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G = D249C4order 192 = 26·3

9th semidirect product of D24 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D249C4, C84(C4×S3), C248(C2×C4), C4.Q85S3, D126(C2×C4), C24⋊C43C2, (C2×C8).63D6, C6.51(C4×D4), C33(D8⋊C4), C4⋊C4.163D6, Dic35D46C2, (C2×D24).13C2, C6.D818C2, C2.6(Q83D6), C22.86(S3×D4), C12.34(C4○D4), C6.72(C8⋊C22), C12.45(C22×C4), C4.6(Q83S3), (C2×C24).112C22, (C2×C12).285C23, (C2×Dic3).165D4, (C2×D12).77C22, C2.11(Dic35D4), (C4×Dic3).31C22, C4.42(S3×C2×C4), (C3×C4.Q8)⋊5C2, (C2×C6).290(C2×D4), (C2×C3⋊C8).62C22, (C3×C4⋊C4).78C22, (C2×C4).388(C22×S3), SmallGroup(192,428)

Series: Derived Chief Lower central Upper central

C1C12 — D249C4
C1C3C6C2×C6C2×C12C2×D12C2×D24 — D249C4
C3C6C12 — D249C4
C1C22C2×C4C4.Q8

Generators and relations for D249C4
 G = < a,b,c | a24=b2=c4=1, bab=a-1, cac-1=a19, cbc-1=a18b >

Subgroups: 448 in 132 conjugacy classes, 49 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×4], C22, C22 [×8], S3 [×4], C6, C6 [×2], C8 [×2], C8, C2×C4, C2×C4 [×8], D4 [×6], C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×8], C2×C6, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, D8 [×4], C22×C4 [×2], C2×D4 [×2], C3⋊C8, C24 [×2], C4×S3 [×4], D12 [×4], D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3 [×2], C8⋊C4, D4⋊C4 [×2], C4.Q8, C4×D4 [×2], C2×D8, D24 [×4], C2×C3⋊C8, C4×Dic3, D6⋊C4 [×2], C3×C4⋊C4 [×2], C2×C24, S3×C2×C4 [×2], C2×D12 [×2], D8⋊C4, C6.D8 [×2], C24⋊C4, C3×C4.Q8, Dic35D4 [×2], C2×D24, D249C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C22×S3, C4×D4, C8⋊C22 [×2], S3×C2×C4, S3×D4, Q83S3, D8⋊C4, Dic35D4, Q83D6 [×2], D249C4

Smallest permutation representation of D249C4
On 96 points
Generators in S96
(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)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(49 51)(52 72)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(61 63)(73 87)(74 86)(75 85)(76 84)(77 83)(78 82)(79 81)(88 96)(89 95)(90 94)(91 93)
(1 82 55 27)(2 77 56 46)(3 96 57 41)(4 91 58 36)(5 86 59 31)(6 81 60 26)(7 76 61 45)(8 95 62 40)(9 90 63 35)(10 85 64 30)(11 80 65 25)(12 75 66 44)(13 94 67 39)(14 89 68 34)(15 84 69 29)(16 79 70 48)(17 74 71 43)(18 93 72 38)(19 88 49 33)(20 83 50 28)(21 78 51 47)(22 73 52 42)(23 92 53 37)(24 87 54 32)

G:=sub<Sym(96)| (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(49,51)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(61,63)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81)(88,96)(89,95)(90,94)(91,93), (1,82,55,27)(2,77,56,46)(3,96,57,41)(4,91,58,36)(5,86,59,31)(6,81,60,26)(7,76,61,45)(8,95,62,40)(9,90,63,35)(10,85,64,30)(11,80,65,25)(12,75,66,44)(13,94,67,39)(14,89,68,34)(15,84,69,29)(16,79,70,48)(17,74,71,43)(18,93,72,38)(19,88,49,33)(20,83,50,28)(21,78,51,47)(22,73,52,42)(23,92,53,37)(24,87,54,32)>;

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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(49,51)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)(61,63)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81)(88,96)(89,95)(90,94)(91,93), (1,82,55,27)(2,77,56,46)(3,96,57,41)(4,91,58,36)(5,86,59,31)(6,81,60,26)(7,76,61,45)(8,95,62,40)(9,90,63,35)(10,85,64,30)(11,80,65,25)(12,75,66,44)(13,94,67,39)(14,89,68,34)(15,84,69,29)(16,79,70,48)(17,74,71,43)(18,93,72,38)(19,88,49,33)(20,83,50,28)(21,78,51,47)(22,73,52,42)(23,92,53,37)(24,87,54,32) );

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),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,24),(17,23),(18,22),(19,21),(26,48),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(49,51),(52,72),(53,71),(54,70),(55,69),(56,68),(57,67),(58,66),(59,65),(60,64),(61,63),(73,87),(74,86),(75,85),(76,84),(77,83),(78,82),(79,81),(88,96),(89,95),(90,94),(91,93)], [(1,82,55,27),(2,77,56,46),(3,96,57,41),(4,91,58,36),(5,86,59,31),(6,81,60,26),(7,76,61,45),(8,95,62,40),(9,90,63,35),(10,85,64,30),(11,80,65,25),(12,75,66,44),(13,94,67,39),(14,89,68,34),(15,84,69,29),(16,79,70,48),(17,74,71,43),(18,93,72,38),(19,88,49,33),(20,83,50,28),(21,78,51,47),(22,73,52,42),(23,92,53,37),(24,87,54,32)])

36 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J6A6B6C8A8B8C8D12A12B12C12D12E12F24A24B24C24D
order1222222234444444444666888812121212121224242424
size111112121212222444466662224412124488884444

36 irreducible representations

dim11111112222224444
type++++++++++++++
imageC1C2C2C2C2C2C4S3D4D6D6C4○D4C4×S3C8⋊C22Q83S3S3×D4Q83D6
kernelD249C4C6.D8C24⋊C4C3×C4.Q8Dic35D4C2×D24D24C4.Q8C2×Dic3C4⋊C4C2×C8C12C8C6C4C22C2
# reps12112181221242114

Matrix representation of D249C4 in GL8(𝔽73)

484854350000
271919540000
563854250000
365646250000
00000005
000000680
000029055
000029296810
,
10000000
172000000
955010000
955100000
0000015048
0000104848
00000010
000000172
,
007210000
272771720000
004600000
7204600000
00003060013
00001343600
00000173013
00005606043

G:=sub<GL(8,GF(73))| [48,27,56,36,0,0,0,0,48,19,38,56,0,0,0,0,54,19,54,46,0,0,0,0,35,54,25,25,0,0,0,0,0,0,0,0,0,0,29,29,0,0,0,0,0,0,0,29,0,0,0,0,0,68,5,68,0,0,0,0,5,0,5,10],[1,1,9,9,0,0,0,0,0,72,55,55,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,50,48,1,1,0,0,0,0,48,48,0,72],[0,27,0,72,0,0,0,0,0,27,0,0,0,0,0,0,72,71,46,46,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,30,13,0,56,0,0,0,0,60,43,17,0,0,0,0,0,0,60,30,60,0,0,0,0,13,0,13,43] >;

D249C4 in GAP, Magma, Sage, TeX

D_{24}\rtimes_9C_4
% in TeX

G:=Group("D24:9C4");
// GroupNames label

G:=SmallGroup(192,428);
// by ID

G=gap.SmallGroup(192,428);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,219,58,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^18*b>;
// generators/relations

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