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G = C4.Dic6order 96 = 25·3

3rd non-split extension by C4 of Dic6 acting via Dic6/Dic3=C2

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 — C4.Dic6
C1C3C6C2×C6C2×Dic3C4×Dic3 — C4.Dic6
C3C2×C6 — C4.Dic6
C1C22C4⋊C4

Generators and relations for C4.Dic6
 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, C4.Dic6
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, C4.Dic6

Character table of C4.Dic6

 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 C4.Dic6
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 95 44 63)(2 90 45 70)(3 85 46 65)(4 92 47 72)(5 87 48 67)(6 94 37 62)(7 89 38 69)(8 96 39 64)(9 91 40 71)(10 86 41 66)(11 93 42 61)(12 88 43 68)(13 27 84 60)(14 34 73 55)(15 29 74 50)(16 36 75 57)(17 31 76 52)(18 26 77 59)(19 33 78 54)(20 28 79 49)(21 35 80 56)(22 30 81 51)(23 25 82 58)(24 32 83 53)
(1 81 44 22)(2 74 45 15)(3 79 46 20)(4 84 47 13)(5 77 48 18)(6 82 37 23)(7 75 38 16)(8 80 39 21)(9 73 40 14)(10 78 41 19)(11 83 42 24)(12 76 43 17)(25 68 58 88)(26 61 59 93)(27 66 60 86)(28 71 49 91)(29 64 50 96)(30 69 51 89)(31 62 52 94)(32 67 53 87)(33 72 54 92)(34 65 55 85)(35 70 56 90)(36 63 57 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), (1,95,44,63)(2,90,45,70)(3,85,46,65)(4,92,47,72)(5,87,48,67)(6,94,37,62)(7,89,38,69)(8,96,39,64)(9,91,40,71)(10,86,41,66)(11,93,42,61)(12,88,43,68)(13,27,84,60)(14,34,73,55)(15,29,74,50)(16,36,75,57)(17,31,76,52)(18,26,77,59)(19,33,78,54)(20,28,79,49)(21,35,80,56)(22,30,81,51)(23,25,82,58)(24,32,83,53), (1,81,44,22)(2,74,45,15)(3,79,46,20)(4,84,47,13)(5,77,48,18)(6,82,37,23)(7,75,38,16)(8,80,39,21)(9,73,40,14)(10,78,41,19)(11,83,42,24)(12,76,43,17)(25,68,58,88)(26,61,59,93)(27,66,60,86)(28,71,49,91)(29,64,50,96)(30,69,51,89)(31,62,52,94)(32,67,53,87)(33,72,54,92)(34,65,55,85)(35,70,56,90)(36,63,57,95)>;

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

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

C4.Dic6 is a maximal subgroup of
D4.Dic6  C4⋊C4.D6  D4.2Dic6  (C2×C8).200D6  Q8.3Dic6  (C2×Q8).36D6  Q8.4Dic6  Q8⋊C4⋊S3  C243Q8  Dic6.Q8  C8.8Dic6  D12.Q8  C244Q8  Dic6.2Q8  C8.6Dic6  D12.2Q8  C6.72+ 1+4  C6.52- 1+4  C6.112+ 1+4  C42.88D6  C42.90D6  C42.94D6  C42.95D6  D45Dic6  D46Dic6  C42.229D6  C42.116D6  Q86Dic6  Q87Dic6  C42.131D6  C42.134D6  C4⋊C4.178D6  C6.432+ 1+4  C6.1152+ 1+4  C6.472+ 1+4  C4⋊C4.187D6  C6.152- 1+4  C6.212- 1+4  C6.772- 1+4  C6.782- 1+4  C6.252- 1+4  C6.802- 1+4  C6.632+ 1+4  C6.852- 1+4  C6.692+ 1+4  Dic67Q8  C42.147D6  S3×C42.C2  C42.148D6  D127Q8  C42.152D6  C42.154D6  C42.156D6  C42.159D6  C42.161D6  C42.162D6  C42.165D6  Dic68Q8  C42.241D6  C42.174D6  D129Q8  C36.3Q8  Dic3.Dic6  C62.16C23  C62.39C23  C62.42C23  C62.234C23  Dic5.1Dic6  Dic5.2Dic6  C60.6Q8  C12.Dic10  C4.Dic30
C4.Dic6 is a maximal quotient of
C2.(C4×Dic6)  Dic3⋊C4⋊C4  (C2×C4).Dic6  (C22×C4).85D6  C12⋊(C4⋊C4)  (C4×Dic3)⋊9C4  (C2×C12).54D4  C4⋊C46Dic3  (C2×C12).55D4  C36.3Q8  Dic3.Dic6  C62.16C23  C62.39C23  C62.42C23  C62.234C23  Dic5.1Dic6  Dic5.2Dic6  C60.6Q8  C12.Dic10  C4.Dic30

Matrix representation of C4.Dic6 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] >;

C4.Dic6 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_6
% in TeX

G:=Group("C4.Dic6");
// 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

Export

Character table of C4.Dic6 in TeX

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