direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D24⋊C2, Q16⋊11D6, D24⋊17C22, C12.12C24, C24.34C23, D12.7C23, C6⋊4(C4○D8), (C6×Q16)⋊8C2, C4.48(S3×D4), (C2×D24)⋊20C2, (C2×Q16)⋊13S3, (C4×S3).31D4, D6.11(C2×D4), C12.87(C2×D4), (C2×C8).247D6, C3⋊C8.23C23, (S3×C8)⋊15C22, C4.12(S3×C23), C8.40(C22×S3), (C2×Q8).178D6, (C3×Q16)⋊9C22, (C3×Q8).6C23, (C4×S3).28C23, (C2×C24).99C22, Dic3.70(C2×D4), Q8⋊3S3⋊7C22, (C22×S3).63D4, C6.113(C22×D4), C22.144(S3×D4), Q8.16(C22×S3), (C2×C12).529C23, (C2×Dic3).217D4, Q8⋊2S3⋊10C22, (C6×Q8).151C22, (C2×D12).180C22, (S3×C2×C8)⋊6C2, C3⋊4(C2×C4○D8), C2.86(C2×S3×D4), (C2×C6).402(C2×D4), (C2×Q8⋊2S3)⋊28C2, (C2×Q8⋊3S3)⋊16C2, (C2×C3⋊C8).286C22, (S3×C2×C4).260C22, (C2×C4).617(C22×S3), SmallGroup(192,1324)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 728 in 266 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×12], S3 [×6], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×14], Q8 [×4], Q8 [×2], C23 [×3], Dic3 [×2], C12 [×2], C12 [×4], D6 [×2], D6 [×10], C2×C6, C2×C8, C2×C8 [×5], D8 [×4], SD16 [×8], Q16 [×4], C22×C4 [×3], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×12], C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C4×S3 [×8], D12 [×4], D12 [×10], C2×Dic3, C2×C12, C2×C12 [×2], C3×Q8 [×4], C3×Q8 [×2], C22×S3, C22×S3 [×2], C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], S3×C8 [×4], D24 [×4], C2×C3⋊C8, Q8⋊2S3 [×8], C2×C24, C3×Q16 [×4], S3×C2×C4, S3×C2×C4 [×2], C2×D12 [×2], C2×D12 [×2], Q8⋊3S3 [×8], Q8⋊3S3 [×4], C6×Q8 [×2], C2×C4○D8, S3×C2×C8, C2×D24, D24⋊C2 [×8], C2×Q8⋊2S3 [×2], C6×Q16, C2×Q8⋊3S3 [×2], C2×D24⋊C2
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C4○D8 [×2], C22×D4, S3×D4 [×2], S3×C23, C2×C4○D8, D24⋊C2 [×2], C2×S3×D4, C2×D24⋊C2
Generators and relations
G = < a,b,c,d | a2=b24=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b17, dcd=b4c >
►On 96 points
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)(57 94)(58 95)(59 96)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 85)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 36)(49 95)(50 94)(51 93)(52 92)(53 91)(54 90)(55 89)(56 88)(57 87)(58 86)(59 85)(60 84)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(72 96)
(1 63)(2 56)(3 49)(4 66)(5 59)(6 52)(7 69)(8 62)(9 55)(10 72)(11 65)(12 58)(13 51)(14 68)(15 61)(16 54)(17 71)(18 64)(19 57)(20 50)(21 67)(22 60)(23 53)(24 70)(25 75)(26 92)(27 85)(28 78)(29 95)(30 88)(31 81)(32 74)(33 91)(34 84)(35 77)(36 94)(37 87)(38 80)(39 73)(40 90)(41 83)(42 76)(43 93)(44 86)(45 79)(46 96)(47 89)(48 82)
G:=sub<Sym(96)| (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,36)(49,95)(50,94)(51,93)(52,92)(53,91)(54,90)(55,89)(56,88)(57,87)(58,86)(59,85)(60,84)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(72,96), (1,63)(2,56)(3,49)(4,66)(5,59)(6,52)(7,69)(8,62)(9,55)(10,72)(11,65)(12,58)(13,51)(14,68)(15,61)(16,54)(17,71)(18,64)(19,57)(20,50)(21,67)(22,60)(23,53)(24,70)(25,75)(26,92)(27,85)(28,78)(29,95)(30,88)(31,81)(32,74)(33,91)(34,84)(35,77)(36,94)(37,87)(38,80)(39,73)(40,90)(41,83)(42,76)(43,93)(44,86)(45,79)(46,96)(47,89)(48,82)>;
G:=Group( (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,36)(49,95)(50,94)(51,93)(52,92)(53,91)(54,90)(55,89)(56,88)(57,87)(58,86)(59,85)(60,84)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(72,96), (1,63)(2,56)(3,49)(4,66)(5,59)(6,52)(7,69)(8,62)(9,55)(10,72)(11,65)(12,58)(13,51)(14,68)(15,61)(16,54)(17,71)(18,64)(19,57)(20,50)(21,67)(22,60)(23,53)(24,70)(25,75)(26,92)(27,85)(28,78)(29,95)(30,88)(31,81)(32,74)(33,91)(34,84)(35,77)(36,94)(37,87)(38,80)(39,73)(40,90)(41,83)(42,76)(43,93)(44,86)(45,79)(46,96)(47,89)(48,82) );
G=PermutationGroup([(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,41),(49,86),(50,87),(51,88),(52,89),(53,90),(54,91),(55,92),(56,93),(57,94),(58,95),(59,96),(60,73),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,81),(69,82),(70,83),(71,84),(72,85)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,48),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,40),(21,39),(22,38),(23,37),(24,36),(49,95),(50,94),(51,93),(52,92),(53,91),(54,90),(55,89),(56,88),(57,87),(58,86),(59,85),(60,84),(61,83),(62,82),(63,81),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(72,96)], [(1,63),(2,56),(3,49),(4,66),(5,59),(6,52),(7,69),(8,62),(9,55),(10,72),(11,65),(12,58),(13,51),(14,68),(15,61),(16,54),(17,71),(18,64),(19,57),(20,50),(21,67),(22,60),(23,53),(24,70),(25,75),(26,92),(27,85),(28,78),(29,95),(30,88),(31,81),(32,74),(33,91),(34,84),(35,77),(36,94),(37,87),(38,80),(39,73),(40,90),(41,83),(42,76),(43,93),(44,86),(45,79),(46,96),(47,89),(48,82)])
Matrix representation ►G ⊆ GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 41 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 25 | 0 | 0 |
| 0 | 0 | 35 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 25 | 0 | 0 |
| 0 | 0 | 35 | 41 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 46 | 46 | 0 | 0 | 0 | 0 |
| 54 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 8 | 0 | 0 |
| 0 | 0 | 55 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,41,0,0,0,0,16,41,0,0,0,0,0,0,32,35,0,0,0,0,25,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[0,32,0,0,0,0,16,0,0,0,0,0,0,0,32,35,0,0,0,0,25,41,0,0,0,0,0,0,72,0,0,0,0,0,1,1],[46,54,0,0,0,0,46,27,0,0,0,0,0,0,46,55,0,0,0,0,8,27,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4○D8 | S3×D4 | S3×D4 | D24⋊C2 |
| kernel | C2×D24⋊C2 | S3×C2×C8 | C2×D24 | D24⋊C2 | C2×Q8⋊2S3 | C6×Q16 | C2×Q8⋊3S3 | C2×Q16 | C4×S3 | C2×Dic3 | C22×S3 | C2×C8 | Q16 | C2×Q8 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 8 | 1 | 1 | 4 |
In GAP, Magma, Sage, TeX
C_2\times D_{24}\rtimes C_2 % in TeX
G:=Group("C2xD24:C2"); // GroupNames label
G:=SmallGroup(192,1324);
// by ID
G=gap.SmallGroup(192,1324);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,1123,185,136,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^24=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^17,d*c*d=b^4*c>;
// generators/relations