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G = C24.60D6order 288 = 25·32

13rd non-split extension by C24 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, A-group

Aliases: C24.60D6, C3⋊C166S3, C8.21S32, C3⋊S32C16, C31(S3×C16), C6.1(S3×C8), C325(C2×C16), C12.43(C4×S3), C3⋊Dic3.3C8, C324C8.4C4, (C3×C24).42C22, C4.11(C6.D6), C2.1(C12.29D6), (C3×C3⋊C16)⋊8C2, (C8×C3⋊S3).3C2, (C4×C3⋊S3).7C4, (C2×C3⋊S3).3C8, (C3×C6).13(C2×C8), (C3×C12).78(C2×C4), SmallGroup(288,190)

Series: Derived Chief Lower central Upper central

C1C32 — C24.60D6
C1C3C32C3×C6C3×C12C3×C24C3×C3⋊C16 — C24.60D6
C32 — C24.60D6
C1C8

Generators and relations for C24.60D6
 G = < a,b,c | a24=1, b6=a3, c2=a18, bab-1=cac-1=a17, cbc-1=a6b5 >

Subgroups: 210 in 67 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4, C22, S3 [×6], C6 [×2], C6, C8, C8, C2×C4, C32, Dic3 [×3], C12 [×2], C12, D6 [×3], C16 [×2], C2×C8, C3⋊S3 [×2], C3×C6, C3⋊C8 [×3], C24 [×2], C24, C4×S3 [×3], C2×C16, C3⋊Dic3, C3×C12, C2×C3⋊S3, C3⋊C16 [×2], C48 [×2], S3×C8 [×3], C324C8, C3×C24, C4×C3⋊S3, S3×C16 [×2], C3×C3⋊C16 [×2], C8×C3⋊S3, C24.60D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C8 [×2], C2×C4, D6 [×2], C16 [×2], C2×C8, C4×S3 [×2], C2×C16, S32, S3×C8 [×2], C6.D6, S3×C16 [×2], C12.29D6, C24.60D6

Smallest permutation representation of C24.60D6
On 48 points
Generators in S48
(1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47)(2 36 22 8 42 28 14 48 34 20 6 40 26 12 46 32 18 4 38 24 10 44 30 16)
(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 3 37 39 25 27 13 15)(2 20 38 8 26 44 14 32)(4 6 40 42 28 30 16 18)(5 23 41 11 29 47 17 35)(7 9 43 45 31 33 19 21)(10 12 46 48 34 36 22 24)

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

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

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

72 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D6A6B6C8A8B8C8D8E8F8G8H12A12B12C12D12E12F16A···16P24A···24H24I24J24K24L48A···48P
order122233344446668888888812121212121216···1624···242424242448···48
size11992241199224111199992222443···32···244446···6

72 irreducible representations

dim11111111222224444
type+++++++
imageC1C2C2C4C4C8C8C16S3D6C4×S3S3×C8S3×C16S32C6.D6C12.29D6C24.60D6
kernelC24.60D6C3×C3⋊C16C8×C3⋊S3C324C8C4×C3⋊S3C3⋊Dic3C2×C3⋊S3C3⋊S3C3⋊C16C24C12C6C3C8C4C2C1
# reps1212244162248161124

Matrix representation of C24.60D6 in GL4(𝔽97) generated by

47000
04700
00961
00960
,
128500
12000
0001
0010
,
04700
47000
0001
0010
G:=sub<GL(4,GF(97))| [47,0,0,0,0,47,0,0,0,0,96,96,0,0,1,0],[12,12,0,0,85,0,0,0,0,0,0,1,0,0,1,0],[0,47,0,0,47,0,0,0,0,0,0,1,0,0,1,0] >;

C24.60D6 in GAP, Magma, Sage, TeX

C_{24}._{60}D_6
% in TeX

G:=Group("C24.60D6");
// GroupNames label

G:=SmallGroup(288,190);
// by ID

G=gap.SmallGroup(288,190);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,36,58,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=1,b^6=a^3,c^2=a^18,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^6*b^5>;
// generators/relations

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