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G = D24⋊C4order 192 = 26·3

3rd semidirect product of D24 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D243C4, C42.17D6, C82(C4×S3), C242(C2×C4), C8⋊C42S3, (C4×D12)⋊2C2, D129(C2×C4), C8⋊Dic32C2, C6.12(C4×D4), (C2×C8).54D6, C31(D8⋊C4), (C2×D24).7C2, C2.15(C4×D12), C2.D2437C2, (C2×C12).237D4, (C2×C4).115D12, C2.2(C8⋊D6), C6.3(C8⋊C22), (C2×C24).55C22, (C4×C12).15C22, C22.31(C2×D12), C4.107(C4○D12), C12.223(C4○D4), C12.106(C22×C4), (C2×C12).732C23, (C2×D12).190C22, C4⋊Dic3.266C22, C4.64(S3×C2×C4), (C3×C8⋊C4)⋊3C2, (C2×C6).115(C2×D4), (C2×C4).676(C22×S3), SmallGroup(192,270)

Series: Derived Chief Lower central Upper central

C1C12 — D24⋊C4
C1C3C6C2×C6C2×C12C2×D12C2×D24 — D24⋊C4
C3C6C12 — D24⋊C4
C1C22C42C8⋊C4

Generators and relations for D24⋊C4
 G = < a,b,c | a24=b2=c4=1, bab=a-1, cac-1=a13, cbc-1=a12b >

Subgroups: 456 in 132 conjugacy classes, 51 normal (23 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 [×2], C2×C4 [×6], D4 [×6], C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×8], C2×C6, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4 [×2], C24 [×2], C24, 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], C4⋊Dic3 [×2], D6⋊C4 [×2], C4×C12, C2×C24 [×2], S3×C2×C4 [×2], C2×D12 [×2], D8⋊C4, C8⋊Dic3, C2.D24 [×2], C3×C8⋊C4, C4×D12 [×2], C2×D24, D24⋊C4
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], D12 [×2], C22×S3, C4×D4, C8⋊C22 [×2], S3×C2×C4, C2×D12, C4○D12, D8⋊C4, C4×D12, C8⋊D6 [×2], D24⋊C4

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

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A···4F4G4H4I4J6A6B6C8A8B8C8D12A12B12C12D12E12F12G12H24A···24H
order1222222234···444446668888121212121212121224···24
size11111212121222···2121212122224444222244444···4

42 irreducible representations

dim11111112222222244
type+++++++++++++
imageC1C2C2C2C2C2C4S3D4D6D6C4○D4C4×S3D12C4○D12C8⋊C22C8⋊D6
kernelD24⋊C4C8⋊Dic3C2.D24C3×C8⋊C4C4×D12C2×D24D24C8⋊C4C2×C12C42C2×C8C12C8C2×C4C4C6C2
# reps11212181212244424

Matrix representation of D24⋊C4 in GL6(𝔽73)

7210000
7200000
0049682816
005545712
005419245
0054356819
,
100000
1720000
0014700
00665900
0036185966
005537714
,
2700000
0270000
006123710
005038071
002671250
0066192335

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,49,5,54,54,0,0,68,54,19,35,0,0,28,57,24,68,0,0,16,12,5,19],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,14,66,36,55,0,0,7,59,18,37,0,0,0,0,59,7,0,0,0,0,66,14],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,61,50,26,66,0,0,23,38,7,19,0,0,71,0,12,23,0,0,0,71,50,35] >;

D24⋊C4 in GAP, Magma, Sage, TeX

D_{24}\rtimes C_4
% in TeX

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

G:=SmallGroup(192,270);
// by ID

G=gap.SmallGroup(192,270);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,387,58,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^12*b>;
// generators/relations

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