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G = C4⋊D24order 192 = 26·3

The semidirect product of C4 and D24 acting via D24/D12=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C121D8, C42D24, D128D4, C42.34D6, C4⋊C83S3, C6.7(C2×D8), (C2×D24)⋊7C2, C32(C4⋊D8), (C2×C8).21D6, C2.9(C2×D24), (C4×D12)⋊17C2, C4⋊D127C2, C4.130(S3×D4), C2.D247C2, (C2×C4).133D12, (C2×C12).122D4, C12.339(C2×D4), C6.38(C4⋊D4), C2.17(C8⋊D6), C6.14(C8⋊C22), (C4×C12).69C22, (C2×C24).22C22, C12.328(C4○D4), C2.11(C12⋊D4), (C2×C12).753C23, C4.44(Q83S3), (C2×D12).14C22, C22.116(C2×D12), C4⋊Dic3.273C22, (C3×C4⋊C8)⋊5C2, (C2×C6).136(C2×D4), (C2×C4).698(C22×S3), SmallGroup(192,402)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C4⋊D24
C1C3C6C12C2×C12C2×D12C4×D12 — C4⋊D24
C3C6C2×C12 — C4⋊D24
C1C22C42C4⋊C8

Generators and relations for C4⋊D24
 G = < a,b,c | a4=b24=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 584 in 140 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C24, C4×S3, D12, D12, C2×Dic3, C2×C12, C22×S3, D4⋊C4, C4⋊C8, C4×D4, C41D4, C2×D8, D24, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C2×D12, C2×D12, C2×D12, C4⋊D8, C2.D24, C3×C4⋊C8, C4×D12, C4⋊D12, C2×D24, C4⋊D24
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×D8, C8⋊C22, D24, C2×D12, S3×D4, Q83S3, C4⋊D8, C12⋊D4, C2×D24, C8⋊D6, C4⋊D24

Smallest permutation representation of C4⋊D24
On 96 points
Generators in S96
(1 95 30 56)(2 57 31 96)(3 73 32 58)(4 59 33 74)(5 75 34 60)(6 61 35 76)(7 77 36 62)(8 63 37 78)(9 79 38 64)(10 65 39 80)(11 81 40 66)(12 67 41 82)(13 83 42 68)(14 69 43 84)(15 85 44 70)(16 71 45 86)(17 87 46 72)(18 49 47 88)(19 89 48 50)(20 51 25 90)(21 91 26 52)(22 53 27 92)(23 93 28 54)(24 55 29 94)
(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 92)(50 91)(51 90)(52 89)(53 88)(54 87)(55 86)(56 85)(57 84)(58 83)(59 82)(60 81)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 96)(70 95)(71 94)(72 93)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G6A6B6C8A8B8C8D12A12B12C12D12E12F12G12H24A···24H
order12222222344444446668888121212121212121224···24
size11111212242422222412122224444222244444···4

39 irreducible representations

dim1111112222222224444
type++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D8C4○D4D12D24C8⋊C22S3×D4Q83S3C8⋊D6
kernelC4⋊D24C2.D24C3×C4⋊C8C4×D12C4⋊D12C2×D24C4⋊C8D12C2×C12C42C2×C8C12C12C2×C4C4C6C4C4C2
# reps1211121221242481112

Matrix representation of C4⋊D24 in GL6(𝔽73)

7200000
0720000
001000
000100
0000027
0000270
,
0410000
16410000
00666600
0075900
000001
0000720
,
7220000
010000
001000
0017200
0000720
000001

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,0,0,27,0],[0,16,0,0,0,0,41,41,0,0,0,0,0,0,66,7,0,0,0,0,66,59,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[72,0,0,0,0,0,2,1,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1] >;

C4⋊D24 in GAP, Magma, Sage, TeX

C_4\rtimes D_{24}
% in TeX

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

G:=SmallGroup(192,402);
// by ID

G=gap.SmallGroup(192,402);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,226,1123,136,6278]);
// Polycyclic

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

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