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G = D487C2order 192 = 26·3

The semidirect product of D48 and C2 acting through Inn(D48)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D487C2, C4.20D24, C8.13D12, C24.63D4, C16.16D6, C12.38D8, Dic247C2, C22.1D24, C24.57C23, C48.18C22, D24.7C22, Dic12.7C22, (C2×C16)⋊6S3, (C2×C48)⋊10C2, C4○D241C2, C31(C4○D16), C48⋊C27C2, C6.11(C2×D8), (C2×C6).20D8, C2.13(C2×D24), (C2×C4).85D12, C4.38(C2×D12), (C2×C8).313D6, C12.281(C2×D4), (C2×C12).395D4, C8.47(C22×S3), (C2×C24).385C22, SmallGroup(192,463)

Series: Derived Chief Lower central Upper central

C1C24 — D487C2
C1C3C6C12C24D24C4○D24 — D487C2
C3C6C12C24 — D487C2
C1C4C2×C4C2×C8C2×C16

Generators and relations for D487C2
 G = < a,b,c | a48=b2=c2=1, bab=a-1, ac=ca, cbc=a24b >

Subgroups: 328 in 84 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C16 [×2], C2×C8, D8 [×2], SD16 [×2], Q16 [×2], C4○D4 [×2], C24 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C2×C12, C2×C16, D16, SD32 [×2], Q32, C4○D8 [×2], C48 [×2], C24⋊C2 [×2], D24 [×2], Dic12 [×2], C2×C24, C4○D12 [×2], C4○D16, D48, C48⋊C2 [×2], Dic24, C2×C48, C4○D24 [×2], D487C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], D8 [×2], C2×D4, D12 [×2], C22×S3, C2×D8, D24 [×2], C2×D12, C4○D16, C2×D24, D487C2

Smallest permutation representation of D487C2
On 96 points
Generators in S96
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(49 87)(50 86)(51 85)(52 84)(53 83)(54 82)(55 81)(56 80)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(88 96)(89 95)(90 94)(91 93)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)

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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(49,87)(50,86)(51,85)(52,84)(53,83)(54,82)(55,81)(56,80)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(88,96)(89,95)(90,94)(91,93), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)>;

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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(49,87)(50,86)(51,85)(52,84)(53,83)(54,82)(55,81)(56,80)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(88,96)(89,95)(90,94)(91,93), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72) );

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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(49,87),(50,86),(51,85),(52,84),(53,83),(54,82),(55,81),(56,80),(57,79),(58,78),(59,77),(60,76),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69),(88,96),(89,95),(90,94),(91,93)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72)])

54 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E6A6B6C8A8B8C8D12A12B12C12D16A···16H24A···24H48A···48P
order1222234444466688881212121216···1624···2448···48
size112242421122424222222222222···22···22···2

54 irreducible representations

dim1111112222222222222
type+++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D8D8D12D12D24D24C4○D16D487C2
kernelD487C2D48C48⋊C2Dic24C2×C48C4○D24C2×C16C24C2×C12C16C2×C8C12C2×C6C8C2×C4C4C22C3C1
# reps11211211121222244816

Matrix representation of D487C2 in GL4(𝔽7) generated by

2550
1405
6140
6023
,
0241
4631
3301
2411
,
4052
5003
2543
5516
G:=sub<GL(4,GF(7))| [2,1,6,6,5,4,1,0,5,0,4,2,0,5,0,3],[0,4,3,2,2,6,3,4,4,3,0,1,1,1,1,1],[4,5,2,5,0,0,5,5,5,0,4,1,2,3,3,6] >;

D487C2 in GAP, Magma, Sage, TeX

D_{48}\rtimes_7C_2
% in TeX

G:=Group("D48:7C2");
// GroupNames label

G:=SmallGroup(192,463);
// by ID

G=gap.SmallGroup(192,463);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,142,675,192,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^48=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^24*b>;
// generators/relations

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