metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.25D6, (C2×Dic3)⋊4D4, C6.39C22≀C2, (C22×C4).50D6, (C22×C6).69D4, C2.7(C23⋊2D6), C6.32(C4⋊D4), (C22×S3).31D4, C22.241(S3×D4), C2.33(Dic3⋊D4), C6.C42⋊17C2, C6.35(C4.4D4), C3⋊2(C23.10D4), C23.25(C3⋊D4), (C23×C6).42C22, C2.22(C23.9D6), C22.99(C4○D12), (S3×C23).15C22, C23.381(C22×S3), (C22×C6).333C23, (C22×C12).27C22, C2.10(C23.14D6), C22.97(D4⋊2S3), C6.34(C22.D4), C2.7(C23.28D6), C2.22(C23.11D6), (C22×Dic3).45C22, (C2×D6⋊C4)⋊7C2, (C6×C22⋊C4)⋊4C2, (C2×C22⋊C4)⋊6S3, (C2×C6).434(C2×D4), (C2×Dic3⋊C4)⋊12C2, (C2×C6.D4)⋊5C2, (C22×C3⋊D4).5C2, (C2×C6).148(C4○D4), C22.127(C2×C3⋊D4), SmallGroup(192,518)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.25D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=cb=bc, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be5 >
Subgroups: 728 in 238 conjugacy classes, 61 normal (51 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22×D4, Dic3⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C22×C12, S3×C23, C23×C6, C23.10D4, C6.C42, C2×Dic3⋊C4, C2×D6⋊C4, C2×C6.D4, C6×C22⋊C4, C22×C3⋊D4, C24.25D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C23.10D4, C23.9D6, Dic3⋊D4, C23.11D6, C23.28D6, C23⋊2D6, C23.14D6, C24.25D6
►On 96 points
Generators in S96
(1 62)(2 93)(3 64)(4 95)(5 66)(6 85)(7 68)(8 87)(9 70)(10 89)(11 72)(12 91)(13 96)(14 67)(15 86)(16 69)(17 88)(18 71)(19 90)(20 61)(21 92)(22 63)(23 94)(24 65)(25 49)(26 39)(27 51)(28 41)(29 53)(30 43)(31 55)(32 45)(33 57)(34 47)(35 59)(36 37)(38 82)(40 84)(42 74)(44 76)(46 78)(48 80)(50 83)(52 73)(54 75)(56 77)(58 79)(60 81)
(1 45)(2 46)(3 47)(4 48)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 60)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 85)(83 86)(84 87)
(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 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(25 82)(26 83)(27 84)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 81)(37 60)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)
(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 6 39 44)(2 43 40 5)(3 4 41 42)(7 12 45 38)(8 37 46 11)(9 10 47 48)(13 22 54 51)(14 50 55 21)(15 20 56 49)(16 60 57 19)(17 18 58 59)(23 24 52 53)(25 62 61 26)(27 72 63 36)(28 35 64 71)(29 70 65 34)(30 33 66 69)(31 68 67 32)(73 80 94 89)(74 88 95 79)(75 78 96 87)(76 86 85 77)(81 84 90 93)(82 92 91 83)
G:=sub<Sym(96)| (1,62)(2,93)(3,64)(4,95)(5,66)(6,85)(7,68)(8,87)(9,70)(10,89)(11,72)(12,91)(13,96)(14,67)(15,86)(16,69)(17,88)(18,71)(19,90)(20,61)(21,92)(22,63)(23,94)(24,65)(25,49)(26,39)(27,51)(28,41)(29,53)(30,43)(31,55)(32,45)(33,57)(34,47)(35,59)(36,37)(38,82)(40,84)(42,74)(44,76)(46,78)(48,80)(50,83)(52,73)(54,75)(56,77)(58,79)(60,81), (1,45)(2,46)(3,47)(4,48)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,60)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,85)(83,86)(84,87), (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), (1,21)(2,22)(3,23)(4,24)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)(25,82)(26,83)(27,84)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(37,60)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90), (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), (1,6,39,44)(2,43,40,5)(3,4,41,42)(7,12,45,38)(8,37,46,11)(9,10,47,48)(13,22,54,51)(14,50,55,21)(15,20,56,49)(16,60,57,19)(17,18,58,59)(23,24,52,53)(25,62,61,26)(27,72,63,36)(28,35,64,71)(29,70,65,34)(30,33,66,69)(31,68,67,32)(73,80,94,89)(74,88,95,79)(75,78,96,87)(76,86,85,77)(81,84,90,93)(82,92,91,83)>;
G:=Group( (1,62)(2,93)(3,64)(4,95)(5,66)(6,85)(7,68)(8,87)(9,70)(10,89)(11,72)(12,91)(13,96)(14,67)(15,86)(16,69)(17,88)(18,71)(19,90)(20,61)(21,92)(22,63)(23,94)(24,65)(25,49)(26,39)(27,51)(28,41)(29,53)(30,43)(31,55)(32,45)(33,57)(34,47)(35,59)(36,37)(38,82)(40,84)(42,74)(44,76)(46,78)(48,80)(50,83)(52,73)(54,75)(56,77)(58,79)(60,81), (1,45)(2,46)(3,47)(4,48)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,60)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,85)(83,86)(84,87), (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), (1,21)(2,22)(3,23)(4,24)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)(25,82)(26,83)(27,84)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(37,60)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90), (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), (1,6,39,44)(2,43,40,5)(3,4,41,42)(7,12,45,38)(8,37,46,11)(9,10,47,48)(13,22,54,51)(14,50,55,21)(15,20,56,49)(16,60,57,19)(17,18,58,59)(23,24,52,53)(25,62,61,26)(27,72,63,36)(28,35,64,71)(29,70,65,34)(30,33,66,69)(31,68,67,32)(73,80,94,89)(74,88,95,79)(75,78,96,87)(76,86,85,77)(81,84,90,93)(82,92,91,83) );
G=PermutationGroup([[(1,62),(2,93),(3,64),(4,95),(5,66),(6,85),(7,68),(8,87),(9,70),(10,89),(11,72),(12,91),(13,96),(14,67),(15,86),(16,69),(17,88),(18,71),(19,90),(20,61),(21,92),(22,63),(23,94),(24,65),(25,49),(26,39),(27,51),(28,41),(29,53),(30,43),(31,55),(32,45),(33,57),(34,47),(35,59),(36,37),(38,82),(40,84),(42,74),(44,76),(46,78),(48,80),(50,83),(52,73),(54,75),(56,77),(58,79),(60,81)], [(1,45),(2,46),(3,47),(4,48),(5,37),(6,38),(7,39),(8,40),(9,41),(10,42),(11,43),(12,44),(13,60),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,57),(23,58),(24,59),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(73,88),(74,89),(75,90),(76,91),(77,92),(78,93),(79,94),(80,95),(81,96),(82,85),(83,86),(84,87)], [(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)], [(1,21),(2,22),(3,23),(4,24),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20),(25,82),(26,83),(27,84),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,79),(35,80),(36,81),(37,60),(38,49),(39,50),(40,51),(41,52),(42,53),(43,54),(44,55),(45,56),(46,57),(47,58),(48,59),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,85),(68,86),(69,87),(70,88),(71,89),(72,90)], [(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)], [(1,6,39,44),(2,43,40,5),(3,4,41,42),(7,12,45,38),(8,37,46,11),(9,10,47,48),(13,22,54,51),(14,50,55,21),(15,20,56,49),(16,60,57,19),(17,18,58,59),(23,24,52,53),(25,62,61,26),(27,72,63,36),(28,35,64,71),(29,70,65,34),(30,33,66,69),(31,68,67,32),(73,80,94,89),(74,88,95,79),(75,78,96,87),(76,86,85,77),(81,84,90,93),(82,92,91,83)]])
42 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 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 | 12 | 12 | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C4○D12 | S3×D4 | D4⋊2S3 |
| kernel | C24.25D6 | C6.C42 | C2×Dic3⋊C4 | C2×D6⋊C4 | C2×C6.D4 | C6×C22⋊C4 | C22×C3⋊D4 | C2×C22⋊C4 | C2×Dic3 | C22×S3 | C22×C6 | C22×C4 | C24 | C2×C6 | C23 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 6 | 4 | 8 | 3 | 1 |
Matrix representation of C24.25D6 ►in GL6(𝔽13)
| 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 | 2 | 9 |
| 0 | 0 | 0 | 0 | 4 | 11 |
| 12 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 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 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 6 | 10 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,2,4,0,0,0,0,9,11],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,7,10,0,0,0,0,3,10],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,2,12,0,0,0,0,0,0,3,6,0,0,0,0,3,10] >;
C24.25D6 in GAP, Magma, Sage, TeX
C_2^4._{25}D_6 % in TeX
G:=Group("C2^4.25D6"); // GroupNames label
G:=SmallGroup(192,518);
// by ID
G=gap.SmallGroup(192,518);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,387,100,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=c,f^2=c*b=b*c,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,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