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G = C24.62D6order 288 = 25·32

15th non-split extension by C24 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: C24.62D6, C325M5(2), C3⋊C165S3, C8.23S32, C6.2(S3×C8), C12.44(C4×S3), C31(D6.C8), C3⋊Dic3.4C8, C324C8.5C4, (C3×C24).44C22, C4.12(C6.D6), C2.3(C12.29D6), (C3×C3⋊C16)⋊10C2, (C8×C3⋊S3).4C2, (C2×C3⋊S3).4C8, (C4×C3⋊S3).8C4, (C3×C6).15(C2×C8), (C3×C12).80(C2×C4), SmallGroup(288,192)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C24.62D6
C1C3C32C3×C6C3×C12C3×C24C3×C3⋊C16 — C24.62D6
C32C3×C6 — C24.62D6
C1C8

Generators and relations for C24.62D6
 G = < a,b,c | a24=1, b6=a3, c2=a6, bab-1=cac-1=a17, cbc-1=a18b5 >

Subgroups: 210 in 63 conjugacy classes, 26 normal (14 characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, S3 [×4], C6 [×2], C6, C8, C8, C2×C4, C32, Dic3 [×3], C12 [×2], C12, D6 [×3], C16 [×2], C2×C8, C3⋊S3, C3×C6, C3⋊C8 [×3], C24 [×2], C24, C4×S3 [×3], M5(2), C3⋊Dic3, C3×C12, C2×C3⋊S3, C3⋊C16 [×2], C48 [×2], S3×C8 [×3], C324C8, C3×C24, C4×C3⋊S3, D6.C8 [×2], C3×C3⋊C16 [×2], C8×C3⋊S3, C24.62D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C8 [×2], C2×C4, D6 [×2], C2×C8, C4×S3 [×2], M5(2), S32, S3×C8 [×2], C6.D6, D6.C8 [×2], C12.29D6, C24.62D6

Smallest permutation representation of C24.62D6
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 39 13 3 25 15 37 27)(2 32 14 44 26 8 38 20)(4 18 16 30 28 42 40 6)(5 11 17 23 29 35 41 47)(7 45 19 9 31 21 43 33)(10 24 22 36 34 48 46 12)

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

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

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

60 conjugacy classes

class 1 2A2B3A3B3C4A4B4C6A6B6C8A8B8C8D8E8F12A12B12C12D12E12F16A···16H24A···24H24I24J24K24L48A···48P
order12233344466688888812121212121216···1624···242424242448···48
size11182241118224111118182222446···62···244446···6

60 irreducible representations

dim11111112222224444
type+++++++
imageC1C2C2C4C4C8C8S3D6C4×S3M5(2)S3×C8D6.C8S32C6.D6C12.29D6C24.62D6
kernelC24.62D6C3×C3⋊C16C8×C3⋊S3C324C8C4×C3⋊S3C3⋊Dic3C2×C3⋊S3C3⋊C16C24C12C32C6C3C8C4C2C1
# reps121224422448161124

Matrix representation of C24.62D6 in GL6(𝔽97)

0960000
110000
0050000
0005000
0000750
0000075
,
2200000
75750000
00504700
00494700
00005047
0000500
,
9600000
110000
0064000
00313300
0000022
0000220

G:=sub<GL(6,GF(97))| [0,1,0,0,0,0,96,1,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,75,0,0,0,0,0,0,75],[22,75,0,0,0,0,0,75,0,0,0,0,0,0,50,49,0,0,0,0,47,47,0,0,0,0,0,0,50,50,0,0,0,0,47,0],[96,1,0,0,0,0,0,1,0,0,0,0,0,0,64,31,0,0,0,0,0,33,0,0,0,0,0,0,0,22,0,0,0,0,22,0] >;

C24.62D6 in GAP, Magma, Sage, TeX

C_{24}._{62}D_6
% in TeX

G:=Group("C24.62D6");
// GroupNames label

G:=SmallGroup(288,192);
// by ID

G=gap.SmallGroup(288,192);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,36,58,80,1356,9414]);
// Polycyclic

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

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