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G = C12.7S4order 288 = 25·32

7th non-split extension by C12 of S4 acting via S4/A4=C2

non-abelian, soluble

Aliases: C12.7S4, SL2(F3):4D6, C4.A4:1S3, C4.3(C3:S4), C6.36(C2xS4), C6.6S4:6C2, (C3xQ8).18D6, C3:2(C4.3S4), (C3xSL2(F3)):4C22, C2.10(C2xC3:S4), (C3xC4.A4):2C2, (C3xC4oD4):2S3, Q8.5(C2xC3:S3), C4oD4:2(C3:S3), SmallGroup(288,915)

Series: Derived Chief Lower central Upper central

C1C2Q8C3xSL2(F3) — C12.7S4
C1C2Q8C3xQ8C3xSL2(F3)C6.6S4 — C12.7S4
C3xSL2(F3) — C12.7S4
C1C2C4

Generators and relations for C12.7S4
 G = < a,b,c,d,e | a12=d3=e2=1, b2=c2=a6, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=a6b, dbd-1=a6bc, ebe=bc, dcd-1=b, ece=a6c, ede=d-1 >

Subgroups: 832 in 110 conjugacy classes, 21 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2xC4, D4, Q8, C23, C32, C12, C12, D6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, C3:S3, C3xC6, C3:C8, SL2(F3), D12, C2xC12, C3xD4, C3xQ8, C22xS3, C8:C22, C3xC12, C2xC3:S3, C4.Dic3, D4:S3, Q8:2S3, GL2(F3), C4.A4, C2xD12, C3xC4oD4, C3xSL2(F3), C12:S3, D4:D6, C4.3S4, C6.6S4, C3xC4.A4, C12.7S4
Quotients: C1, C2, C22, S3, D6, C3:S3, S4, C2xC3:S3, C2xS4, C3:S4, C4.3S4, C2xC3:S4, C12.7S4

Character table of C12.7S4

 class 12A2B2C2D3A3B3C3D4A4B6A6B6C6D6E8A8B12A12B12C12D12E12F12G12H12I
 size 116363628882628881236362288888812
ρ1111111111111111111111111111    trivial
ρ211-11-11111-111111-11-1-1-1-1-1-1-1-1-11    linear of order 2
ρ3111-1-111111111111-1-1111111111    linear of order 2
ρ411-1-111111-111111-1-11-1-1-1-1-1-1-1-11    linear of order 2
ρ522-200-1-1-12-22-12-1-1100111-2-2111-1    orthogonal lifted from D6
ρ622-200-12-1-1-22-1-12-110011-21111-2-1    orthogonal lifted from D6
ρ722-200-1-12-1-22-1-1-1210011111-2-21-1    orthogonal lifted from D6
ρ822200-1-1-1222-12-1-1-100-1-1-122-1-1-1-1    orthogonal lifted from S3
ρ922-2002-1-1-1-222-1-1-1-200-2-21111112    orthogonal lifted from D6
ρ1022200-1-12-122-1-1-12-100-1-1-1-1-122-1-1    orthogonal lifted from S3
ρ1122200-12-1-122-1-12-1-100-1-12-1-1-1-12-1    orthogonal lifted from S3
ρ12222002-1-1-1222-1-1-120022-1-1-1-1-1-12    orthogonal lifted from S3
ρ1333-1-1-130003-13000-11133000000-1    orthogonal lifted from S4
ρ143311-13000-3-130001-11-3-3000000-1    orthogonal lifted from C2xS4
ρ1533-11130003-13000-1-1-133000000-1    orthogonal lifted from S4
ρ16331-113000-3-1300011-1-3-3000000-1    orthogonal lifted from C2xS4
ρ174-40004-2-2-200-4222000000000000    orthogonal lifted from C4.3S4
ρ184-4000-2-211002-12-100023-230-333-300    orthogonal faithful
ρ194-4000411100-4-1-1-100000-33-33-330    orthogonal lifted from C4.3S4
ρ204-4000411100-4-1-1-1000003-33-33-30    orthogonal lifted from C4.3S4
ρ214-4000-211-20022-1-1000-23233003-3-30    orthogonal faithful
ρ224-4000-2-211002-12-1000-232303-3-3300    orthogonal faithful
ρ234-4000-21-21002-1-12000-2323-3-330030    orthogonal faithful
ρ244-4000-211-20022-1-100023-23-300-3330    orthogonal faithful
ρ254-4000-21-21002-1-1200023-2333-300-30    orthogonal faithful
ρ2666200-3000-6-2-3000-100330000001    orthogonal lifted from C2xC3:S4
ρ2766-200-30006-2-3000100-3-30000001    orthogonal lifted from C3:S4

Smallest permutation representation of C12.7S4
On 48 points
Generators in S48
(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)
(1 45 7 39)(2 46 8 40)(3 47 9 41)(4 48 10 42)(5 37 11 43)(6 38 12 44)(13 36 19 30)(14 25 20 31)(15 26 21 32)(16 27 22 33)(17 28 23 34)(18 29 24 35)
(1 23 7 17)(2 24 8 18)(3 13 9 19)(4 14 10 20)(5 15 11 21)(6 16 12 22)(25 48 31 42)(26 37 32 43)(27 38 33 44)(28 39 34 45)(29 40 35 46)(30 41 36 47)
(13 36 47)(14 25 48)(15 26 37)(16 27 38)(17 28 39)(18 29 40)(19 30 41)(20 31 42)(21 32 43)(22 33 44)(23 34 45)(24 35 46)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 15)(16 24)(17 23)(18 22)(19 21)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)

G:=sub<Sym(48)| (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), (1,45,7,39)(2,46,8,40)(3,47,9,41)(4,48,10,42)(5,37,11,43)(6,38,12,44)(13,36,19,30)(14,25,20,31)(15,26,21,32)(16,27,22,33)(17,28,23,34)(18,29,24,35), (1,23,7,17)(2,24,8,18)(3,13,9,19)(4,14,10,20)(5,15,11,21)(6,16,12,22)(25,48,31,42)(26,37,32,43)(27,38,33,44)(28,39,34,45)(29,40,35,46)(30,41,36,47), (13,36,47)(14,25,48)(15,26,37)(16,27,38)(17,28,39)(18,29,40)(19,30,41)(20,31,42)(21,32,43)(22,33,44)(23,34,45)(24,35,46), (2,12)(3,11)(4,10)(5,9)(6,8)(13,15)(16,24)(17,23)(18,22)(19,21)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)>;

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), (1,45,7,39)(2,46,8,40)(3,47,9,41)(4,48,10,42)(5,37,11,43)(6,38,12,44)(13,36,19,30)(14,25,20,31)(15,26,21,32)(16,27,22,33)(17,28,23,34)(18,29,24,35), (1,23,7,17)(2,24,8,18)(3,13,9,19)(4,14,10,20)(5,15,11,21)(6,16,12,22)(25,48,31,42)(26,37,32,43)(27,38,33,44)(28,39,34,45)(29,40,35,46)(30,41,36,47), (13,36,47)(14,25,48)(15,26,37)(16,27,38)(17,28,39)(18,29,40)(19,30,41)(20,31,42)(21,32,43)(22,33,44)(23,34,45)(24,35,46), (2,12)(3,11)(4,10)(5,9)(6,8)(13,15)(16,24)(17,23)(18,22)(19,21)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37) );

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)], [(1,45,7,39),(2,46,8,40),(3,47,9,41),(4,48,10,42),(5,37,11,43),(6,38,12,44),(13,36,19,30),(14,25,20,31),(15,26,21,32),(16,27,22,33),(17,28,23,34),(18,29,24,35)], [(1,23,7,17),(2,24,8,18),(3,13,9,19),(4,14,10,20),(5,15,11,21),(6,16,12,22),(25,48,31,42),(26,37,32,43),(27,38,33,44),(28,39,34,45),(29,40,35,46),(30,41,36,47)], [(13,36,47),(14,25,48),(15,26,37),(16,27,38),(17,28,39),(18,29,40),(19,30,41),(20,31,42),(21,32,43),(22,33,44),(23,34,45),(24,35,46)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,15),(16,24),(17,23),(18,22),(19,21),(25,48),(26,47),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37)]])

Matrix representation of C12.7S4 in GL4(F73) generated by

5906666
6659066
0665966
7770
,
0010
72727271
72000
1101
,
1112
0010
07200
7207272
,
1000
72727271
07200
0111
,
7207272
0727272
7272072
1112
G:=sub<GL(4,GF(73))| [59,66,0,7,0,59,66,7,66,0,59,7,66,66,66,0],[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],[1,72,0,0,0,72,72,1,0,72,0,1,0,71,0,1],[72,0,72,1,0,72,72,1,72,72,0,1,72,72,72,2] >;

C12.7S4 in GAP, Magma, Sage, TeX

C_{12}._7S_4
% in TeX

G:=Group("C12.7S4");
// GroupNames label

G:=SmallGroup(288,915);
// by ID

G=gap.SmallGroup(288,915);
# by ID

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

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

Export

Character table of C12.7S4 in TeX

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