metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.51D12, (C2×C8).187D6, C4.11(D6⋊C4), (C2×Dic6)⋊14C4, (C2×C4).148D12, C12.415(C2×D4), (C2×C12).166D4, (C22×C6).99D4, C2.Dic12⋊39C2, Dic6.25(C2×C4), C2.3(C8.D6), (C22×C4).148D6, C22.55(C2×D12), C12.24(C22⋊C4), (C2×C24).317C22, C12.114(C22×C4), (C2×C12).771C23, C22.26(D6⋊C4), C6.19(C8.C22), C3⋊3(C23.38D4), (C6×M4(2)).25C2, (C2×M4(2)).14S3, C4⋊Dic3.283C22, (C22×Dic6).14C2, (C22×C12).179C22, (C2×Dic6).220C22, C23.26D6.16C2, C4.72(S3×C2×C4), (C2×C4).48(C4×S3), C2.26(C2×D6⋊C4), (C2×C12).99(C2×C4), (C2×C6).161(C2×D4), C4.108(C2×C3⋊D4), C6.54(C2×C22⋊C4), (C2×C4).75(C3⋊D4), (C2×C6).18(C22⋊C4), (C2×C4).719(C22×S3), SmallGroup(192,679)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.51D12
G = < a,b,c,d,e | a2=b2=c2=1, d12=e2=c, ab=ba, dad-1=ac=ca, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd11 >
Subgroups: 376 in 150 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C24, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, Q8⋊C4, C42⋊C2, C2×M4(2), C22×Q8, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C23.38D4, C2.Dic12, C23.26D6, C6×M4(2), C22×Dic6, C23.51D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C8.C22, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C23.38D4, C8.D6, C2×D6⋊C4, C23.51D12
►On 96 points
Generators in S96
(1 90)(2 79)(3 92)(4 81)(5 94)(6 83)(7 96)(8 85)(9 74)(10 87)(11 76)(12 89)(13 78)(14 91)(15 80)(16 93)(17 82)(18 95)(19 84)(20 73)(21 86)(22 75)(23 88)(24 77)(25 57)(26 70)(27 59)(28 72)(29 61)(30 50)(31 63)(32 52)(33 65)(34 54)(35 67)(36 56)(37 69)(38 58)(39 71)(40 60)(41 49)(42 62)(43 51)(44 64)(45 53)(46 66)(47 55)(48 68)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 81)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 69)(26 70)(27 71)(28 72)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 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 42 13 30)(2 49 14 61)(3 40 15 28)(4 71 16 59)(5 38 17 26)(6 69 18 57)(7 36 19 48)(8 67 20 55)(9 34 21 46)(10 65 22 53)(11 32 23 44)(12 63 24 51)(25 83 37 95)(27 81 39 93)(29 79 41 91)(31 77 43 89)(33 75 45 87)(35 73 47 85)(50 90 62 78)(52 88 64 76)(54 86 66 74)(56 84 68 96)(58 82 70 94)(60 80 72 92)
G:=sub<Sym(96)| (1,90)(2,79)(3,92)(4,81)(5,94)(6,83)(7,96)(8,85)(9,74)(10,87)(11,76)(12,89)(13,78)(14,91)(15,80)(16,93)(17,82)(18,95)(19,84)(20,73)(21,86)(22,75)(23,88)(24,77)(25,57)(26,70)(27,59)(28,72)(29,61)(30,50)(31,63)(32,52)(33,65)(34,54)(35,67)(36,56)(37,69)(38,58)(39,71)(40,60)(41,49)(42,62)(43,51)(44,64)(45,53)(46,66)(47,55)(48,68), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,69)(26,70)(27,71)(28,72)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,42,13,30)(2,49,14,61)(3,40,15,28)(4,71,16,59)(5,38,17,26)(6,69,18,57)(7,36,19,48)(8,67,20,55)(9,34,21,46)(10,65,22,53)(11,32,23,44)(12,63,24,51)(25,83,37,95)(27,81,39,93)(29,79,41,91)(31,77,43,89)(33,75,45,87)(35,73,47,85)(50,90,62,78)(52,88,64,76)(54,86,66,74)(56,84,68,96)(58,82,70,94)(60,80,72,92)>;
G:=Group( (1,90)(2,79)(3,92)(4,81)(5,94)(6,83)(7,96)(8,85)(9,74)(10,87)(11,76)(12,89)(13,78)(14,91)(15,80)(16,93)(17,82)(18,95)(19,84)(20,73)(21,86)(22,75)(23,88)(24,77)(25,57)(26,70)(27,59)(28,72)(29,61)(30,50)(31,63)(32,52)(33,65)(34,54)(35,67)(36,56)(37,69)(38,58)(39,71)(40,60)(41,49)(42,62)(43,51)(44,64)(45,53)(46,66)(47,55)(48,68), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,69)(26,70)(27,71)(28,72)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,42,13,30)(2,49,14,61)(3,40,15,28)(4,71,16,59)(5,38,17,26)(6,69,18,57)(7,36,19,48)(8,67,20,55)(9,34,21,46)(10,65,22,53)(11,32,23,44)(12,63,24,51)(25,83,37,95)(27,81,39,93)(29,79,41,91)(31,77,43,89)(33,75,45,87)(35,73,47,85)(50,90,62,78)(52,88,64,76)(54,86,66,74)(56,84,68,96)(58,82,70,94)(60,80,72,92) );
G=PermutationGroup([[(1,90),(2,79),(3,92),(4,81),(5,94),(6,83),(7,96),(8,85),(9,74),(10,87),(11,76),(12,89),(13,78),(14,91),(15,80),(16,93),(17,82),(18,95),(19,84),(20,73),(21,86),(22,75),(23,88),(24,77),(25,57),(26,70),(27,59),(28,72),(29,61),(30,50),(31,63),(32,52),(33,65),(34,54),(35,67),(36,56),(37,69),(38,58),(39,71),(40,60),(41,49),(42,62),(43,51),(44,64),(45,53),(46,66),(47,55),(48,68)], [(1,90),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,73),(9,74),(10,75),(11,76),(12,77),(13,78),(14,79),(15,80),(16,81),(17,82),(18,83),(19,84),(20,85),(21,86),(22,87),(23,88),(24,89),(25,69),(26,70),(27,71),(28,72),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(36,56),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(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)], [(1,42,13,30),(2,49,14,61),(3,40,15,28),(4,71,16,59),(5,38,17,26),(6,69,18,57),(7,36,19,48),(8,67,20,55),(9,34,21,46),(10,65,22,53),(11,32,23,44),(12,63,24,51),(25,83,37,95),(27,81,39,93),(29,79,41,91),(31,77,43,89),(33,75,45,87),(35,73,47,85),(50,90,62,78),(52,88,64,76),(54,86,66,74),(56,84,68,96),(58,82,70,94),(60,80,72,92)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | D12 | C8.C22 | C8.D6 |
| kernel | C23.51D12 | C2.Dic12 | C23.26D6 | C6×M4(2) | C22×Dic6 | C2×Dic6 | C2×M4(2) | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 3 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 2 | 4 |
Matrix representation of C23.51D12 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 48 | 48 | 1 | 0 |
| 0 | 0 | 48 | 48 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 46 | 46 | 0 | 0 | 0 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 2 | 1 |
| 0 | 0 | 72 | 72 | 1 | 2 |
| 0 | 0 | 41 | 48 | 1 | 1 |
| 0 | 0 | 55 | 41 | 1 | 1 |
| 68 | 18 | 0 | 0 | 0 | 0 |
| 23 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 30 | 0 | 0 |
| 0 | 0 | 60 | 43 | 0 | 0 |
| 0 | 0 | 12 | 55 | 27 | 27 |
| 0 | 0 | 39 | 9 | 0 | 46 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,48,48,0,0,0,72,48,48,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[46,27,0,0,0,0,46,0,0,0,0,0,0,0,72,72,41,55,0,0,72,72,48,41,0,0,2,1,1,1,0,0,1,2,1,1],[68,23,0,0,0,0,18,5,0,0,0,0,0,0,30,60,12,39,0,0,30,43,55,9,0,0,0,0,27,0,0,0,0,0,27,46] >;
C23.51D12 in GAP, Magma, Sage, TeX
C_2^3._{51}D_{12} % in TeX
G:=Group("C2^3.51D12"); // GroupNames label
G:=SmallGroup(192,679);
// by ID
G=gap.SmallGroup(192,679);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,254,387,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=e^2=c,a*b=b*a,d*a*d^-1=a*c=c*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^11>;
// generators/relations