Copied to
clipboard

G = C249D6order 288 = 25·32

9th semidirect product of C24 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C249D6, Dic61D6, D12.1D6, C88S32, C24⋊C25S3, C3⋊S33SD16, C32(S3×SD16), C6.28(S3×D4), C325(C2×SD16), (C3×C24)⋊15C22, D6⋊D6.3C2, C3⋊Dic3.39D4, Dic3.D61C2, Dic6⋊S31C2, C2.5(D6⋊D6), C12.67(C22×S3), (C3×C12).44C23, (C3×Dic6)⋊3C22, (C3×D12).3C22, C324C816C22, (C8×C3⋊S3)⋊6C2, C4.65(C2×S32), (C2×C3⋊S3).40D4, (C3×C6).28(C2×D4), (C3×C24⋊C2)⋊10C2, (C4×C3⋊S3).65C22, SmallGroup(288,444)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C249D6
C1C3C32C3×C6C3×C12C3×D12D6⋊D6 — C249D6
C32C3×C6C3×C12 — C249D6
C1C2C4C8

Generators and relations for C249D6
 G = < a,b,c | a24=b6=c2=1, bab-1=a11, cac=a17, cbc=b-1 >

Subgroups: 690 in 147 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×3], C22 [×5], S3 [×8], C6 [×2], C6 [×3], C8, C8, C2×C4 [×2], D4 [×3], Q8 [×3], C23, C32, Dic3 [×5], C12 [×2], C12 [×3], D6 [×9], C2×C6 [×2], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8 [×3], C24 [×2], C24, Dic6 [×2], Dic6 [×2], C4×S3 [×5], D12 [×2], C3⋊D4 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3 [×2], C2×SD16, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S32 [×2], S3×C6 [×2], C2×C3⋊S3, S3×C8 [×3], C24⋊C2 [×2], D4.S3 [×2], Q82S3 [×2], C3×SD16 [×2], S3×D4 [×2], S3×Q8 [×2], C324C8, C3×C24, C6.D6, D6⋊S3, C322Q8, C3×Dic6 [×2], C3×D12 [×2], C4×C3⋊S3, C2×S32, S3×SD16 [×2], Dic6⋊S3 [×2], C3×C24⋊C2 [×2], C8×C3⋊S3, Dic3.D6, D6⋊D6, C249D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], SD16 [×2], C2×D4, C22×S3 [×2], C2×SD16, S32, S3×D4 [×2], C2×S32, S3×SD16 [×2], D6⋊D6, C249D6

Smallest permutation representation of C249D6
On 48 points
Generators in S48
(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 32 17 40 9 48)(2 43 18 27 10 35)(3 30 19 38 11 46)(4 41 20 25 12 33)(5 28 21 36 13 44)(6 39 22 47 14 31)(7 26 23 34 15 42)(8 37 24 45 16 29)
(1 17)(2 10)(4 20)(5 13)(7 23)(8 16)(11 19)(14 22)(25 33)(27 43)(28 36)(30 46)(31 39)(34 42)(37 45)(40 48)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F24A···24H
order122222333444466666888812121212121224···24
size1199121222421212182242424221818444424244···4

36 irreducible representations

dim1111112222222444444
type+++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6SD16S32S3×D4C2×S32S3×SD16D6⋊D6C249D6
kernelC249D6Dic6⋊S3C3×C24⋊C2C8×C3⋊S3Dic3.D6D6⋊D6C24⋊C2C3⋊Dic3C2×C3⋊S3C24Dic6D12C3⋊S3C8C6C4C3C2C1
# reps1221112112224121424

Matrix representation of C249D6 in GL6(𝔽73)

63550000
0220000
0072000
0007200
0000721
0000720
,
5100000
56680000
0072100
0072000
000001
000010
,
7200000
0720000
0072000
0072100
000001
000010

G:=sub<GL(6,GF(73))| [63,0,0,0,0,0,55,22,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[5,56,0,0,0,0,10,68,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C249D6 in GAP, Magma, Sage, TeX

C_{24}\rtimes_9D_6
% in TeX

G:=Group("C24:9D6");
// GroupNames label

G:=SmallGroup(288,444);
// by ID

G=gap.SmallGroup(288,444);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,135,58,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^11,c*a*c=a^17,c*b*c=b^-1>;
// generators/relations

׿
×
𝔽