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 [×3], C3 [×4], C4 [×3], C4, C22 [×3], S3 [×12], C6 [×4], C2×C4 [×3], D4 [×3], Q8, C32, Dic3 [×4], C12 [×12], D6 [×12], C4○D4, C3⋊S3 [×3], C3×C6, C4×S3 [×12], D12 [×12], C3×Q8 [×4], C3⋊Dic3, C3×C12 [×3], C2×C3⋊S3 [×3], Q8⋊3S3 [×4], C4×C3⋊S3 [×3], C12⋊S3 [×3], Q8×C32, C12.26D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], C23, D6 [×12], C4○D4, C3⋊S3, C22×S3 [×4], C2×C3⋊S3 [×3], Q8⋊3S3 [×4], 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
(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 13 37 62 57 29 7 19 43 68 51 35)(2 20 38 69 58 36 8 14 44 63 52 30)(3 15 39 64 59 31 9 21 45 70 53 25)(4 22 40 71 60 26 10 16 46 65 54 32)(5 17 41 66 49 33 11 23 47 72 55 27)(6 24 42 61 50 28 12 18 48 67 56 34)
(1 29 7 35)(2 34 8 28)(3 27 9 33)(4 32 10 26)(5 25 11 31)(6 30 12 36)(13 51 19 57)(14 56 20 50)(15 49 21 55)(16 54 22 60)(17 59 23 53)(18 52 24 58)(37 62 43 68)(38 67 44 61)(39 72 45 66)(40 65 46 71)(41 70 47 64)(42 63 48 69)
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), (1,13,37,62,57,29,7,19,43,68,51,35)(2,20,38,69,58,36,8,14,44,63,52,30)(3,15,39,64,59,31,9,21,45,70,53,25)(4,22,40,71,60,26,10,16,46,65,54,32)(5,17,41,66,49,33,11,23,47,72,55,27)(6,24,42,61,50,28,12,18,48,67,56,34), (1,29,7,35)(2,34,8,28)(3,27,9,33)(4,32,10,26)(5,25,11,31)(6,30,12,36)(13,51,19,57)(14,56,20,50)(15,49,21,55)(16,54,22,60)(17,59,23,53)(18,52,24,58)(37,62,43,68)(38,67,44,61)(39,72,45,66)(40,65,46,71)(41,70,47,64)(42,63,48,69)>;
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), (1,13,37,62,57,29,7,19,43,68,51,35)(2,20,38,69,58,36,8,14,44,63,52,30)(3,15,39,64,59,31,9,21,45,70,53,25)(4,22,40,71,60,26,10,16,46,65,54,32)(5,17,41,66,49,33,11,23,47,72,55,27)(6,24,42,61,50,28,12,18,48,67,56,34), (1,29,7,35)(2,34,8,28)(3,27,9,33)(4,32,10,26)(5,25,11,31)(6,30,12,36)(13,51,19,57)(14,56,20,50)(15,49,21,55)(16,54,22,60)(17,59,23,53)(18,52,24,58)(37,62,43,68)(38,67,44,61)(39,72,45,66)(40,65,46,71)(41,70,47,64)(42,63,48,69) );
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)], [(1,13,37,62,57,29,7,19,43,68,51,35),(2,20,38,69,58,36,8,14,44,63,52,30),(3,15,39,64,59,31,9,21,45,70,53,25),(4,22,40,71,60,26,10,16,46,65,54,32),(5,17,41,66,49,33,11,23,47,72,55,27),(6,24,42,61,50,28,12,18,48,67,56,34)], [(1,29,7,35),(2,34,8,28),(3,27,9,33),(4,32,10,26),(5,25,11,31),(6,30,12,36),(13,51,19,57),(14,56,20,50),(15,49,21,55),(16,54,22,60),(17,59,23,53),(18,52,24,58),(37,62,43,68),(38,67,44,61),(39,72,45,66),(40,65,46,71),(41,70,47,64),(42,63,48,69)])
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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