metabelian, supersoluble, monomial
Aliases: C12.18D6, C32:11SD16, (C3xQ8):3S3, Q8:2(C3:S3), (C3xC6).36D4, C32:4C8:5C2, (Q8xC32):2C2, C12:S3.3C2, C3:3(Q8:2S3), C6.24(C3:D4), (C3xC12).14C22, C2.6(C32:7D4), C4.3(C2xC3:S3), SmallGroup(144,98)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32:11SD16
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c3 >
Subgroups: 234 in 60 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, D4, Q8, C32, C12, C12, D6, SD16, C3:S3, C3xC6, C3:C8, D12, C3xQ8, C3xC12, C3xC12, C2xC3:S3, Q8:2S3, C32:4C8, C12:S3, Q8xC32, C32:11SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3:S3, C3:D4, C2xC3:S3, Q8:2S3, C32:7D4, C32:11SD16
Character table of C32:11SD16
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | |
| size | 1 | 1 | 36 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 18 | 18 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | -2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ8 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | 2 | -2 | -1 | -1 | -1 | 2 | 0 | 0 | 2 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ9 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | -2 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | -1 | -1 | 2 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | -2 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | 1 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | -1 | 2 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | 2 | -1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | -2 | 0 | -1 | -1 | 2 | -1 | 0 | 0 | 1 | √-3 | -√-3 | √-3 | -√-3 | 0 | -√-3 | 0 | √-3 | 1 | 1 | -2 | complex lifted from C3:D4 |
| ρ15 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | -2 | 0 | -1 | 2 | -1 | -1 | 0 | 0 | 1 | -√-3 | √-3 | √-3 | 0 | -√-3 | -√-3 | √-3 | 0 | 1 | -2 | 1 | complex lifted from C3:D4 |
| ρ16 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | -2 | 0 | -1 | -1 | -1 | 2 | 0 | 0 | -2 | 0 | 0 | -√-3 | -√-3 | -√-3 | √-3 | √-3 | √-3 | 1 | 1 | 1 | complex lifted from C3:D4 |
| ρ17 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | -2 | 0 | -1 | -1 | 2 | -1 | 0 | 0 | 1 | -√-3 | √-3 | -√-3 | √-3 | 0 | √-3 | 0 | -√-3 | 1 | 1 | -2 | complex lifted from C3:D4 |
| ρ18 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 1 | √-3 | -√-3 | 0 | √-3 | -√-3 | 0 | √-3 | -√-3 | -2 | 1 | 1 | complex lifted from C3:D4 |
| ρ19 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | -2 | 0 | -1 | 2 | -1 | -1 | 0 | 0 | 1 | √-3 | -√-3 | -√-3 | 0 | √-3 | √-3 | -√-3 | 0 | 1 | -2 | 1 | complex lifted from C3:D4 |
| ρ20 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 1 | -√-3 | √-3 | 0 | -√-3 | √-3 | 0 | -√-3 | √-3 | -2 | 1 | 1 | complex lifted from C3:D4 |
| ρ21 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | -2 | 0 | -1 | -1 | -1 | 2 | 0 | 0 | -2 | 0 | 0 | √-3 | √-3 | √-3 | -√-3 | -√-3 | -√-3 | 1 | 1 | 1 | complex lifted from C3:D4 |
| ρ22 | 2 | -2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ23 | 2 | -2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ24 | 4 | -4 | 0 | -2 | -2 | 4 | -2 | 0 | 0 | 2 | 2 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8:2S3 |
| ρ25 | 4 | -4 | 0 | -2 | -2 | -2 | 4 | 0 | 0 | -4 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8:2S3 |
| ρ26 | 4 | -4 | 0 | -2 | 4 | -2 | -2 | 0 | 0 | 2 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8:2S3 |
| ρ27 | 4 | -4 | 0 | 4 | -2 | -2 | -2 | 0 | 0 | 2 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8:2S3 |
►On 72 points
Generators in S72
(1 56 71)(2 72 49)(3 50 65)(4 66 51)(5 52 67)(6 68 53)(7 54 69)(8 70 55)(9 40 57)(10 58 33)(11 34 59)(12 60 35)(13 36 61)(14 62 37)(15 38 63)(16 64 39)(17 30 48)(18 41 31)(19 32 42)(20 43 25)(21 26 44)(22 45 27)(23 28 46)(24 47 29)
(1 41 15)(2 16 42)(3 43 9)(4 10 44)(5 45 11)(6 12 46)(7 47 13)(8 14 48)(17 70 62)(18 63 71)(19 72 64)(20 57 65)(21 66 58)(22 59 67)(23 68 60)(24 61 69)(25 40 50)(26 51 33)(27 34 52)(28 53 35)(29 36 54)(30 55 37)(31 38 56)(32 49 39)
(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)
(2 4)(3 7)(6 8)(9 47)(10 42)(11 45)(12 48)(13 43)(14 46)(15 41)(16 44)(17 35)(18 38)(19 33)(20 36)(21 39)(22 34)(23 37)(24 40)(25 61)(26 64)(27 59)(28 62)(29 57)(30 60)(31 63)(32 58)(49 66)(50 69)(51 72)(52 67)(53 70)(54 65)(55 68)(56 71)
G:=sub<Sym(72)| (1,56,71)(2,72,49)(3,50,65)(4,66,51)(5,52,67)(6,68,53)(7,54,69)(8,70,55)(9,40,57)(10,58,33)(11,34,59)(12,60,35)(13,36,61)(14,62,37)(15,38,63)(16,64,39)(17,30,48)(18,41,31)(19,32,42)(20,43,25)(21,26,44)(22,45,27)(23,28,46)(24,47,29), (1,41,15)(2,16,42)(3,43,9)(4,10,44)(5,45,11)(6,12,46)(7,47,13)(8,14,48)(17,70,62)(18,63,71)(19,72,64)(20,57,65)(21,66,58)(22,59,67)(23,68,60)(24,61,69)(25,40,50)(26,51,33)(27,34,52)(28,53,35)(29,36,54)(30,55,37)(31,38,56)(32,49,39), (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), (2,4)(3,7)(6,8)(9,47)(10,42)(11,45)(12,48)(13,43)(14,46)(15,41)(16,44)(17,35)(18,38)(19,33)(20,36)(21,39)(22,34)(23,37)(24,40)(25,61)(26,64)(27,59)(28,62)(29,57)(30,60)(31,63)(32,58)(49,66)(50,69)(51,72)(52,67)(53,70)(54,65)(55,68)(56,71)>;
G:=Group( (1,56,71)(2,72,49)(3,50,65)(4,66,51)(5,52,67)(6,68,53)(7,54,69)(8,70,55)(9,40,57)(10,58,33)(11,34,59)(12,60,35)(13,36,61)(14,62,37)(15,38,63)(16,64,39)(17,30,48)(18,41,31)(19,32,42)(20,43,25)(21,26,44)(22,45,27)(23,28,46)(24,47,29), (1,41,15)(2,16,42)(3,43,9)(4,10,44)(5,45,11)(6,12,46)(7,47,13)(8,14,48)(17,70,62)(18,63,71)(19,72,64)(20,57,65)(21,66,58)(22,59,67)(23,68,60)(24,61,69)(25,40,50)(26,51,33)(27,34,52)(28,53,35)(29,36,54)(30,55,37)(31,38,56)(32,49,39), (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), (2,4)(3,7)(6,8)(9,47)(10,42)(11,45)(12,48)(13,43)(14,46)(15,41)(16,44)(17,35)(18,38)(19,33)(20,36)(21,39)(22,34)(23,37)(24,40)(25,61)(26,64)(27,59)(28,62)(29,57)(30,60)(31,63)(32,58)(49,66)(50,69)(51,72)(52,67)(53,70)(54,65)(55,68)(56,71) );
G=PermutationGroup([[(1,56,71),(2,72,49),(3,50,65),(4,66,51),(5,52,67),(6,68,53),(7,54,69),(8,70,55),(9,40,57),(10,58,33),(11,34,59),(12,60,35),(13,36,61),(14,62,37),(15,38,63),(16,64,39),(17,30,48),(18,41,31),(19,32,42),(20,43,25),(21,26,44),(22,45,27),(23,28,46),(24,47,29)], [(1,41,15),(2,16,42),(3,43,9),(4,10,44),(5,45,11),(6,12,46),(7,47,13),(8,14,48),(17,70,62),(18,63,71),(19,72,64),(20,57,65),(21,66,58),(22,59,67),(23,68,60),(24,61,69),(25,40,50),(26,51,33),(27,34,52),(28,53,35),(29,36,54),(30,55,37),(31,38,56),(32,49,39)], [(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)], [(2,4),(3,7),(6,8),(9,47),(10,42),(11,45),(12,48),(13,43),(14,46),(15,41),(16,44),(17,35),(18,38),(19,33),(20,36),(21,39),(22,34),(23,37),(24,40),(25,61),(26,64),(27,59),(28,62),(29,57),(30,60),(31,63),(32,58),(49,66),(50,69),(51,72),(52,67),(53,70),(54,65),(55,68),(56,71)]])
C32:11SD16 is a maximal subgroup of
S3xQ8:2S3 D12:6D6 Dic6.22D6 D12.13D6 SD16xC3:S3 C24:7D6 C24.35D6 C24.28D6 C62.134D4 C62.73D4 C62.74D4 He3:10SD16 C36.20D6 C32.3GL2(F3) C33:13SD16 C33:15SD16 C33:27SD16 C32:5GL2(F3)
C32:11SD16 is a maximal quotient of
C12.10Dic6 C62.113D4 C62.117D4 C36.20D6 He3:11SD16 C33:13SD16 C33:15SD16 C33:27SD16
Matrix representation of C32:11SD16 ►in GL6(F73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 13 | 0 | 0 |
| 0 | 0 | 43 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 | 55 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 48 | 72 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,43,43,0,0,0,0,13,30,0,0,0,0,0,0,61,4,0,0,0,0,55,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,48,0,0,0,0,0,72] >;
C32:11SD16 in GAP, Magma, Sage, TeX
C_3^2\rtimes_{11}{\rm SD}_{16} % in TeX
G:=Group("C3^2:11SD16"); // GroupNames label
G:=SmallGroup(144,98);
// by ID
G=gap.SmallGroup(144,98);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,55,218,116,50,964,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^3>;
// generators/relations
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