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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(C2×Q8), (C4×C12).3C2, (C2×C4).74D6, C6.2(C4○D4), C4⋊Dic3.5C2, C2.5(C2×Dic6), C31(C42.C2), C2.6(C4○D12), Dic3⋊C4.1C2, (C2×C6).11C23, (C2×C12).72C22, C22.35(C22×S3), (C2×Dic3).2C22, SmallGroup(96,77)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C12.6Q8
C1C3C6C2×C6C2×Dic3Dic3⋊C4 — C12.6Q8
C3C2×C6 — 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 [×2], C3, C4 [×2], C4 [×6], C22, C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×4], Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C42, C4⋊C4 [×6], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C42.C2, Dic3⋊C4 [×4], C4⋊Dic3 [×2], C4×C12, C12.6Q8
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D6 [×3], C2×Q8, C4○D4 [×2], Dic6 [×2], C22×S3, C42.C2, C2×Dic6, C4○D12 [×2], 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 C4○D4
ρ202-22-220-2i0002i0000-22-2002i002i000-2i-2i0    complex lifted from C4○D4
ρ212-22-2202i000-2i0000-22-200-2i00-2i0002i2i0    complex lifted from C4○D4
ρ2222-2-22002i-2i000000-2-220-2i002i0002i00-2i    complex lifted from C4○D4
ρ2322-2-2-1002i-2i00000011-1-3i--33-i-33-3-i--3-3i    complex lifted from C4○D12
ρ242-22-2-10-2i0002i00001-113--3-i-3-3-i3-3--3ii-3    complex lifted from C4○D12
ρ252-22-2-10-2i0002i00001-11-3-3-i3--3-i-33-3ii--3    complex lifted from C4○D12
ρ2622-2-2-100-2i2i00000011-1-3-i-33i--33-3i-3--3-i    complex lifted from C4○D12
ρ2722-2-2-1002i-2i00000011-13i-3-3-i--3-33-i-3--3i    complex lifted from C4○D12
ρ282-22-2-102i000-2i00001-113-3i-3--3i3-3-3-i-i--3    complex lifted from C4○D12
ρ2922-2-2-100-2i2i00000011-13-i--3-3i-3-33i--3-3-i    complex lifted from C4○D12
ρ302-22-2-102i000-2i00001-11-3--3i3-3i-33--3-i-i-3    complex lifted from C4○D12

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

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

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

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

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  D45Dic6  C42.105D6  C42.113D6  C42.118D6  C42.119D6  Dic610Q8  Q86Dic6  D1210Q8  C42.132D6  C42.134D6  C42.140D6  C42.234D6  C42.145D6  C42.147D6  S3×C42.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)  (C2×C4).Dic6  C124(C4⋊C4)  (C2×C42).6S3  C4211Dic3  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

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