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G = C12.3Q8order 96 = 25·3

3rd non-split extension by C12 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.3Q8, C4.3Dic6, C4⋊C4.6S3, C6.6(C2×Q8), (C2×C4).43D6, C4⋊Dic3.7C2, C2.8(C2×Dic6), C33(C42.C2), C6.25(C4○D4), Dic3⋊C4.3C2, (C2×C6).31C23, (C4×Dic3).2C2, (C2×C12).22C22, C2.4(Q83S3), C2.12(D42S3), C22.48(C22×S3), (C2×Dic3).10C22, (C3×C4⋊C4).7C2, SmallGroup(96,97)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C12.3Q8
C1C3C6C2×C6C2×Dic3C4×Dic3 — C12.3Q8
C3C2×C6 — C12.3Q8
C1C22C4⋊C4

Generators and relations for C12.3Q8
 G = < a,b,c | a12=b4=1, c2=b2, bab-1=a7, cac-1=a5, cbc-1=a6b-1 >

Subgroups: 106 in 56 conjugacy classes, 33 normal (19 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×6], C22, C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C42, C4⋊C4, C4⋊C4 [×5], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C42.C2, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C4⋊Dic3 [×2], C3×C4⋊C4, C12.3Q8
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, D42S3, Q83S3, C12.3Q8

Character table of C12.3Q8

 class 12A2B2C34A4B4C4D4E4F4G4H4I4J6A6B6C12A12B12C12D12E12F
 size 11112224466661212222444444
ρ1111111111111111111111111    trivial
ρ211111-1-11-11-11-11-1111-11-1-1-11    linear of order 2
ρ311111-1-1-111-11-1-11111-1-11-11-1    linear of order 2
ρ41111111-1-11111-1-11111-1-11-1-1    linear of order 2
ρ511111-1-11-1-11-11-11111-11-1-1-11    linear of order 2
ρ6111111111-1-1-1-1-1-1111111111    linear of order 2
ρ71111111-1-1-1-1-1-1111111-1-11-1-1    linear of order 2
ρ811111-1-1-11-11-111-1111-1-11-11-1    linear of order 2
ρ92222-1-2-2-22000000-1-1-111-11-11    orthogonal lifted from D6
ρ102222-12222000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ112222-122-2-2000000-1-1-1-111-111    orthogonal lifted from D6
ρ122222-1-2-22-2000000-1-1-11-1111-1    orthogonal lifted from D6
ρ1322-2-22-2200000000-22-2200-200    symplectic lifted from Q8, Schur index 2
ρ1422-2-222-200000000-22-2-200200    symplectic lifted from Q8, Schur index 2
ρ1522-2-2-12-2000000001-1113-3-13-3    symplectic lifted from Dic6, Schur index 2
ρ1622-2-2-12-2000000001-111-33-1-33    symplectic lifted from Dic6, Schur index 2
ρ1722-2-2-1-22000000001-11-1331-3-3    symplectic lifted from Dic6, Schur index 2
ρ1822-2-2-1-22000000001-11-1-3-3133    symplectic lifted from Dic6, Schur index 2
ρ192-2-22200000-2i02i00-2-22000000    complex lifted from C4○D4
ρ202-22-220000-2i02i0002-2-2000000    complex lifted from C4○D4
ρ212-22-2200002i0-2i0002-2-2000000    complex lifted from C4○D4
ρ222-2-222000002i0-2i00-2-22000000    complex lifted from C4○D4
ρ234-4-44-2000000000022-2000000    orthogonal lifted from Q83S3, Schur index 2
ρ244-44-4-20000000000-222000000    symplectic lifted from D42S3, Schur index 2

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

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

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

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

Matrix representation of C12.3Q8 in GL4(𝔽13) generated by

4000
31000
0083
0005
,
8000
5500
001211
0011
,
121100
1100
0050
0005
G:=sub<GL(4,GF(13))| [4,3,0,0,0,10,0,0,0,0,8,0,0,0,3,5],[8,5,0,0,0,5,0,0,0,0,12,1,0,0,11,1],[12,1,0,0,11,1,0,0,0,0,5,0,0,0,0,5] >;

C12.3Q8 in GAP, Magma, Sage, TeX

C_{12}._3Q_8
% in TeX

G:=Group("C12.3Q8");
// GroupNames label

G:=SmallGroup(96,97);
// by ID

G=gap.SmallGroup(96,97);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,103,506,188,50,2309]);
// Polycyclic

G:=Group<a,b,c|a^12=b^4=1,c^2=b^2,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^6*b^-1>;
// generators/relations

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