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## G = C24.60D6order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C24.60D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C3×C3⋊C16 — C24.60D6
 Lower central C32 — C24.60D6
 Upper central C1 — C8

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 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E 12F 16A ··· 16P 24A ··· 24H 24I 24J 24K 24L 48A ··· 48P order 1 2 2 2 3 3 3 4 4 4 4 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 12 16 ··· 16 24 ··· 24 24 24 24 24 48 ··· 48 size 1 1 9 9 2 2 4 1 1 9 9 2 2 4 1 1 1 1 9 9 9 9 2 2 2 2 4 4 3 ··· 3 2 ··· 2 4 4 4 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + image C1 C2 C2 C4 C4 C8 C8 C16 S3 D6 C4×S3 S3×C8 S3×C16 S32 C6.D6 C12.29D6 C24.60D6 kernel C24.60D6 C3×C3⋊C16 C8×C3⋊S3 C32⋊4C8 C4×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C3⋊S3 C3⋊C16 C24 C12 C6 C3 C8 C4 C2 C1 # reps 1 2 1 2 2 4 4 16 2 2 4 8 16 1 1 2 4

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

 47 0 0 0 0 47 0 0 0 0 96 1 0 0 96 0
,
 12 85 0 0 12 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
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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