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G = C3⋊D48order 288 = 25·32

The semidirect product of C3 and D48 acting via D48/D24=C2

metabelian, supersoluble, monomial

Aliases: C32D48, D241S3, C323D16, C12.9D12, C6.12D24, C24.48D6, C8.4S32, C3⋊C161S3, (C3×C6).7D8, (C3×D24)⋊6C2, C325D84C2, C31(C3⋊D16), C6.1(D4⋊S3), (C3×C12).22D4, (C3×C24).7C22, C4.1(C3⋊D12), C2.4(C3⋊D24), C12.66(C3⋊D4), (C3×C3⋊C16)⋊1C2, SmallGroup(288,194)

Series: Derived Chief Lower central Upper central

C1C3×C24 — C3⋊D48
C1C3C32C3×C6C3×C12C3×C24C3×D24 — C3⋊D48
C32C3×C6C3×C12C3×C24 — C3⋊D48
C1C2C4C8

Generators and relations for C3⋊D48
 G = < a,b,c | a3=b48=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 498 in 67 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C22 [×2], S3 [×5], C6 [×2], C6 [×2], C8, D4 [×2], C32, C12 [×2], C12, D6 [×5], C2×C6, C16, D8 [×2], C3×S3, C3⋊S3, C3×C6, C24 [×2], C24, D12 [×5], C3×D4, D16, C3×C12, S3×C6, C2×C3⋊S3, C3⋊C16, C48, D24, D24 [×3], C3×D8, C3×C24, C3×D12, C12⋊S3, D48, C3⋊D16, C3×C3⋊C16, C3×D24, C325D8, C3⋊D48
Quotients: C1, C2 [×3], C22, S3 [×2], D4, D6 [×2], D8, D12, C3⋊D4, D16, S32, D24, D4⋊S3, C3⋊D12, D48, C3⋊D16, C3⋊D24, C3⋊D48

Smallest permutation representation of C3⋊D48
On 48 points
Generators in S48
(1 33 17)(2 18 34)(3 35 19)(4 20 36)(5 37 21)(6 22 38)(7 39 23)(8 24 40)(9 41 25)(10 26 42)(11 43 27)(12 28 44)(13 45 29)(14 30 46)(15 47 31)(16 32 48)
(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)
(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)

G:=sub<Sym(48)| (1,33,17)(2,18,34)(3,35,19)(4,20,36)(5,37,21)(6,22,38)(7,39,23)(8,24,40)(9,41,25)(10,26,42)(11,43,27)(12,28,44)(13,45,29)(14,30,46)(15,47,31)(16,32,48), (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), (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)>;

G:=Group( (1,33,17)(2,18,34)(3,35,19)(4,20,36)(5,37,21)(6,22,38)(7,39,23)(8,24,40)(9,41,25)(10,26,42)(11,43,27)(12,28,44)(13,45,29)(14,30,46)(15,47,31)(16,32,48), (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), (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) );

G=PermutationGroup([(1,33,17),(2,18,34),(3,35,19),(4,20,36),(5,37,21),(6,22,38),(7,39,23),(8,24,40),(9,41,25),(10,26,42),(11,43,27),(12,28,44),(13,45,29),(14,30,46),(15,47,31),(16,32,48)], [(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)], [(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)])

42 conjugacy classes

class 1 2A2B2C3A3B3C 4 6A6B6C6D6E8A8B12A12B12C12D12E16A16B16C16D24A24B24C24D24E···24J48A···48H
order1222333466666881212121212161616162424242424···2448···48
size112472224222424242222444666622224···46···6

42 irreducible representations

dim11112222222222444444
type+++++++++++++++++++
imageC1C2C2C2S3S3D4D6D8D12C3⋊D4D16D24D48S32D4⋊S3C3⋊D12C3⋊D16C3⋊D24C3⋊D48
kernelC3⋊D48C3×C3⋊C16C3×D24C325D8C3⋊C16D24C3×C12C24C3×C6C12C12C32C6C3C8C6C4C3C2C1
# reps11111112222448111224

Matrix representation of C3⋊D48 in GL4(𝔽97) generated by

09600
19600
0010
0001
,
09600
96000
003284
001319
,
0100
1000
009518
00162
G:=sub<GL(4,GF(97))| [0,1,0,0,96,96,0,0,0,0,1,0,0,0,0,1],[0,96,0,0,96,0,0,0,0,0,32,13,0,0,84,19],[0,1,0,0,1,0,0,0,0,0,95,16,0,0,18,2] >;

C3⋊D48 in GAP, Magma, Sage, TeX

C_3\rtimes D_{48}
% in TeX

G:=Group("C3:D48");
// GroupNames label

G:=SmallGroup(288,194);
// by ID

G=gap.SmallGroup(288,194);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,254,142,675,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^3=b^48=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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