Copied to
clipboard

G = C12.6Q8order 96 = 25·3

3rd non-split extension by C12 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.6Q8, C42.5S3, C4.6Dic6, C6.3(C2xQ8), (C4xC12).3C2, (C2xC4).74D6, C6.2(C4oD4), C4:Dic3.5C2, C2.5(C2xDic6), C3:1(C42.C2), C2.6(C4oD12), Dic3:C4.1C2, (C2xC6).11C23, (C2xC12).72C22, C22.35(C22xS3), (C2xDic3).2C22, SmallGroup(96,77)

Series: Derived Chief Lower central Upper central

C1C2xC6 — C12.6Q8
C1C3C6C2xC6C2xDic3Dic3:C4 — C12.6Q8
C3C2xC6 — C12.6Q8
C1C22C42

Generators and relations for C12.6Q8
 G = < a,b,c | a12=b4=1, c2=a6b2, ab=ba, cac-1=a-1, cbc-1=a6b-1 >

Subgroups: 106 in 56 conjugacy classes, 33 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2xC4, C2xC4, C2xC4, Dic3, C12, C12, C2xC6, C42, C4:C4, C2xDic3, C2xC12, C2xC12, C42.C2, Dic3:C4, C4:Dic3, C4xC12, C12.6Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2xQ8, C4oD4, Dic6, C22xS3, C42.C2, C2xDic6, C4oD12, C12.6Q8

Character table of C12.6Q8

 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
ρ92222-12-2-2-22-20000-1-1-1-111-111-1-11111    orthogonal lifted from D6
ρ102222-12222220000-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ112222-1-2-222-2-20000-1-1-11-111-1111-111-1    orthogonal lifted from D6
ρ122222-1-22-2-2-220000-1-1-111-111-1111-1-11    orthogonal lifted from D6
ρ132-2-2222000-2000002-2-2-200-200220000    symplectic lifted from Q8, Schur index 2
ρ142-2-222-20002000002-2-2200200-2-20000    symplectic lifted from Q8, Schur index 2
ρ152-2-22-1-2000200000-111-1-33-1-3-3113-333    symplectic lifted from Dic6, Schur index 2
ρ162-2-22-1-2000200000-111-13-3-13311-33-3-3    symplectic lifted from Dic6, Schur index 2
ρ172-2-22-12000-200000-1111-3-31-33-1-133-33    symplectic lifted from Dic6, Schur index 2
ρ182-2-22-12000-200000-11113313-3-1-1-3-33-3    symplectic lifted from Dic6, Schur index 2
ρ1922-2-2200-2i2i000000-2-2202i00-2i000-2i002i    complex lifted from C4oD4
ρ202-22-220-2i0002i0000-22-2002i002i000-2i-2i0    complex lifted from C4oD4
ρ212-22-2202i000-2i0000-22-200-2i00-2i0002i2i0    complex lifted from C4oD4
ρ2222-2-22002i-2i000000-2-220-2i002i0002i00-2i    complex lifted from C4oD4
ρ2322-2-2-1002i-2i00000011-1-3i--33-i-33-3-i--3-3i    complex lifted from C4oD12
ρ242-22-2-10-2i0002i00001-113--3-i-3-3-i3-3--3ii-3    complex lifted from C4oD12
ρ252-22-2-10-2i0002i00001-11-3-3-i3--3-i-33-3ii--3    complex lifted from C4oD12
ρ2622-2-2-100-2i2i00000011-1-3-i-33i--33-3i-3--3-i    complex lifted from C4oD12
ρ2722-2-2-1002i-2i00000011-13i-3-3-i--3-33-i-3--3i    complex lifted from C4oD12
ρ282-22-2-102i000-2i00001-113-3i-3--3i3-3-3-i-i--3    complex lifted from C4oD12
ρ2922-2-2-100-2i2i00000011-13-i--3-3i-3-33i--3-3-i    complex lifted from C4oD12
ρ302-22-2-102i000-2i00001-11-3--3i3-3i-33--3-i-i-3    complex lifted from C4oD12

Smallest permutation representation of C12.6Q8
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 85 29 84)(2 86 30 73)(3 87 31 74)(4 88 32 75)(5 89 33 76)(6 90 34 77)(7 91 35 78)(8 92 36 79)(9 93 25 80)(10 94 26 81)(11 95 27 82)(12 96 28 83)(13 40 54 63)(14 41 55 64)(15 42 56 65)(16 43 57 66)(17 44 58 67)(18 45 59 68)(19 46 60 69)(20 47 49 70)(21 48 50 71)(22 37 51 72)(23 38 52 61)(24 39 53 62)
(1 68 35 39)(2 67 36 38)(3 66 25 37)(4 65 26 48)(5 64 27 47)(6 63 28 46)(7 62 29 45)(8 61 30 44)(9 72 31 43)(10 71 32 42)(11 70 33 41)(12 69 34 40)(13 90 60 83)(14 89 49 82)(15 88 50 81)(16 87 51 80)(17 86 52 79)(18 85 53 78)(19 96 54 77)(20 95 55 76)(21 94 56 75)(22 93 57 74)(23 92 58 73)(24 91 59 84)

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

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

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

C12.6Q8 is a maximal subgroup of
C42.2D6  C24.13Q8  C42.264D6  C8:Dic6  C42.14D6  C42.19D6  Dic6.3Q8  D12.3Q8  D4.3Dic6  Q8.5Dic6  C42.213D6  C42.215D6  C42.72D6  C42.76D6  C42.77D6  C42.274D6  C42.277D6  C42.89D6  C42.90D6  C42.94D6  C42.96D6  C42.100D6  D4:5Dic6  C42.105D6  C42.113D6  C42.118D6  C42.119D6  Dic6:10Q8  Q8:6Dic6  D12:10Q8  C42.132D6  C42.134D6  C42.140D6  C42.234D6  C42.145D6  C42.147D6  S3xC42.C2  C42.236D6  C42.157D6  C42.159D6  C42.161D6  C42.168D6  C42.174D6  C42.176D6  C36.6Q8  C62.37C23  C62.39C23  C12.25Dic6  Dic5.7Dic6  C20.Dic6  C60.24Q8
C12.6Q8 is a maximal quotient of
C6.(C4:Q8)  (C2xC4).Dic6  C12:4(C4:C4)  (C2xC42).6S3  C42:11Dic3  C36.6Q8  C62.37C23  C62.39C23  C12.25Dic6  Dic5.7Dic6  C20.Dic6  C60.24Q8

Matrix representation of C12.6Q8 in GL4(F13) generated by

0100
121200
0060
00411
,
3600
71000
00120
00121
,
8000
5500
00810
0005
G:=sub<GL(4,GF(13))| [0,12,0,0,1,12,0,0,0,0,6,4,0,0,0,11],[3,7,0,0,6,10,0,0,0,0,12,12,0,0,0,1],[8,5,0,0,0,5,0,0,0,0,8,0,0,0,10,5] >;

C12.6Q8 in GAP, Magma, Sage, TeX

C_{12}._6Q_8
% in TeX

G:=Group("C12.6Q8");
// GroupNames label

G:=SmallGroup(96,77);
// by ID

G=gap.SmallGroup(96,77);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,217,55,218,86,2309]);
// Polycyclic

G:=Group<a,b,c|a^12=b^4=1,c^2=a^6*b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;
// generators/relations

Export

Character table of C12.6Q8 in TeX

׿
x
:
Z
F
o
wr
Q
<