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G = C6.17D24order 288 = 25·32

6th non-split extension by C6 of D24 acting via D24/D12=C2

metabelian, supersoluble, monomial

Aliases: C6.17D24, C12.72D12, C62.19D4, C4⋊Dic31S3, (C3×C6).15D8, C12⋊S36C4, C12.15(C4×S3), C6.6(D4⋊S3), C6.4(D6⋊C4), (C2×C12).77D6, (C2×C6).50D12, (C3×C12).34D4, (C3×C6).9SD16, C6.8(C24⋊C2), C31(C6.D8), C31(C2.D24), C2.2(C3⋊D24), C12.30(C3⋊D4), C326(D4⋊C4), (C6×C12).26C22, C6.1(Q82S3), C4.1(C6.D6), C4.14(C3⋊D12), C2.1(C325SD16), C2.5(C6.D12), C22.13(C3⋊D12), (C6×C3⋊C8)⋊4C2, (C2×C3⋊C8)⋊2S3, (C2×C4).53S32, (C3×C4⋊Dic3)⋊2C2, (C3×C12).24(C2×C4), (C2×C12⋊S3).8C2, (C2×C6).31(C3⋊D4), (C3×C6).31(C22⋊C4), SmallGroup(288,212)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C6.17D24
C1C3C32C3×C6C62C6×C12C3×C4⋊Dic3 — C6.17D24
C32C3×C6C3×C12 — C6.17D24
C1C22C2×C4

Generators and relations for C6.17D24
 G = < a,b,c | a6=b24=c2=1, bab-1=cac=a-1, cbc=a3b-1 >

Subgroups: 738 in 127 conjugacy classes, 40 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4, C22, C22 [×4], S3 [×8], C6 [×6], C6 [×3], C8, C2×C4, C2×C4, D4 [×3], C23, C32, Dic3, C12 [×4], C12 [×3], D6 [×16], C2×C6 [×2], C2×C6, C4⋊C4, C2×C8, C2×D4, C3⋊S3 [×2], C3×C6 [×3], C3⋊C8, C24, D12 [×10], C2×Dic3, C2×C12 [×2], C2×C12 [×2], C22×S3 [×4], D4⋊C4, C3×Dic3, C3×C12 [×2], C2×C3⋊S3 [×4], C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×D12 [×3], C3×C3⋊C8, C6×Dic3, C12⋊S3 [×2], C12⋊S3, C6×C12, C22×C3⋊S3, C6.D8, C2.D24, C6×C3⋊C8, C3×C4⋊Dic3, C2×C12⋊S3, C6.17D24
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], D6 [×2], C22⋊C4, D8, SD16, C4×S3 [×2], D12 [×2], C3⋊D4 [×2], D4⋊C4, S32, C24⋊C2, D24, D6⋊C4 [×2], D4⋊S3, Q82S3, C6.D6, C3⋊D12 [×2], C6.D8, C2.D24, C3⋊D24, C325SD16, C6.D12, C6.17D24

Smallest permutation representation of C6.17D24
On 48 points
Generators in S48
(1 41 17 33 9 25)(2 26 10 34 18 42)(3 43 19 35 11 27)(4 28 12 36 20 44)(5 45 21 37 13 29)(6 30 14 38 22 46)(7 47 23 39 15 31)(8 32 16 40 24 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 46)(3 13)(4 44)(5 11)(6 42)(7 9)(8 40)(10 38)(12 36)(14 34)(16 32)(17 23)(18 30)(19 21)(20 28)(22 26)(24 48)(25 31)(27 29)(33 47)(35 45)(37 43)(39 41)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D6A···6F6G6H6I8A8B8C8D12A12B12C12D12E···12J12K12L12M12N24A···24H
order12222233344446···666688881212121212···121212121224···24
size111136362242212122···2444666622224···4121212126···6

48 irreducible representations

dim111112222222222222244444444
type+++++++++++++++++++++
imageC1C2C2C2C4S3S3D4D4D6D8SD16C4×S3D12C3⋊D4D12C3⋊D4C24⋊C2D24S32D4⋊S3Q82S3C6.D6C3⋊D12C3⋊D12C3⋊D24C325SD16
kernelC6.17D24C6×C3⋊C8C3×C4⋊Dic3C2×C12⋊S3C12⋊S3C2×C3⋊C8C4⋊Dic3C3×C12C62C2×C12C3×C6C3×C6C12C12C12C2×C6C2×C6C6C6C2×C4C6C6C4C4C22C2C2
# reps111141111222422224411111122

Matrix representation of C6.17D24 in GL6(𝔽73)

100000
010000
0072000
0007200
0000721
0000720
,
41480000
3800000
00272700
0046000
000001
000010
,
100000
25720000
001100
0007200
000001
000010

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[41,38,0,0,0,0,48,0,0,0,0,0,0,0,27,46,0,0,0,0,27,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,25,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C6.17D24 in GAP, Magma, Sage, TeX

C_6._{17}D_{24}
% in TeX

G:=Group("C6.17D24");
// GroupNames label

G:=SmallGroup(288,212);
// by ID

G=gap.SmallGroup(288,212);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,204,675,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^6=b^24=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=a^3*b^-1>;
// generators/relations

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