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G = C125SD16order 192 = 26·3

5th semidirect product of C12 and SD16 acting via SD16/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C125SD16, D12.24D4, C42.78D6, C4⋊Q81S3, C4.56(S3×D4), C12⋊C833C2, C12.36(C2×D4), (C2×Q8).68D6, C42(Q82S3), (C4×D12).17C2, (C2×C12).154D4, C34(D4.D4), C6.75(C2×SD16), C12.80(C4○D4), Q82Dic323C2, C4.5(D42S3), (C6×Q8).62C22, C2.14(D63D4), C6.105(C4⋊D4), (C2×C12).401C23, (C4×C12).130C22, C6.95(C8.C22), (C2×D12).247C22, C4⋊Dic3.346C22, C2.16(Q8.11D6), (C3×C4⋊Q8)⋊1C2, (C2×C6).532(C2×D4), (C2×C3⋊C8).135C22, (C2×Q82S3).6C2, C2.13(C2×Q82S3), (C2×C4).187(C3⋊D4), (C2×C4).498(C22×S3), C22.204(C2×C3⋊D4), SmallGroup(192,642)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C125SD16
C1C3C6C12C2×C12C2×D12C4×D12 — C125SD16
C3C6C2×C12 — C125SD16
C1C22C42C4⋊Q8

Generators and relations for C125SD16
 G = < a,b,c | a12=b8=c2=1, bab-1=a-1, cac=a5, cbc=b3 >

Subgroups: 352 in 120 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, Q82S3, C4×C12, C3×C4⋊C4, S3×C2×C4, C2×D12, C6×Q8, D4.D4, C12⋊C8, Q82Dic3, C4×D12, C2×Q82S3, C3×C4⋊Q8, C125SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C2×SD16, C8.C22, Q82S3, S3×D4, D42S3, C2×C3⋊D4, D4.D4, D63D4, C2×Q82S3, Q8.11D6, C125SD16

Smallest permutation representation of C125SD16
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 40 89 28 20 71 50 81)(2 39 90 27 21 70 51 80)(3 38 91 26 22 69 52 79)(4 37 92 25 23 68 53 78)(5 48 93 36 24 67 54 77)(6 47 94 35 13 66 55 76)(7 46 95 34 14 65 56 75)(8 45 96 33 15 64 57 74)(9 44 85 32 16 63 58 73)(10 43 86 31 17 62 59 84)(11 42 87 30 18 61 60 83)(12 41 88 29 19 72 49 82)
(2 6)(3 11)(5 9)(8 12)(13 21)(15 19)(16 24)(18 22)(25 37)(26 42)(27 47)(28 40)(29 45)(30 38)(31 43)(32 48)(33 41)(34 46)(35 39)(36 44)(49 96)(50 89)(51 94)(52 87)(53 92)(54 85)(55 90)(56 95)(57 88)(58 93)(59 86)(60 91)(61 79)(62 84)(63 77)(64 82)(65 75)(66 80)(67 73)(68 78)(69 83)(70 76)(71 81)(72 74)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I6A6B6C8A8B8C8D12A···12F12G12H12I12J
order1222223444444444666888812···1212121212
size11111212222224881212222121212124···48888

33 irreducible representations

dim1111112222222244444
type+++++++++++-++-
imageC1C2C2C2C2C2S3D4D4D6D6SD16C4○D4C3⋊D4C8.C22Q82S3S3×D4D42S3Q8.11D6
kernelC125SD16C12⋊C8Q82Dic3C4×D12C2×Q82S3C3×C4⋊Q8C4⋊Q8D12C2×C12C42C2×Q8C12C12C2×C4C6C4C4C4C2
# reps1121211221242412112

Matrix representation of C125SD16 in GL6(𝔽73)

1720000
100000
0020600
00675300
000010
000001
,
0720000
7200000
000100
001000
00006112
0000670
,
010000
100000
001000
000100
000010
0000172

G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,72,0,0,0,0,0,0,0,20,67,0,0,0,0,6,53,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,61,67,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,72] >;

C125SD16 in GAP, Magma, Sage, TeX

C_{12}\rtimes_5{\rm SD}_{16}
% in TeX

G:=Group("C12:5SD16");
// GroupNames label

G:=SmallGroup(192,642);
// by ID

G=gap.SmallGroup(192,642);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,184,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^5,c*b*c=b^3>;
// generators/relations

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