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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, (C3xQ8):8D6, C24:S3:3C2, C3:5(Q8:3D6), (C3xSD16):1S3, (C3xD4).17D6, C6.122(S3xD4), C32:5D8:10C2, C32:7D8:7C2, SD16:1(C3:S3), (C3xC24):11C22, C3:Dic3.66D4, C12:S3:8C22, C32:11SD16:6C2, C12.26D6:3C2, C32:21(C8:C22), (C3xC12).95C23, C12.91(C22xS3), (C32xSD16):3C2, (Q8xC32):8C22, C32:4C8:11C22, (D4xC32).18C22, C8:3(C2xC3:S3), (D4xC3:S3):5C2, Q8:3(C2xC3:S3), D4.3(C2xC3:S3), C2.19(D4xC3:S3), (C2xC3:S3).66D4, C4.5(C22xC3:S3), (C3xC6).243(C2xD4), (C4xC3:S3).24C22, SmallGroup(288,771)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC12 — C24:7D6
Chief series
C1 — C3 — C32 — C3xC6 — C3xC12 — C4xC3:S3 — D4xC3:S3 — C24:7D6
Lower central
C32 — C3xC6 — C3xC12 — 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, C2xC4, D4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2xC6, M4(2), D8, SD16, SD16, C2xD4, C4oD4, C3:S3, C3xC6, C3xC6, C3:C8, C24, C4xS3, D12, C3:D4, C3xD4, C3xQ8, C22xS3, C8:C22, C3:Dic3, C3xC12, C3xC12, C2xC3:S3, C2xC3:S3, C62, C8:S3, D24, D4:S3, Q8:2S3, C3xSD16, S3xD4, Q8:3S3, C32:4C8, C3xC24, C4xC3:S3, C4xC3:S3, C12:S3, C12:S3, C32:7D4, D4xC32, Q8xC32, C22xC3:S3, Q8:3D6, C24:S3, C32:5D8, C32:7D8, C32:11SD16, C32xSD16, D4xC3:S3, C12.26D6, C24:7D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:S3, C22xS3, C8:C22, C2xC3:S3, S3xD4, C22xC3:S3, Q8:3D6, D4xC3: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:C22S3xD4Q8:3D6
kernelC24:7D6C24:S3C32:5D8C32:7D8C32:11SD16C32xSD16D4xC3:S3C12.26D6C3xSD16C3:Dic3C2xC3:S3C24C3xD4C3xQ8C32C6C3
# 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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