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G = C247D6order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: C247D6, (C3×Q8)⋊8D6, C24⋊S33C2, C35(Q83D6), (C3×SD16)⋊1S3, (C3×D4).17D6, C6.122(S3×D4), C325D810C2, C327D87C2, SD161(C3⋊S3), (C3×C24)⋊11C22, C3⋊Dic3.66D4, C12⋊S38C22, C3211SD166C2, C12.26D63C2, C3221(C8⋊C22), (C3×C12).95C23, C12.91(C22×S3), (C32×SD16)⋊3C2, (Q8×C32)⋊8C22, C324C811C22, (D4×C32).18C22, C83(C2×C3⋊S3), (D4×C3⋊S3)⋊5C2, Q83(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

C1C3×C12 — C247D6
C1C3C32C3×C6C3×C12C4×C3⋊S3D4×C3⋊S3 — C247D6
C32C3×C6C3×C12 — C247D6
C1C2C4SD16

Generators and relations for C247D6
 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 [×4], C3 [×4], C4, C4 [×2], C22 [×6], S3 [×12], C6 [×4], C6 [×4], C8, C8, C2×C4 [×2], D4, D4 [×4], Q8, C23, C32, Dic3 [×4], C12 [×4], C12 [×4], D6 [×20], C2×C6 [×4], M4(2), D8 [×2], SD16, SD16, C2×D4, C4○D4, C3⋊S3 [×3], C3×C6, C3×C6, C3⋊C8 [×4], C24 [×4], C4×S3 [×8], D12 [×12], C3⋊D4 [×4], C3×D4 [×4], C3×Q8 [×4], C22×S3 [×4], C8⋊C22, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3 [×4], C62, C8⋊S3 [×4], D24 [×4], D4⋊S3 [×4], Q82S3 [×4], C3×SD16 [×4], S3×D4 [×4], Q83S3 [×4], C324C8, C3×C24, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3 [×2], C12⋊S3, C327D4, D4×C32, Q8×C32, C22×C3⋊S3, Q83D6 [×4], C24⋊S3, C325D8, C327D8, C3211SD16, C32×SD16, D4×C3⋊S3, C12.26D6, C247D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C22×S3 [×4], C8⋊C22, C2×C3⋊S3 [×3], S3×D4 [×4], C22×C3⋊S3, Q83D6 [×4], D4×C3⋊S3, C247D6

Smallest permutation representation of C247D6
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 60 39)(2 55 40 20 61 34)(3 50 41 15 62 29)(4 69 42 10 63 48)(5 64 43)(6 59 44 24 65 38)(7 54 45 19 66 33)(8 49 46 14 67 28)(9 68 47)(11 58 25 23 70 37)(12 53 26 18 71 32)(13 72 27)(16 57 30 22 51 36)(17 52 31)(21 56 35)
(1 27)(2 26)(3 25)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(49 59)(50 58)(51 57)(52 56)(53 55)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)

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

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

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

39 conjugacy classes

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

39 irreducible representations

dim11111111222222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6C8⋊C22S3×D4Q83D6
kernelC247D6C24⋊S3C325D8C327D8C3211SD16C32×SD16D4×C3⋊S3C12.26D6C3×SD16C3⋊Dic3C2×C3⋊S3C24C3×D4C3×Q8C32C6C3
# reps11111111411444148

Matrix representation of C247D6 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] >;

C247D6 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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