Copied to
clipboard

G = C244D6order 288 = 25·32

4th semidirect product of C24 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C244D6, D244S3, D121D6, C87S32, C3⋊S32D8, C32(S3×D8), C324(C2×D8), (C3×D24)⋊9C2, C6.29(S3×D4), D6⋊D61C2, (C3×C24)⋊7C22, C322D82C2, (C3×D12)⋊3C22, C3⋊Dic3.40D4, C2.6(D6⋊D6), (C3×C12).45C23, C12.68(C22×S3), C324C817C22, (C8×C3⋊S3)⋊1C2, C4.66(C2×S32), (C2×C3⋊S3).41D4, (C3×C6).29(C2×D4), (C4×C3⋊S3).66C22, SmallGroup(288,445)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C244D6
C1C3C32C3×C6C3×C12C3×D12D6⋊D6 — C244D6
C32C3×C6C3×C12 — C244D6
C1C2C4C8

Generators and relations for C244D6
 G = < a,b,c | a24=b6=c2=1, bab-1=a-1, cac=a17, cbc=b-1 >

Subgroups: 882 in 163 conjugacy classes, 40 normal (14 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4, C4, C22 [×9], S3 [×10], C6 [×2], C6 [×5], C8, C8, C2×C4, D4 [×6], C23 [×2], C32, Dic3 [×3], C12 [×2], C12, D6 [×15], C2×C6 [×4], C2×C8, D8 [×4], C2×D4 [×2], C3×S3 [×4], C3⋊S3 [×2], C3×C6, C3⋊C8 [×3], C24 [×2], C24, C4×S3 [×3], D12 [×4], C3⋊D4 [×4], C3×D4 [×4], C22×S3 [×4], C2×D8, C3⋊Dic3, C3×C12, S32 [×4], S3×C6 [×4], C2×C3⋊S3, S3×C8 [×3], D24 [×2], D4⋊S3 [×4], C3×D8 [×2], S3×D4 [×4], C324C8, C3×C24, D6⋊S3 [×2], C3×D12 [×4], C4×C3⋊S3, C2×S32 [×2], S3×D8 [×2], C322D8 [×2], C3×D24 [×2], C8×C3⋊S3, D6⋊D6 [×2], C244D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], D8 [×2], C2×D4, C22×S3 [×2], C2×D8, S32, S3×D4 [×2], C2×S32, S3×D8 [×2], D6⋊D6, C244D6

Smallest permutation representation of C244D6
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 41 17 25 9 33)(2 40 18 48 10 32)(3 39 19 47 11 31)(4 38 20 46 12 30)(5 37 21 45 13 29)(6 36 22 44 14 28)(7 35 23 43 15 27)(8 34 24 42 16 26)
(1 5)(2 22)(3 15)(4 8)(6 18)(7 11)(9 21)(10 14)(12 24)(13 17)(16 20)(19 23)(25 45)(26 38)(27 31)(28 48)(29 41)(30 34)(32 44)(33 37)(35 47)(36 40)(39 43)(42 46)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D24A···24H
order1222222233344666666688881212121224···24
size1199121212122242182242424242422181844444···4

36 irreducible representations

dim11111222222444444
type+++++++++++++++
imageC1C2C2C2C2S3D4D4D6D6D8S32S3×D4C2×S32S3×D8D6⋊D6C244D6
kernelC244D6C322D8C3×D24C8×C3⋊S3D6⋊D6D24C3⋊Dic3C2×C3⋊S3C24D12C3⋊S3C8C6C4C3C2C1
# reps12212211244121424

Matrix representation of C244D6 in GL6(𝔽73)

16570000
16160000
0072000
0007200
000001
00007272
,
14430000
43590000
0017200
001000
000010
00007272
,
100000
010000
0072000
0072100
000010
00007272

G:=sub<GL(6,GF(73))| [16,16,0,0,0,0,57,16,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[14,43,0,0,0,0,43,59,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

C244D6 in GAP, Magma, Sage, TeX

C_{24}\rtimes_4D_6
% in TeX

G:=Group("C24:4D6");
// GroupNames label

G:=SmallGroup(288,445);
// by ID

G=gap.SmallGroup(288,445);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,135,142,675,346,80,1356,9414]);
// Polycyclic

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

׿
×
𝔽