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G = C2.D48order 192 = 26·3

2nd central extension by C2 of D48

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D241C4, C2.2D48, C6.5D16, C24.79D4, C6.3SD32, C12.23SD16, C22.10D24, (C2×C48)⋊3C2, (C2×C16)⋊3S3, C8.19(C4×S3), C241C41C2, (C2×C6).16D8, C32(C2.D16), C24.49(C2×C4), (C2×D24).1C2, (C2×C8).299D6, (C2×C4).74D12, C4.2(C24⋊C2), C4.16(D6⋊C4), (C2×C12).372D4, C8.36(C3⋊D4), C2.3(C48⋊C2), C2.7(C2.D24), C6.15(D4⋊C4), C12.40(C22⋊C4), (C2×C24).372C22, SmallGroup(192,68)

Series: Derived Chief Lower central Upper central

C1C24 — C2.D48
C1C3C6C12C24C2×C24C2×D24 — C2.D48
C3C6C12C24 — C2.D48
C1C22C2×C4C2×C8C2×C16

Generators and relations for C2.D48
 G = < a,b,c | a2=b48=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >

Subgroups: 328 in 66 conjugacy classes, 29 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], S3 [×2], C6 [×3], C8 [×2], C2×C4, C2×C4, D4 [×3], C23, Dic3, C12 [×2], D6 [×4], C2×C6, C16, C4⋊C4, C2×C8, D8 [×3], C2×D4, C24 [×2], D12 [×3], C2×Dic3, C2×C12, C22×S3, C2.D8, C2×C16, C2×D8, C48, D24 [×2], D24, C4⋊Dic3, C2×C24, C2×D12, C2.D16, C241C4, C2×C48, C2×D24, C2.D48
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, D16, SD32, C24⋊C2, D24, D6⋊C4, C2.D16, D48, C48⋊C2, C2.D24, C2.D48

Smallest permutation representation of C2.D48
On 96 points
Generators in S96
(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)
(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 72 73 48)(2 47 74 71)(3 70 75 46)(4 45 76 69)(5 68 77 44)(6 43 78 67)(7 66 79 42)(8 41 80 65)(9 64 81 40)(10 39 82 63)(11 62 83 38)(12 37 84 61)(13 60 85 36)(14 35 86 59)(15 58 87 34)(16 33 88 57)(17 56 89 32)(18 31 90 55)(19 54 91 30)(20 29 92 53)(21 52 93 28)(22 27 94 51)(23 50 95 26)(24 25 96 49)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D6A6B6C8A8B8C8D12A12B12C12D16A···16H24A···24H48A···48P
order1222223444466688881212121216···1624···2448···48
size111124242222424222222222222···22···22···2

54 irreducible representations

dim11111222222222222222
type+++++++++++++
imageC1C2C2C2C4S3D4D4D6SD16D8C4×S3C3⋊D4D12D16SD32C24⋊C2D24D48C48⋊C2
kernelC2.D48C241C4C2×C48C2×D24D24C2×C16C24C2×C12C2×C8C12C2×C6C8C8C2×C4C6C6C4C22C2C2
# reps11114111122222444488

Matrix representation of C2.D48 in GL3(𝔽97) generated by

9600
0960
0096
,
7500
04345
05295
,
2200
04345
0254
G:=sub<GL(3,GF(97))| [96,0,0,0,96,0,0,0,96],[75,0,0,0,43,52,0,45,95],[22,0,0,0,43,2,0,45,54] >;

C2.D48 in GAP, Magma, Sage, TeX

C_2.D_{48}
% in TeX

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

G:=SmallGroup(192,68);
// by ID

G=gap.SmallGroup(192,68);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,204,422,268,1684,102,6278]);
// Polycyclic

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

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