Copied to
clipboard

G = D48:7C2order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D48:7C2, C4.20D24, C8.13D12, C24.63D4, C16.16D6, C12.38D8, Dic24:7C2, C22.1D24, C24.57C23, C48.18C22, D24.7C22, Dic12.7C22, (C2xC16):6S3, (C2xC48):10C2, C4oD24:1C2, C3:1(C4oD16), C48:C2:7C2, C6.11(C2xD8), (C2xC6).20D8, C2.13(C2xD24), (C2xC4).85D12, C4.38(C2xD12), (C2xC8).313D6, C12.281(C2xD4), (C2xC12).395D4, C8.47(C22xS3), (C2xC24).385C22, SmallGroup(192,463)

Series: Derived Chief Lower central Upper central

C1C24 — D48:7C2
C1C3C6C12C24D24C4oD24 — D48:7C2
C3C6C12C24 — D48:7C2
C1C4C2xC4C2xC8C2xC16

Generators and relations for D48:7C2
 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, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, Dic3, C12, D6, C2xC6, C16, C2xC8, D8, SD16, Q16, C4oD4, C24, Dic6, C4xS3, D12, C3:D4, C2xC12, C2xC16, D16, SD32, Q32, C4oD8, C48, C24:C2, D24, Dic12, C2xC24, C4oD12, C4oD16, D48, C48:C2, Dic24, C2xC48, C4oD24, D48:7C2
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, D12, C22xS3, C2xD8, D24, C2xD12, C4oD16, C2xD24, D48:7C2

Smallest permutation representation of D48:7C2
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 93)(50 92)(51 91)(52 90)(53 89)(54 88)(55 87)(56 86)(57 85)(58 84)(59 83)(60 82)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(94 96)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)

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

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

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

54 conjugacy classes

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

54 irreducible representations

dim1111112222222222222
type+++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D8D8D12D12D24D24C4oD16D48:7C2
kernelD48:7C2D48C48:C2Dic24C2xC48C4oD24C2xC16C24C2xC12C16C2xC8C12C2xC6C8C2xC4C4C22C3C1
# reps11211211121222244816

Matrix representation of D48:7C2 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] >;

D48:7C2 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

׿
x
:
Z
F
o
wr
Q
<