metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.28D4, C3⋊C8.22D4, (C6×Q16)⋊6C2, (C2×Q16)⋊6S3, C4.29(S3×D4), (C8×Dic3)⋊7C2, (C2×C8).244D6, (C2×Q8).91D6, (C2×D24).11C2, C6.82(C4○D8), C12.189(C2×D4), C8.19(C3⋊D4), C3⋊5(C8.12D4), C12.23D4⋊6C2, C6.35(C4⋊1D4), (C2×C24).96C22, C22.282(S3×D4), (C6×Q8).94C22, C2.26(C12⋊3D4), (C2×C12).465C23, (C2×Dic3).118D4, C2.19(D24⋊C2), (C2×D12).127C22, (C4×Dic3).245C22, C4.16(C2×C3⋊D4), (C2×C6).376(C2×D4), (C2×Q8⋊2S3)⋊21C2, (C2×C3⋊C8).279C22, (C2×C4).553(C22×S3), SmallGroup(192,750)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.28D4
G = < a,b,c | a24=b4=c2=1, bab-1=a17, cac=a-1, cbc=a12b-1 >
Subgroups: 440 in 130 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, C3⋊C8, C24, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×Q16, D24, C2×C3⋊C8, C4×Dic3, D6⋊C4, Q8⋊2S3, C2×C24, C3×Q16, C2×D12, C6×Q8, C8.12D4, C8×Dic3, C2×D24, C2×Q8⋊2S3, C12.23D4, C6×Q16, C24.28D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, C4○D8, S3×D4, C2×C3⋊D4, C8.12D4, D24⋊C2, C12⋊3D4, C24.28D4
►On 96 points
(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 82 52 26)(2 75 53 43)(3 92 54 36)(4 85 55 29)(5 78 56 46)(6 95 57 39)(7 88 58 32)(8 81 59 25)(9 74 60 42)(10 91 61 35)(11 84 62 28)(12 77 63 45)(13 94 64 38)(14 87 65 31)(15 80 66 48)(16 73 67 41)(17 90 68 34)(18 83 69 27)(19 76 70 44)(20 93 71 37)(21 86 72 30)(22 79 49 47)(23 96 50 40)(24 89 51 33)
(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(25 95)(26 94)(27 93)(28 92)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 84)(37 83)(38 82)(39 81)(40 80)(41 79)(42 78)(43 77)(44 76)(45 75)(46 74)(47 73)(48 96)(49 55)(50 54)(51 53)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)
G:=sub<Sym(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), (1,82,52,26)(2,75,53,43)(3,92,54,36)(4,85,55,29)(5,78,56,46)(6,95,57,39)(7,88,58,32)(8,81,59,25)(9,74,60,42)(10,91,61,35)(11,84,62,28)(12,77,63,45)(13,94,64,38)(14,87,65,31)(15,80,66,48)(16,73,67,41)(17,90,68,34)(18,83,69,27)(19,76,70,44)(20,93,71,37)(21,86,72,30)(22,79,49,47)(23,96,50,40)(24,89,51,33), (2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,96)(49,55)(50,54)(51,53)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)>;
G:=Group( (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), (1,82,52,26)(2,75,53,43)(3,92,54,36)(4,85,55,29)(5,78,56,46)(6,95,57,39)(7,88,58,32)(8,81,59,25)(9,74,60,42)(10,91,61,35)(11,84,62,28)(12,77,63,45)(13,94,64,38)(14,87,65,31)(15,80,66,48)(16,73,67,41)(17,90,68,34)(18,83,69,27)(19,76,70,44)(20,93,71,37)(21,86,72,30)(22,79,49,47)(23,96,50,40)(24,89,51,33), (2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,96)(49,55)(50,54)(51,53)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65) );
G=PermutationGroup([[(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)], [(1,82,52,26),(2,75,53,43),(3,92,54,36),(4,85,55,29),(5,78,56,46),(6,95,57,39),(7,88,58,32),(8,81,59,25),(9,74,60,42),(10,91,61,35),(11,84,62,28),(12,77,63,45),(13,94,64,38),(14,87,65,31),(15,80,66,48),(16,73,67,41),(17,90,68,34),(18,83,69,27),(19,76,70,44),(20,93,71,37),(21,86,72,30),(22,79,49,47),(23,96,50,40),(24,89,51,33)], [(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(25,95),(26,94),(27,93),(28,92),(29,91),(30,90),(31,89),(32,88),(33,87),(34,86),(35,85),(36,84),(37,83),(38,82),(39,81),(40,80),(41,79),(42,78),(43,77),(44,76),(45,75),(46,74),(47,73),(48,96),(49,55),(50,54),(51,53),(56,72),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 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 | 3 | 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 | 24 | 24 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 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 | S3 | D4 | D4 | D4 | D6 | D6 | C3⋊D4 | C4○D8 | S3×D4 | S3×D4 | D24⋊C2 |
| kernel | C24.28D4 | C8×Dic3 | C2×D24 | C2×Q8⋊2S3 | C12.23D4 | C6×Q16 | C2×Q16 | C3⋊C8 | C24 | C2×Dic3 | C2×C8 | C2×Q8 | C8 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 8 | 1 | 1 | 4 |
Matrix representation of C24.28D4 ►in GL6(𝔽73)
| 11 | 3 | 0 | 0 | 0 | 0 |
| 8 | 62 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 57 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 62 | 70 | 0 | 0 | 0 | 0 |
| 65 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 17 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [11,8,0,0,0,0,3,62,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,16,16,0,0,0,0,57,16],[62,65,0,0,0,0,70,11,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[1,17,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;
C24.28D4 in GAP, Magma, Sage, TeX
C_{24}._{28}D_4 % in TeX
G:=Group("C24.28D4"); // GroupNames label
G:=SmallGroup(192,750);
// by ID
G=gap.SmallGroup(192,750);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,555,184,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^17,c*a*c=a^-1,c*b*c=a^12*b^-1>;
// generators/relations