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G = C12.6S4order 288 = 25·32

6th non-split extension by C12 of S4 acting via S4/A4=C2

non-abelian, soluble

Aliases: C12.6S4, SL2(F3).9D6, C4.2(C3:S4), C6.34(C2xS4), C4.A4.1S3, C6.5S4:6C2, (C3xQ8).16D6, C3:2(C4.S4), (C3xSL2(F3)).9C22, C2.8(C2xC3:S4), Q8.3(C2xC3:S3), (C3xC4.A4).3C2, (C3xC4oD4).6S3, C4oD4.2(C3:S3), SmallGroup(288,913)

Series: Derived Chief Lower central Upper central

C1C2Q8C3xSL2(F3) — C12.6S4
C1C2Q8C3xQ8C3xSL2(F3)C6.5S4 — C12.6S4
C3xSL2(F3) — C12.6S4
C1C2C4

Generators and relations for C12.6S4
 G = < a,b,c,d,e | a12=d3=1, b2=c2=e2=a6, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a6b, dbd-1=a6bc, ebe-1=bc, dcd-1=b, ece-1=a6c, ede-1=d-1 >

Subgroups: 480 in 94 conjugacy classes, 21 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2xC4, D4, Q8, Q8, C32, Dic3, C12, C12, C2xC6, M4(2), SD16, Q16, C2xQ8, C4oD4, C3xC6, C3:C8, SL2(F3), Dic6, C2xDic3, C2xC12, C3xD4, C3xQ8, C8.C22, C3:Dic3, C3xC12, C4.Dic3, D4.S3, C3:Q16, CSU2(F3), C4.A4, C2xDic6, C3xC4oD4, C3xSL2(F3), C32:4Q8, Q8.14D6, C4.S4, C6.5S4, C3xC4.A4, C12.6S4
Quotients: C1, C2, C22, S3, D6, C3:S3, S4, C2xC3:S3, C2xS4, C3:S4, C4.S4, C2xC3:S4, C12.6S4

Character table of C12.6S4

 class 12A2B3A3B3C3D4A4B4C4D6A6B6C6D6E8A8B12A12B12C12D12E12F12G12H12I
 size 116288826363628881236362288888812
ρ1111111111111111111111111111    trivial
ρ2111111111-1-111111-1-1111111111    linear of order 2
ρ311-11111-11-111111-1-11-1-1-1-1-1-1-1-11    linear of order 2
ρ411-11111-111-11111-11-1-1-1-1-1-1-1-1-11    linear of order 2
ρ522-2-1-12-1-2200-1-1-1210011111-2-21-1    orthogonal lifted from D6
ρ6222-1-12-12200-1-1-12-100-1-1-1-1-122-1-1    orthogonal lifted from S3
ρ722-2-12-1-1-2200-1-12-1100111-2-2111-1    orthogonal lifted from D6
ρ8222-12-1-12200-1-12-1-100-1-1-122-1-1-1-1    orthogonal lifted from S3
ρ92222-1-1-122002-1-1-120022-1-1-1-1-1-12    orthogonal lifted from S3
ρ1022-2-1-1-12-2200-12-1-110011-21111-2-1    orthogonal lifted from D6
ρ11222-1-1-122200-12-1-1-100-1-12-1-1-1-12-1    orthogonal lifted from S3
ρ1222-22-1-1-1-22002-1-1-1-200-2-21111112    orthogonal lifted from D6
ρ133313000-3-11-130001-11-3-3000000-1    orthogonal lifted from C2xS4
ρ1433-130003-1-1-13000-11133000000-1    orthogonal lifted from S4
ρ1533-130003-1113000-1-1-133000000-1    orthogonal lifted from S4
ρ163313000-3-1-11300011-1-3-3000000-1    orthogonal lifted from C2xS4
ρ174-404-2-2-20000-4222000000000000    symplectic lifted from C4.S4, Schur index 2
ρ184-40-21-2100002-1-1200023-2333-300-30    symplectic faithful, Schur index 2
ρ194-40-2-21100002-12-100023-23-300-3330    symplectic faithful, Schur index 2
ρ204-40-21-2100002-1-12000-2323-3-330030    symplectic faithful, Schur index 2
ρ214-40-211-2000022-1-100023-230-333-300    symplectic faithful, Schur index 2
ρ224-40-2-21100002-12-1000-23233003-3-30    symplectic faithful, Schur index 2
ρ234-4041110000-4-1-1-100000-33-33-330    symplectic lifted from C4.S4, Schur index 2
ρ244-4041110000-4-1-1-1000003-33-33-30    symplectic lifted from C4.S4, Schur index 2
ρ254-40-211-2000022-1-1000-232303-3-3300    symplectic faithful, Schur index 2
ρ2666-2-30006-200-3000100-3-30000001    orthogonal lifted from C3:S4
ρ27662-3000-6-200-3000-100330000001    orthogonal lifted from C2xC3:S4

Smallest permutation representation of C12.6S4
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 28 7 34)(2 29 8 35)(3 30 9 36)(4 31 10 25)(5 32 11 26)(6 33 12 27)(13 78 19 84)(14 79 20 73)(15 80 21 74)(16 81 22 75)(17 82 23 76)(18 83 24 77)(37 89 43 95)(38 90 44 96)(39 91 45 85)(40 92 46 86)(41 93 47 87)(42 94 48 88)(49 70 55 64)(50 71 56 65)(51 72 57 66)(52 61 58 67)(53 62 59 68)(54 63 60 69)
(1 86 7 92)(2 87 8 93)(3 88 9 94)(4 89 10 95)(5 90 11 96)(6 91 12 85)(13 63 19 69)(14 64 20 70)(15 65 21 71)(16 66 22 72)(17 67 23 61)(18 68 24 62)(25 43 31 37)(26 44 32 38)(27 45 33 39)(28 46 34 40)(29 47 35 41)(30 48 36 42)(49 79 55 73)(50 80 56 74)(51 81 57 75)(52 82 58 76)(53 83 59 77)(54 84 60 78)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 52 65)(14 53 66)(15 54 67)(16 55 68)(17 56 69)(18 57 70)(19 58 71)(20 59 72)(21 60 61)(22 49 62)(23 50 63)(24 51 64)(25 87 45)(26 88 46)(27 89 47)(28 90 48)(29 91 37)(30 92 38)(31 93 39)(32 94 40)(33 95 41)(34 96 42)(35 85 43)(36 86 44)(73 77 81)(74 78 82)(75 79 83)(76 80 84)
(1 81 7 75)(2 80 8 74)(3 79 9 73)(4 78 10 84)(5 77 11 83)(6 76 12 82)(13 37 19 43)(14 48 20 42)(15 47 21 41)(16 46 22 40)(17 45 23 39)(18 44 24 38)(25 63 31 69)(26 62 32 68)(27 61 33 67)(28 72 34 66)(29 71 35 65)(30 70 36 64)(49 94 55 88)(50 93 56 87)(51 92 57 86)(52 91 58 85)(53 90 59 96)(54 89 60 95)

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

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

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

Matrix representation of C12.6S4 in GL4(F73) generated by

66077
76607
07667
66666652
,
0010
72727271
72000
1101
,
1112
0010
07200
7207272
,
7207272
0001
1101
072072
,
2201212
0122212
1222012
51515139
G:=sub<GL(4,GF(73))| [66,7,0,66,0,66,7,66,7,0,66,66,7,7,7,52],[0,72,72,1,0,72,0,1,1,72,0,0,0,71,0,1],[1,0,0,72,1,0,72,0,1,1,0,72,2,0,0,72],[72,0,1,0,0,0,1,72,72,0,0,0,72,1,1,72],[22,0,12,51,0,12,22,51,12,22,0,51,12,12,12,39] >;

C12.6S4 in GAP, Magma, Sage, TeX

C_{12}._6S_4
% in TeX

G:=Group("C12.6S4");
// GroupNames label

G:=SmallGroup(288,913);
// by ID

G=gap.SmallGroup(288,913);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,2045,1016,170,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^12=d^3=1,b^2=c^2=e^2=a^6,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c^-1=a^6*b,d*b*d^-1=a^6*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^6*c,e*d*e^-1=d^-1>;
// generators/relations

Export

Character table of C12.6S4 in TeX

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