direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C24⋊C2, C24⋊10C23, C12.54C24, Dic6⋊4C23, C23.65D12, D12.20C23, (C2×C8)⋊35D6, C8⋊9(C22×S3), C6⋊1(C2×SD16), (C2×C6)⋊9SD16, (C22×C8)⋊14S3, (C2×C4).99D12, C4.44(C2×D12), C3⋊1(C22×SD16), (C2×C24)⋊46C22, (C22×C24)⋊14C2, C12.289(C2×D4), (C2×C12).390D4, C4.51(S3×C23), C6.21(C22×D4), (C22×D12).9C2, C2.23(C22×D12), C22.69(C2×D12), (C22×C4).458D6, (C22×C6).144D4, (C2×C12).786C23, (C2×Dic6)⋊56C22, (C22×Dic6)⋊11C2, (C2×D12).228C22, (C22×C12).525C22, (C2×C6).177(C2×D4), (C2×C4).735(C22×S3), SmallGroup(192,1298)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×C24⋊C2
G = < a,b,c,d | a2=b2=c24=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c11 >
Subgroups: 920 in 298 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C24, Dic6, Dic6, D12, D12, C2×Dic3, C2×C12, C22×S3, C22×C6, C22×C8, C2×SD16, C22×D4, C22×Q8, C24⋊C2, C2×C24, C2×Dic6, C2×Dic6, C2×D12, C2×D12, C22×Dic3, C22×C12, S3×C23, C22×SD16, C2×C24⋊C2, C22×C24, C22×Dic6, C22×D12, C22×C24⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C24, D12, C22×S3, C2×SD16, C22×D4, C24⋊C2, C2×D12, S3×C23, C22×SD16, C2×C24⋊C2, C22×D12, C22×C24⋊C2
►On 96 points
Generators in S96
(1 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 91)(9 92)(10 93)(11 94)(12 95)(13 96)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 49)(45 50)(46 51)(47 52)(48 53)
(1 71)(2 72)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 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)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 47)(26 34)(27 45)(28 32)(29 43)(31 41)(33 39)(35 37)(36 48)(38 46)(40 44)(49 69)(50 56)(51 67)(52 54)(53 65)(55 63)(57 61)(58 72)(60 70)(62 68)(64 66)(73 83)(74 94)(75 81)(76 92)(77 79)(78 90)(80 88)(82 86)(85 95)(87 93)(89 91)
G:=sub<Sym(96)| (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,49)(45,50)(46,51)(47,52)(48,53), (1,71)(2,72)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90), (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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,47)(26,34)(27,45)(28,32)(29,43)(31,41)(33,39)(35,37)(36,48)(38,46)(40,44)(49,69)(50,56)(51,67)(52,54)(53,65)(55,63)(57,61)(58,72)(60,70)(62,68)(64,66)(73,83)(74,94)(75,81)(76,92)(77,79)(78,90)(80,88)(82,86)(85,95)(87,93)(89,91)>;
G:=Group( (1,84)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,54)(26,55)(27,56)(28,57)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,49)(45,50)(46,51)(47,52)(48,53), (1,71)(2,72)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90), (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), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,47)(26,34)(27,45)(28,32)(29,43)(31,41)(33,39)(35,37)(36,48)(38,46)(40,44)(49,69)(50,56)(51,67)(52,54)(53,65)(55,63)(57,61)(58,72)(60,70)(62,68)(64,66)(73,83)(74,94)(75,81)(76,92)(77,79)(78,90)(80,88)(82,86)(85,95)(87,93)(89,91) );
G=PermutationGroup([[(1,84),(2,85),(3,86),(4,87),(5,88),(6,89),(7,90),(8,91),(9,92),(10,93),(11,94),(12,95),(13,96),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(25,54),(26,55),(27,56),(28,57),(29,58),(30,59),(31,60),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,68),(40,69),(41,70),(42,71),(43,72),(44,49),(45,50),(46,51),(47,52),(48,53)], [(1,71),(2,72),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,67),(22,68),(23,69),(24,70),(25,91),(26,92),(27,93),(28,94),(29,95),(30,96),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,89),(48,90)], [(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)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,47),(26,34),(27,45),(28,32),(29,43),(31,41),(33,39),(35,37),(36,48),(38,46),(40,44),(49,69),(50,56),(51,67),(52,54),(53,65),(55,63),(57,61),(58,72),(60,70),(62,68),(64,66),(73,83),(74,94),(75,81),(76,92),(77,79),(78,90),(80,88),(82,86),(85,95),(87,93),(89,91)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | SD16 | D12 | D12 | C24⋊C2 |
| kernel | C22×C24⋊C2 | C2×C24⋊C2 | C22×C24 | C22×Dic6 | C22×D12 | C22×C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 1 | 1 | 1 | 3 | 1 | 6 | 1 | 8 | 6 | 2 | 16 |
Matrix representation of C22×C24⋊C2 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 11 | 48 |
| 0 | 0 | 25 | 36 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,72,0,0,0,0,11,25,0,0,48,36],[72,0,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;
C22×C24⋊C2 in GAP, Magma, Sage, TeX
C_2^2\times C_{24}\rtimes C_2 % in TeX
G:=Group("C2^2xC24:C2"); // GroupNames label
G:=SmallGroup(192,1298);
// by ID
G=gap.SmallGroup(192,1298);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,80,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^24=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^11>;
// generators/relations