-10-2000200001-11-3-3-133-1-33-3113    symplectic lifted from Dic6, Schur index 2

Smallest permutation representation of C122Q8
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 13 52 94)(2 14 53 95)(3 15 54 96)(4 16 55 85)(5 17 56 86)(6 18 57 87)(7 19 58 88)(8 20 59 89)(9 21 60 90)(10 22 49 91)(11 23 50 92)(12 24 51 93)(25 41 69 81)(26 42 70 82)(27 43 71 83)(28 44 72 84)(29 45 61 73)(30 46 62 74)(31 47 63 75)(32 48 64 76)(33 37 65 77)(34 38 66 78)(35 39 67 79)(36 40 68 80)
(1 78 52 38)(2 77 53 37)(3 76 54 48)(4 75 55 47)(5 74 56 46)(6 73 57 45)(7 84 58 44)(8 83 59 43)(9 82 60 42)(10 81 49 41)(11 80 50 40)(12 79 51 39)(13 66 94 34)(14 65 95 33)(15 64 96 32)(16 63 85 31)(17 62 86 30)(18 61 87 29)(19 72 88 28)(20 71 89 27)(21 70 90 26)(22 69 91 25)(23 68 92 36)(24 67 93 35)

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

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

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

C122Q8 is a maximal subgroup of
C4.Dic12  C12.47D8  C12.2D8  C249Q8  C12.14Q16  C248Q8  C85D12  C4.5D24  C124Q16  C8⋊Dic6  C42.14D6  C42.20D6  C8.D12  Q8.14D12  C12⋊SD16  D123Q8  D124Q8  C4⋊Dic12  Dic63Q8  Dic64Q8  C12.50D8  C12.38SD16  D4.2D12  Q84Dic6  Q85Dic6  C127Q16  C42.62D6  C42.65D6  C42.68D6  C42.71D6  C12.16D8  C124SD16  C12.17D8  C12.9Q16  C12.SD16  C123Q16  C12.Q16  C42.274D6  C42.276D6  C42.89D6  C42.90D6  C42.92D6  C42.99D6  D4×Dic6  C42.106D6  D46Dic6  D1224D4  D46D12  C42.117D6  Q8×Dic6  Dic610Q8  Q87Dic6  Q8×D12  D1210Q8  C42.135D6  C42.141D6  C42.144D6  C42.148D6  C42.156D6  C42.165D6  C42.238D6  S3×C4⋊Q8  C42.241D6  C362Q8  Dic3⋊Dic6  C123Dic6  C126Dic6  Dic5⋊Dic6  C60⋊Q8  C608Q8
C122Q8 is a maximal quotient of
(C2×C4)⋊Dic6  (C22×C4).85D6  C249Q8  C248Q8  C24.13Q8  C8⋊Dic6  C124(C4⋊C4)  (C2×Dic6)⋊7C4  C4210Dic3  C362Q8  Dic3⋊Dic6  C123Dic6  C126Dic6  Dic5⋊Dic6  C60⋊Q8  C608Q8

Matrix representation of C122Q8 in GL4(𝔽13) generated by

01200
1000
0070
0062
,
01200
1000
00120
00012
,
10400
4300
0068
0077
G:=sub<GL(4,GF(13))| [0,1,0,0,12,0,0,0,0,0,7,6,0,0,0,2],[0,1,0,0,12,0,0,0,0,0,12,0,0,0,0,12],[10,4,0,0,4,3,0,0,0,0,6,7,0,0,8,7] >;

C122Q8 in GAP, Magma, Sage, TeX

C_{12}\rtimes_2Q_8
% in TeX

G:=Group("C12:2Q8");
// GroupNames label

G:=SmallGroup(96,76);
// by ID

G=gap.SmallGroup(96,76);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,217,103,218,50,2309]);
// Polycyclic

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

Export

Character table of C122Q8 in TeX

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