metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.30D6, (C6×D4)⋊7C4, (C2×D4)⋊7Dic3, C6.102(C4×D4), C12⋊3(C22⋊C4), (C2×Dic3)⋊12D4, (C2×C12).191D4, C2.19(D4×Dic3), (C22×D4).8S3, C2.5(D6⋊3D4), C6.36(C4⋊1D4), C4⋊1(C6.D4), C2.4(C12⋊3D4), (C22×C4).369D6, C22.121(S3×D4), C6.129(C4⋊D4), C6.47(C4.4D4), (C23×C6).47C22, C23.15(C2×Dic3), C2.4(C23.12D6), C23.316(C22×S3), (C22×C6).366C23, C3⋊4(C24.3C22), C22.62(D4⋊2S3), (C22×C12).199C22, C22.52(C22×Dic3), (C22×Dic3).197C22, (D4×C2×C6).5C2, (C2×C4×Dic3)⋊3C2, (C2×C4⋊Dic3)⋊35C2, (C2×C6).555(C2×D4), C6.76(C2×C22⋊C4), (C2×C12).118(C2×C4), (C22×C6).73(C2×C4), (C2×C4).50(C2×Dic3), C22.92(C2×C3⋊D4), (C2×C6).162(C4○D4), (C2×C6.D4)⋊10C2, (C2×C4).148(C3⋊D4), (C2×C6).197(C22×C4), C2.12(C2×C6.D4), SmallGroup(192,780)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.30D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=d, f2=db=bd, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
Subgroups: 616 in 258 conjugacy classes, 91 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C4×Dic3, C4⋊Dic3, C6.D4, C22×Dic3, C22×C12, C6×D4, C6×D4, C23×C6, C24.3C22, C2×C4×Dic3, C2×C4⋊Dic3, C2×C6.D4, D4×C2×C6, C24.30D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C6.D4, S3×D4, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C24.3C22, D4×Dic3, C23.12D6, D6⋊3D4, C12⋊3D4, C2×C6.D4, C24.30D6
►On 96 points
Generators in S96
(1 4)(2 11)(3 6)(5 8)(7 10)(9 12)(13 22)(14 17)(15 24)(16 19)(18 21)(20 23)(25 50)(26 57)(27 52)(28 59)(29 54)(30 49)(31 56)(32 51)(33 58)(34 53)(35 60)(36 55)(37 96)(38 91)(39 86)(40 93)(41 88)(42 95)(43 90)(44 85)(45 92)(46 87)(47 94)(48 89)(61 70)(62 65)(63 72)(64 67)(66 69)(68 71)(73 82)(74 77)(75 84)(76 79)(78 81)(80 83)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)(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 95)(50 96)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)(61 79)(62 80)(63 81)(64 82)(65 83)(66 84)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 61)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 49)(34 50)(35 51)(36 52)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 85)(48 86)
(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 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 37 14 31)(2 42 15 36)(3 47 16 29)(4 40 17 34)(5 45 18 27)(6 38 19 32)(7 43 20 25)(8 48 21 30)(9 41 22 35)(10 46 23 28)(11 39 24 33)(12 44 13 26)(49 72 89 84)(50 65 90 77)(51 70 91 82)(52 63 92 75)(53 68 93 80)(54 61 94 73)(55 66 95 78)(56 71 96 83)(57 64 85 76)(58 69 86 81)(59 62 87 74)(60 67 88 79)
G:=sub<Sym(96)| (1,4)(2,11)(3,6)(5,8)(7,10)(9,12)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23)(25,50)(26,57)(27,52)(28,59)(29,54)(30,49)(31,56)(32,51)(33,58)(34,53)(35,60)(36,55)(37,96)(38,91)(39,86)(40,93)(41,88)(42,95)(43,90)(44,85)(45,92)(46,87)(47,94)(48,89)(61,70)(62,65)(63,72)(64,67)(66,69)(68,71)(73,82)(74,77)(75,84)(76,79)(78,81)(80,83), (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,95)(50,96)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,61)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,49)(34,50)(35,51)(36,52)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,85)(48,86), (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), (1,2,3,4,5,6,7,8,9,10,11,12)(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), (1,37,14,31)(2,42,15,36)(3,47,16,29)(4,40,17,34)(5,45,18,27)(6,38,19,32)(7,43,20,25)(8,48,21,30)(9,41,22,35)(10,46,23,28)(11,39,24,33)(12,44,13,26)(49,72,89,84)(50,65,90,77)(51,70,91,82)(52,63,92,75)(53,68,93,80)(54,61,94,73)(55,66,95,78)(56,71,96,83)(57,64,85,76)(58,69,86,81)(59,62,87,74)(60,67,88,79)>;
G:=Group( (1,4)(2,11)(3,6)(5,8)(7,10)(9,12)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23)(25,50)(26,57)(27,52)(28,59)(29,54)(30,49)(31,56)(32,51)(33,58)(34,53)(35,60)(36,55)(37,96)(38,91)(39,86)(40,93)(41,88)(42,95)(43,90)(44,85)(45,92)(46,87)(47,94)(48,89)(61,70)(62,65)(63,72)(64,67)(66,69)(68,71)(73,82)(74,77)(75,84)(76,79)(78,81)(80,83), (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,95)(50,96)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,91)(58,92)(59,93)(60,94)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,61)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,49)(34,50)(35,51)(36,52)(37,87)(38,88)(39,89)(40,90)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,85)(48,86), (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), (1,2,3,4,5,6,7,8,9,10,11,12)(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), (1,37,14,31)(2,42,15,36)(3,47,16,29)(4,40,17,34)(5,45,18,27)(6,38,19,32)(7,43,20,25)(8,48,21,30)(9,41,22,35)(10,46,23,28)(11,39,24,33)(12,44,13,26)(49,72,89,84)(50,65,90,77)(51,70,91,82)(52,63,92,75)(53,68,93,80)(54,61,94,73)(55,66,95,78)(56,71,96,83)(57,64,85,76)(58,69,86,81)(59,62,87,74)(60,67,88,79) );
G=PermutationGroup([[(1,4),(2,11),(3,6),(5,8),(7,10),(9,12),(13,22),(14,17),(15,24),(16,19),(18,21),(20,23),(25,50),(26,57),(27,52),(28,59),(29,54),(30,49),(31,56),(32,51),(33,58),(34,53),(35,60),(36,55),(37,96),(38,91),(39,86),(40,93),(41,88),(42,95),(43,90),(44,85),(45,92),(46,87),(47,94),(48,89),(61,70),(62,65),(63,72),(64,67),(66,69),(68,71),(73,82),(74,77),(75,84),(76,79),(78,81),(80,83)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,13),(7,14),(8,15),(9,16),(10,17),(11,18),(12,19),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(49,95),(50,96),(51,85),(52,86),(53,87),(54,88),(55,89),(56,90),(57,91),(58,92),(59,93),(60,94),(61,79),(62,80),(63,81),(64,82),(65,83),(66,84),(67,73),(68,74),(69,75),(70,76),(71,77),(72,78)], [(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,61),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,49),(34,50),(35,51),(36,52),(37,87),(38,88),(39,89),(40,90),(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,85),(48,86)], [(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)], [(1,2,3,4,5,6,7,8,9,10,11,12),(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)], [(1,37,14,31),(2,42,15,36),(3,47,16,29),(4,40,17,34),(5,45,18,27),(6,38,19,32),(7,43,20,25),(8,48,21,30),(9,41,22,35),(10,46,23,28),(11,39,24,33),(12,44,13,26),(49,72,89,84),(50,65,90,77),(51,70,91,82),(52,63,92,75),(53,68,93,80),(54,61,94,73),(55,66,95,78),(56,71,96,83),(57,64,85,76),(58,69,86,81),(59,62,87,74),(60,67,88,79)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 6A | ··· | 6G | 6H | ··· | 6O | 12A | 12B | 12C | 12D |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | Dic3 | D6 | C4○D4 | C3⋊D4 | S3×D4 | D4⋊2S3 |
| kernel | C24.30D6 | C2×C4×Dic3 | C2×C4⋊Dic3 | C2×C6.D4 | D4×C2×C6 | C6×D4 | C22×D4 | C2×Dic3 | C2×C12 | C22×C4 | C2×D4 | C24 | C2×C6 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 4 | 1 | 8 | 1 | 4 | 4 | 1 | 4 | 2 | 4 | 8 | 2 | 2 |
Matrix representation of C24.30D6 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 4 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 7 | 3 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,1,1],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,10,4],[5,0,0,0,0,0,0,8,0,0,0,5,0,0,0,0,0,0,10,7,0,0,0,6,3] >;
C24.30D6 in GAP, Magma, Sage, TeX
C_2^4._{30}D_6 % in TeX
G:=Group("C2^4.30D6"); // GroupNames label
G:=SmallGroup(192,780);
// by ID
G=gap.SmallGroup(192,780);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,422,387,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=d,f^2=d*b=b*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*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=e^5>;
// generators/relations