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G = C24.23D6order 288 = 25·32

23rd non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C24.23D6, Dic124S3, Dic6.1D6, C8.16S32, C3⋊S32Q16, C32(S3×Q16), C6.31(S3×D4), C324(C2×Q16), (C3×Dic12)⋊9C2, C3⋊Dic3.41D4, C322Q162C2, C2.8(D6⋊D6), (C3×C12).50C23, (C3×C24).22C22, C12.70(C22×S3), Dic3.D6.3C2, (C3×Dic6).4C22, C324C8.21C22, C4.68(C2×S32), (C8×C3⋊S3).1C2, (C2×C3⋊S3).42D4, (C3×C6).34(C2×D4), (C4×C3⋊S3).67C22, SmallGroup(288,450)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C24.23D6
C1C3C32C3×C6C3×C12C3×Dic6Dic3.D6 — C24.23D6
C32C3×C6C3×C12 — C24.23D6
C1C2C4C8

Generators and relations for C24.23D6
 G = < a,b,c | a24=c2=1, b6=a12, bab-1=a-1, cac=a17, cbc=b5 >

Subgroups: 498 in 131 conjugacy classes, 40 normal (14 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×5], C22, S3 [×6], C6 [×2], C6, C8, C8, C2×C4 [×3], Q8 [×6], C32, Dic3 [×7], C12 [×2], C12 [×5], D6 [×3], C2×C8, Q16 [×4], C2×Q8 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8 [×3], C24 [×2], C24, Dic6 [×4], Dic6 [×4], C4×S3 [×7], C3×Q8 [×4], C2×Q16, C3×Dic3 [×4], C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×C8 [×3], Dic12 [×2], C3⋊Q16 [×4], C3×Q16 [×2], S3×Q8 [×4], C324C8, C3×C24, C6.D6 [×2], C322Q8 [×2], C3×Dic6 [×4], C4×C3⋊S3, S3×Q16 [×2], C322Q16 [×2], C3×Dic12 [×2], C8×C3⋊S3, Dic3.D6 [×2], C24.23D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], Q16 [×2], C2×D4, C22×S3 [×2], C2×Q16, S32, S3×D4 [×2], C2×S32, S3×Q16 [×2], D6⋊D6, C24.23D6

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

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

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

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

36 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F6A6B6C8A8B8C8D12A12B12C12D12E12F12G12H24A···24H
order12223334444446668888121212121212121224···24
size1199224212121212182242218184444242424244···4

36 irreducible representations

dim11111222222444444
type++++++++++-+++-
imageC1C2C2C2C2S3D4D4D6D6Q16S32S3×D4C2×S32S3×Q16D6⋊D6C24.23D6
kernelC24.23D6C322Q16C3×Dic12C8×C3⋊S3Dic3.D6Dic12C3⋊Dic3C2×C3⋊S3C24Dic6C3⋊S3C8C6C4C3C2C1
# reps12212211244121424

Matrix representation of C24.23D6 in GL6(𝔽73)

0410000
16410000
0072000
0007200
000001
00007272
,
2310000
54710000
0007200
001100
000010
00007272
,
7200000
0720000
000100
001000
000010
00007272

G:=sub<GL(6,GF(73))| [0,16,0,0,0,0,41,41,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[2,54,0,0,0,0,31,71,0,0,0,0,0,0,0,1,0,0,0,0,72,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

C24.23D6 in GAP, Magma, Sage, TeX

C_{24}._{23}D_6
% in TeX

G:=Group("C24.23D6");
// GroupNames label

G:=SmallGroup(288,450);
// by ID

G=gap.SmallGroup(288,450);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,135,142,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=c^2=1,b^6=a^12,b*a*b^-1=a^-1,c*a*c=a^17,c*b*c=b^5>;
// generators/relations

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