metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.18D12, C22⋊C8⋊7S3, C24⋊1C4⋊5C2, C8⋊Dic3⋊9C2, (C2×C8).110D6, C6.10(C4○D8), (C2×C12).242D4, (C2×C4).120D12, C2.D24⋊11C2, C12⋊7D4.8C2, (C22×C6).57D4, C2.12(C4○D24), C2.15(C8⋊D6), C6.12(C8⋊C22), (C22×C4).103D6, C12.284(C4○D4), (C2×C24).121C22, (C2×C12).747C23, (C2×D12).12C22, C22.110(C2×D12), C23.26D6⋊1C2, C3⋊1(C23.19D4), C4⋊Dic3.14C22, C4.108(D4⋊2S3), (C22×C12).98C22, C6.19(C22.D4), C2.15(C23.21D6), (C3×C22⋊C8)⋊9C2, (C2×C6).130(C2×D4), (C2×C4).692(C22×S3), SmallGroup(192,296)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.18D12
G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=cb=bc, ab=ba, eae-1=ac=ca, dad-1=abc, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd11 >
Subgroups: 352 in 106 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C24, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C4×Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C2×C24, C2×D12, C2×C3⋊D4, C22×C12, C23.19D4, C8⋊Dic3, C24⋊1C4, C2.D24, C3×C22⋊C8, C23.26D6, C12⋊7D4, C23.18D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C22.D4, C4○D8, C8⋊C22, C2×D12, D4⋊2S3, C23.19D4, C23.21D6, C4○D24, C8⋊D6, C23.18D12
►On 96 points
Generators in S96
(1 41)(2 90)(3 43)(4 92)(5 45)(6 94)(7 47)(8 96)(9 25)(10 74)(11 27)(12 76)(13 29)(14 78)(15 31)(16 80)(17 33)(18 82)(19 35)(20 84)(21 37)(22 86)(23 39)(24 88)(26 58)(28 60)(30 62)(32 64)(34 66)(36 68)(38 70)(40 72)(42 50)(44 52)(46 54)(48 56)(49 89)(51 91)(53 93)(55 95)(57 73)(59 75)(61 77)(63 79)(65 81)(67 83)(69 85)(71 87)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)
(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 72 49 24)(2 23 50 71)(3 70 51 22)(4 21 52 69)(5 68 53 20)(6 19 54 67)(7 66 55 18)(8 17 56 65)(9 64 57 16)(10 15 58 63)(11 62 59 14)(12 13 60 61)(25 44 73 92)(26 91 74 43)(27 42 75 90)(28 89 76 41)(29 40 77 88)(30 87 78 39)(31 38 79 86)(32 85 80 37)(33 36 81 84)(34 83 82 35)(45 48 93 96)(46 95 94 47)
G:=sub<Sym(96)| (1,41)(2,90)(3,43)(4,92)(5,45)(6,94)(7,47)(8,96)(9,25)(10,74)(11,27)(12,76)(13,29)(14,78)(15,31)(16,80)(17,33)(18,82)(19,35)(20,84)(21,37)(22,86)(23,39)(24,88)(26,58)(28,60)(30,62)(32,64)(34,66)(36,68)(38,70)(40,72)(42,50)(44,52)(46,54)(48,56)(49,89)(51,91)(53,93)(55,95)(57,73)(59,75)(61,77)(63,79)(65,81)(67,83)(69,85)(71,87), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84), (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), (1,72,49,24)(2,23,50,71)(3,70,51,22)(4,21,52,69)(5,68,53,20)(6,19,54,67)(7,66,55,18)(8,17,56,65)(9,64,57,16)(10,15,58,63)(11,62,59,14)(12,13,60,61)(25,44,73,92)(26,91,74,43)(27,42,75,90)(28,89,76,41)(29,40,77,88)(30,87,78,39)(31,38,79,86)(32,85,80,37)(33,36,81,84)(34,83,82,35)(45,48,93,96)(46,95,94,47)>;
G:=Group( (1,41)(2,90)(3,43)(4,92)(5,45)(6,94)(7,47)(8,96)(9,25)(10,74)(11,27)(12,76)(13,29)(14,78)(15,31)(16,80)(17,33)(18,82)(19,35)(20,84)(21,37)(22,86)(23,39)(24,88)(26,58)(28,60)(30,62)(32,64)(34,66)(36,68)(38,70)(40,72)(42,50)(44,52)(46,54)(48,56)(49,89)(51,91)(53,93)(55,95)(57,73)(59,75)(61,77)(63,79)(65,81)(67,83)(69,85)(71,87), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84), (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), (1,72,49,24)(2,23,50,71)(3,70,51,22)(4,21,52,69)(5,68,53,20)(6,19,54,67)(7,66,55,18)(8,17,56,65)(9,64,57,16)(10,15,58,63)(11,62,59,14)(12,13,60,61)(25,44,73,92)(26,91,74,43)(27,42,75,90)(28,89,76,41)(29,40,77,88)(30,87,78,39)(31,38,79,86)(32,85,80,37)(33,36,81,84)(34,83,82,35)(45,48,93,96)(46,95,94,47) );
G=PermutationGroup([[(1,41),(2,90),(3,43),(4,92),(5,45),(6,94),(7,47),(8,96),(9,25),(10,74),(11,27),(12,76),(13,29),(14,78),(15,31),(16,80),(17,33),(18,82),(19,35),(20,84),(21,37),(22,86),(23,39),(24,88),(26,58),(28,60),(30,62),(32,64),(34,66),(36,68),(38,70),(40,72),(42,50),(44,52),(46,54),(48,56),(49,89),(51,91),(53,93),(55,95),(57,73),(59,75),(61,77),(63,79),(65,81),(67,83),(69,85),(71,87)], [(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,79),(44,80),(45,81),(46,82),(47,83),(48,84)], [(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)], [(1,72,49,24),(2,23,50,71),(3,70,51,22),(4,21,52,69),(5,68,53,20),(6,19,54,67),(7,66,55,18),(8,17,56,65),(9,64,57,16),(10,15,58,63),(11,62,59,14),(12,13,60,61),(25,44,73,92),(26,91,74,43),(27,42,75,90),(28,89,76,41),(29,40,77,88),(30,87,78,39),(31,38,79,86),(32,85,80,37),(33,36,81,84),(34,83,82,35),(45,48,93,96),(46,95,94,47)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 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 | 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 | 4 | 24 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | D12 | D12 | C4○D8 | C4○D24 | C8⋊C22 | D4⋊2S3 | C8⋊D6 |
| kernel | C23.18D12 | C8⋊Dic3 | C24⋊1C4 | C2.D24 | C3×C22⋊C8 | C23.26D6 | C12⋊7D4 | C22⋊C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C12 | C2×C4 | C23 | C6 | C2 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C23.18D12 ►in GL4(𝔽73) generated by
| 30 | 13 | 0 | 0 |
| 60 | 43 | 0 | 0 |
| 0 | 0 | 1 | 71 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 68 | 23 | 0 | 0 |
| 50 | 18 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 46 | 27 |
| 50 | 18 | 0 | 0 |
| 68 | 23 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
G:=sub<GL(4,GF(73))| [30,60,0,0,13,43,0,0,0,0,1,0,0,0,71,72],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[68,50,0,0,23,18,0,0,0,0,46,46,0,0,0,27],[50,68,0,0,18,23,0,0,0,0,46,0,0,0,0,46] >;
C23.18D12 in GAP, Magma, Sage, TeX
C_2^3._{18}D_{12} % in TeX
G:=Group("C2^3.18D12"); // GroupNames label
G:=SmallGroup(192,296);
// by ID
G=gap.SmallGroup(192,296);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,555,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,d*a*d^-1=a*b*c,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