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G = C24⋊7D6  order 288 = 25·32

7th semidirect product of C24 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C24⋊7D6, (C3×Q8)⋊8D6, C24⋊S3⋊3C2, C3⋊5(Q8⋊3D6), (C3×SD16)⋊1S3, (C3×D4).17D6, C6.122(S3×D4), C32⋊5D8⋊10C2, C32⋊7D8⋊7C2, SD16⋊1(C3⋊S3), (C3×C24)⋊11C22, C3⋊Dic3.66D4, C12⋊S3⋊8C22, C32⋊11SD16⋊6C2, C12.26D6⋊3C2, C32⋊21(C8⋊C22), (C3×C12).95C23, C12.91(C22×S3), (C32×SD16)⋊3C2, (Q8×C32)⋊8C22, C32⋊4C8⋊11C22, (D4×C32).18C22, C8⋊3(C2×C3⋊S3), (D4×C3⋊S3)⋊5C2, Q8⋊3(C2×C3⋊S3), D4.3(C2×C3⋊S3), C2.19(D4×C3⋊S3), (C2×C3⋊S3).66D4, C4.5(C22×C3⋊S3), (C3×C6).243(C2×D4), (C4×C3⋊S3).24C22, SmallGroup(288,771)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — C24⋊7D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C4×C3⋊S3 — D4×C3⋊S3 — C24⋊7D6
Lower central
C32 — C3×C6 — C3×C12 — C24⋊7D6
Upper central
C1 — C2 — C4 — SD16

Generators and relations for C24⋊7D6
 G = < a,b,c | a24=b6=c2=1, bab-1=a19, cac=a-1, cbc=b-1 >

Subgroups: 1036 in 204 conjugacy classes, 55 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, M4(2), D8, SD16, SD16, C2×D4, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C8⋊C22, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C8⋊S3, D24, D4⋊S3, Q8⋊2S3, C3×SD16, S3×D4, Q8⋊3S3, C32⋊4C8, C3×C24, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C12⋊S3, C32⋊7D4, D4×C32, Q8×C32, C22×C3⋊S3, Q8⋊3D6, C24⋊S3, C32⋊5D8, C32⋊7D8, C32⋊11SD16, C32×SD16, D4×C3⋊S3, C12.26D6, C24⋊7D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C22×S3, C8⋊C22, C2×C3⋊S3, S3×D4, C22×C3⋊S3, Q8⋊3D6, D4×C3⋊S3, C24⋊7D6

Smallest permutation representation of C24⋊7D6
►On 72 points
Generators in S72
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)

(1 50 37)(2 69 38 20 51 32)(3 64 39 15 52 27)(4 59 40 10 53 46)(5 54 41)(6 49 42 24 55 36)(7 68 43 19 56 31)(8 63 44 14 57 26)(9 58 45)(11 72 47 23 60 35)(12 67 48 18 61 30)(13 62 25)(16 71 28 22 65 34)(17 66 29)(21 70 33)

(1 25)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(64 72)(65 71)(66 70)(67 69)
 
G:=sub<Sym(72)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,50,37)(2,69,38,20,51,32)(3,64,39,15,52,27)(4,59,40,10,53,46)(5,54,41)(6,49,42,24,55,36)(7,68,43,19,56,31)(8,63,44,14,57,26)(9,58,45)(11,72,47,23,60,35)(12,67,48,18,61,30)(13,62,25)(16,71,28,22,65,34)(17,66,29)(21,70,33), (1,25)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(64,72)(65,71)(66,70)(67,69)>;
 
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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,50,37)(2,69,38,20,51,32)(3,64,39,15,52,27)(4,59,40,10,53,46)(5,54,41)(6,49,42,24,55,36)(7,68,43,19,56,31)(8,63,44,14,57,26)(9,58,45)(11,72,47,23,60,35)(12,67,48,18,61,30)(13,62,25)(16,71,28,22,65,34)(17,66,29)(21,70,33), (1,25)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(64,72)(65,71)(66,70)(67,69) );
 
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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,50,37),(2,69,38,20,51,32),(3,64,39,15,52,27),(4,59,40,10,53,46),(5,54,41),(6,49,42,24,55,36),(7,68,43,19,56,31),(8,63,44,14,57,26),(9,58,45),(11,72,47,23,60,35),(12,67,48,18,61,30),(13,62,25),(16,71,28,22,65,34),(17,66,29),(21,70,33)], [(1,25),(2,48),(3,47),(4,46),(5,45),(6,44),(7,43),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,35),(16,34),(17,33),(18,32),(19,31),(20,30),(21,29),(22,28),(23,27),(24,26),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(64,72),(65,71),(66,70),(67,69)]])
 

39 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D4A4B4C6A6B6C6D6E6F6G6H8A8B12A12B12C12D12E12F12G12H24A···24H
order12222233334446666666688121212121212121224···24
size1141836362222241822228888436444488884···4

39 irreducible representations

dim11111111222222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6C8⋊C22S3×D4Q8⋊3D6
kernelC24⋊7D6C24⋊S3C32⋊5D8C32⋊7D8C32⋊11SD16C32×SD16D4×C3⋊S3C12.26D6C3×SD16C3⋊Dic3C2×C3⋊S3C24C3×D4C3×Q8C32C6C3
# reps11111111411444148

Matrix representation of C24⋊7D6 ►in GL8(ℤ)

-1-1000000
10000000
00-100000
000-10000
00000001
00000010
0000-1000
00000100
,
-10000000
0-1000000
00110000
00-100000
00001000
00000-100
00000001
00000010
,
-10000000
11000000
000-10000
00-100000
0000-1000
00000100
00000001
00000010

G:=sub<GL(8,Integers())| [-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C24⋊7D6 in GAP, Magma, Sage, TeX 

C_{24}\rtimes_7D_6
 
% in TeX
 
G:=Group("C24:7D6");
 
// GroupNames label
 
G:=SmallGroup(288,771);
 
// by ID
 
G=gap.SmallGroup(288,771);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,100,346,185,80,2693,9414]);
 
// Polycyclic
 
G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

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