metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.20D6, (C22×C4).46D6, (C22×C6).64D4, C6.84(C4⋊D4), (C2×Dic3).56D4, C22.240(S3×D4), C6.C42⋊14C2, C6.33(C4.4D4), C3⋊3(C23.11D4), C23.24(C3⋊D4), (C23×C6).35C22, C6.15(C42⋊2C2), C2.9(C23.14D6), C2.21(C23.9D6), C22.97(C4○D12), C23.379(C22×S3), (C22×C12).24C22, (C22×C6).327C23, C22.95(D4⋊2S3), C6.57(C22.D4), C2.13(C23.8D6), C2.6(C23.28D6), C2.7(C23.23D6), C2.21(C23.11D6), (C22×Dic3).41C22, (C2×C6).431(C2×D4), (C2×C22⋊C4).9S3, (C6×C22⋊C4).8C2, (C2×Dic3⋊C4)⋊10C2, (C2×C6).143(C4○D4), C22.125(C2×C3⋊D4), (C2×C6.D4).14C2, SmallGroup(192,511)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.20D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=bcd, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be5 >
Subgroups: 440 in 170 conjugacy classes, 57 normal (51 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, Dic3⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, C22×C12, C23×C6, C23.11D4, C6.C42, C2×Dic3⋊C4, C2×C6.D4, C6×C22⋊C4, C24.20D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C23.11D4, C23.8D6, C23.9D6, C23.11D6, C23.28D6, C23.23D6, C23.14D6, C24.20D6
►On 96 points
Generators in S96
(2 33)(4 35)(6 25)(8 27)(10 29)(12 31)(14 68)(16 70)(18 72)(20 62)(22 64)(24 66)(37 51)(38 44)(39 53)(40 46)(41 55)(42 48)(43 57)(45 59)(47 49)(50 56)(52 58)(54 60)(73 79)(74 87)(75 81)(76 89)(77 83)(78 91)(80 93)(82 95)(84 85)(86 92)(88 94)(90 96)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 13)(12 14)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 61)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 85)(46 86)(47 87)(48 88)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 81)(57 82)(58 83)(59 84)(60 73)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(37 57)(38 58)(39 59)(40 60)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(73 92)(74 93)(75 94)(76 95)(77 96)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)
(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 55 63 93)(2 73 64 46)(3 53 65 91)(4 83 66 44)(5 51 67 89)(6 81 68 42)(7 49 69 87)(8 79 70 40)(9 59 71 85)(10 77 72 38)(11 57 61 95)(12 75 62 48)(13 82 36 43)(14 50 25 88)(15 80 26 41)(16 60 27 86)(17 78 28 39)(18 58 29 96)(19 76 30 37)(20 56 31 94)(21 74 32 47)(22 54 33 92)(23 84 34 45)(24 52 35 90)
G:=sub<Sym(96)| (2,33)(4,35)(6,25)(8,27)(10,29)(12,31)(14,68)(16,70)(18,72)(20,62)(22,64)(24,66)(37,51)(38,44)(39,53)(40,46)(41,55)(42,48)(43,57)(45,59)(47,49)(50,56)(52,58)(54,60)(73,79)(74,87)(75,81)(76,89)(77,83)(78,91)(80,93)(82,95)(84,85)(86,92)(88,94)(90,96), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,13)(12,14)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,61)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,85)(46,86)(47,87)(48,88)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,81)(57,82)(58,83)(59,84)(60,73), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,32)(2,33)(3,34)(4,35)(5,36)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(37,57)(38,58)(39,59)(40,60)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(73,92)(74,93)(75,94)(76,95)(77,96)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91), (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), (1,55,63,93)(2,73,64,46)(3,53,65,91)(4,83,66,44)(5,51,67,89)(6,81,68,42)(7,49,69,87)(8,79,70,40)(9,59,71,85)(10,77,72,38)(11,57,61,95)(12,75,62,48)(13,82,36,43)(14,50,25,88)(15,80,26,41)(16,60,27,86)(17,78,28,39)(18,58,29,96)(19,76,30,37)(20,56,31,94)(21,74,32,47)(22,54,33,92)(23,84,34,45)(24,52,35,90)>;
G:=Group( (2,33)(4,35)(6,25)(8,27)(10,29)(12,31)(14,68)(16,70)(18,72)(20,62)(22,64)(24,66)(37,51)(38,44)(39,53)(40,46)(41,55)(42,48)(43,57)(45,59)(47,49)(50,56)(52,58)(54,60)(73,79)(74,87)(75,81)(76,89)(77,83)(78,91)(80,93)(82,95)(84,85)(86,92)(88,94)(90,96), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,13)(12,14)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,61)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,85)(46,86)(47,87)(48,88)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,81)(57,82)(58,83)(59,84)(60,73), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,32)(2,33)(3,34)(4,35)(5,36)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(37,57)(38,58)(39,59)(40,60)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(73,92)(74,93)(75,94)(76,95)(77,96)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91), (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), (1,55,63,93)(2,73,64,46)(3,53,65,91)(4,83,66,44)(5,51,67,89)(6,81,68,42)(7,49,69,87)(8,79,70,40)(9,59,71,85)(10,77,72,38)(11,57,61,95)(12,75,62,48)(13,82,36,43)(14,50,25,88)(15,80,26,41)(16,60,27,86)(17,78,28,39)(18,58,29,96)(19,76,30,37)(20,56,31,94)(21,74,32,47)(22,54,33,92)(23,84,34,45)(24,52,35,90) );
G=PermutationGroup([[(2,33),(4,35),(6,25),(8,27),(10,29),(12,31),(14,68),(16,70),(18,72),(20,62),(22,64),(24,66),(37,51),(38,44),(39,53),(40,46),(41,55),(42,48),(43,57),(45,59),(47,49),(50,56),(52,58),(54,60),(73,79),(74,87),(75,81),(76,89),(77,83),(78,91),(80,93),(82,95),(84,85),(86,92),(88,94),(90,96)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,13),(12,14),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,61),(37,89),(38,90),(39,91),(40,92),(41,93),(42,94),(43,95),(44,96),(45,85),(46,86),(47,87),(48,88),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,81),(57,82),(58,83),(59,84),(60,73)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(37,57),(38,58),(39,59),(40,60),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56),(73,92),(74,93),(75,94),(76,95),(77,96),(78,85),(79,86),(80,87),(81,88),(82,89),(83,90),(84,91)], [(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)], [(1,55,63,93),(2,73,64,46),(3,53,65,91),(4,83,66,44),(5,51,67,89),(6,81,68,42),(7,49,69,87),(8,79,70,40),(9,59,71,85),(10,77,72,38),(11,57,61,95),(12,75,62,48),(13,82,36,43),(14,50,25,88),(15,80,26,41),(16,60,27,86),(17,78,28,39),(18,58,29,96),(19,76,30,37),(20,56,31,94),(21,74,32,47),(22,54,33,92),(23,84,34,45),(24,52,35,90)]])
42 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C4○D12 | S3×D4 | D4⋊2S3 |
| kernel | C24.20D6 | C6.C42 | C2×Dic3⋊C4 | C2×C6.D4 | C6×C22⋊C4 | C2×C22⋊C4 | C2×Dic3 | C22×C6 | C22×C4 | C24 | C2×C6 | C23 | C22 | C22 | C22 |
| # reps | 1 | 3 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 10 | 4 | 8 | 1 | 3 |
Matrix representation of C24.20D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 11 | 4 | 0 | 0 | 0 | 0 |
| 9 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[6,4,0,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,9,0,0,0,0,4,2,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,8,0,0,0,0,5,0] >;
C24.20D6 in GAP, Magma, Sage, TeX
C_2^4._{20}D_6 % in TeX
G:=Group("C2^4.20D6"); // GroupNames label
G:=SmallGroup(192,511);
// by ID
G=gap.SmallGroup(192,511);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,387,100,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=c,f^2=b*c*d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*e^5>;
// generators/relations