metabelian, supersoluble, monomial
Aliases: C12.26D6, (C3×Q8)⋊5S3, Q8⋊3(C3⋊S3), C12⋊S3⋊7C2, (Q8×C32)⋊6C2, C3⋊3(Q8⋊3S3), C32⋊12(C4○D4), C6.37(C22×S3), (C3×C6).36C23, (C3×C12).26C22, C3⋊Dic3.20C22, (C4×C3⋊S3)⋊5C2, C4.7(C2×C3⋊S3), C2.9(C22×C3⋊S3), (C2×C3⋊S3).19C22, SmallGroup(144,175)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.26D6
G = < a,b,c | a12=1, b6=c2=a6, bab-1=a7, cac-1=a5, cbc-1=b5 >
Subgroups: 410 in 120 conjugacy classes, 47 normal (8 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, D4, Q8, C32, Dic3, C12, D6, C4○D4, C3⋊S3, C3×C6, C4×S3, D12, C3×Q8, C3⋊Dic3, C3×C12, C2×C3⋊S3, Q8⋊3S3, C4×C3⋊S3, C12⋊S3, Q8×C32, C12.26D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, C2×C3⋊S3, Q8⋊3S3, C22×C3⋊S3, C12.26D6
Character table of C12.26D6
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | |
| size | 1 | 1 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 9 | 9 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | -1 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 1 | -2 | -1 | 1 | 1 | 1 | -2 | -1 | 2 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 1 | 1 | -1 | -2 | 1 | 1 | 1 | -1 | -1 | 2 | -2 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 1 | 1 | -1 | -1 | 2 | -1 | -2 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | -2 | 1 | -2 | 1 | 1 | 1 | 1 | 1 | -1 | 2 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ17 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 1 | -1 | 1 | 1 | -2 | 1 | 2 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | 2 | 1 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ19 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | 1 | 1 | -2 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | 1 | -2 | orthogonal lifted from D6 |
| ρ20 | 2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | -2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | 2 | -1 | -1 | -1 | 1 | 1 | -2 | -2 | 1 | orthogonal lifted from D6 |
| ρ21 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ22 | 2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | -1 | 1 | -2 | 1 | 1 | 1 | 1 | 1 | -2 | 2 | -1 | orthogonal lifted from D6 |
| ρ23 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 1 | -2 | 1 | -1 | -1 | -1 | 2 | 1 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ24 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 1 | 1 | 2 | 1 | -2 | 1 | 1 | -1 | -1 | -1 | 1 | -2 | orthogonal lifted from D6 |
| ρ25 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2i | 2i | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2i | -2i | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | -2 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 4 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
| ρ29 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
| ρ30 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
►On 72 points
Generators in S72
(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)
(1 30 52 44 13 62 7 36 58 38 19 68)(2 25 53 39 14 69 8 31 59 45 20 63)(3 32 54 46 15 64 9 26 60 40 21 70)(4 27 55 41 16 71 10 33 49 47 22 65)(5 34 56 48 17 66 11 28 50 42 23 72)(6 29 57 43 18 61 12 35 51 37 24 67)
(1 62 7 68)(2 67 8 61)(3 72 9 66)(4 65 10 71)(5 70 11 64)(6 63 12 69)(13 30 19 36)(14 35 20 29)(15 28 21 34)(16 33 22 27)(17 26 23 32)(18 31 24 25)(37 59 43 53)(38 52 44 58)(39 57 45 51)(40 50 46 56)(41 55 47 49)(42 60 48 54)
G:=sub<Sym(72)| (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), (1,30,52,44,13,62,7,36,58,38,19,68)(2,25,53,39,14,69,8,31,59,45,20,63)(3,32,54,46,15,64,9,26,60,40,21,70)(4,27,55,41,16,71,10,33,49,47,22,65)(5,34,56,48,17,66,11,28,50,42,23,72)(6,29,57,43,18,61,12,35,51,37,24,67), (1,62,7,68)(2,67,8,61)(3,72,9,66)(4,65,10,71)(5,70,11,64)(6,63,12,69)(13,30,19,36)(14,35,20,29)(15,28,21,34)(16,33,22,27)(17,26,23,32)(18,31,24,25)(37,59,43,53)(38,52,44,58)(39,57,45,51)(40,50,46,56)(41,55,47,49)(42,60,48,54)>;
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), (1,30,52,44,13,62,7,36,58,38,19,68)(2,25,53,39,14,69,8,31,59,45,20,63)(3,32,54,46,15,64,9,26,60,40,21,70)(4,27,55,41,16,71,10,33,49,47,22,65)(5,34,56,48,17,66,11,28,50,42,23,72)(6,29,57,43,18,61,12,35,51,37,24,67), (1,62,7,68)(2,67,8,61)(3,72,9,66)(4,65,10,71)(5,70,11,64)(6,63,12,69)(13,30,19,36)(14,35,20,29)(15,28,21,34)(16,33,22,27)(17,26,23,32)(18,31,24,25)(37,59,43,53)(38,52,44,58)(39,57,45,51)(40,50,46,56)(41,55,47,49)(42,60,48,54) );
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)], [(1,30,52,44,13,62,7,36,58,38,19,68),(2,25,53,39,14,69,8,31,59,45,20,63),(3,32,54,46,15,64,9,26,60,40,21,70),(4,27,55,41,16,71,10,33,49,47,22,65),(5,34,56,48,17,66,11,28,50,42,23,72),(6,29,57,43,18,61,12,35,51,37,24,67)], [(1,62,7,68),(2,67,8,61),(3,72,9,66),(4,65,10,71),(5,70,11,64),(6,63,12,69),(13,30,19,36),(14,35,20,29),(15,28,21,34),(16,33,22,27),(17,26,23,32),(18,31,24,25),(37,59,43,53),(38,52,44,58),(39,57,45,51),(40,50,46,56),(41,55,47,49),(42,60,48,54)]])
C12.26D6 is a maximal subgroup of
C32⋊7C4≀C2 D12.10D6 Dic6.10D6 D12.14D6 C24⋊7D6 C24.40D6 C24.35D6 C24.28D6 C12⋊S3.C4 Dic6.26D6 S3×Q8⋊3S3 D12⋊16D6 C32⋊72- 1+4 C4○D4×C3⋊S3 C62.154C23 C6.(S3×A4) (Q8×He3)⋊C2 C36.29D6 C3⋊Dic3.2A4 C12.39S32 C12.40S32 (Q8×C33)⋊C2
C12.26D6 is a maximal quotient of
C62.234C23 C62.236C23 C62.237C23 C62.238C23 C12⋊3D12 C62.242C23 Q8×C3⋊Dic3 C62.261C23 C62.262C23 C36.29D6 He3⋊5D4⋊C2 C12.39S32 C12.40S32 (Q8×C33)⋊C2
Matrix representation of C12.26D6 ►in GL6(𝔽13)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;
C12.26D6 in GAP, Magma, Sage, TeX
C_{12}._{26}D_6 % in TeX
G:=Group("C12.26D6"); // GroupNames label
G:=SmallGroup(144,175);
// by ID
G=gap.SmallGroup(144,175);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,116,50,964,3461]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^6=c^2=a^6,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=b^5>;
// generators/relations
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