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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 [×3], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], S3 [×2], C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×4], C23, Dic3, C12 [×2], C12 [×2], C12 [×3], D6 [×4], C2×C6, C42, C22⋊C4, C4⋊C4 [×3], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8 [×2], C3⋊C8 [×2], C4×S3 [×2], D12 [×2], D12, C2×Dic3, C2×C12 [×3], C2×C12 [×2], C3×Q8 [×4], C22×S3, Q8⋊C4 [×2], C4⋊C8, C4×D4, C4⋊Q8, C2×SD16 [×2], C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, Q82S3 [×4], C4×C12, C3×C4⋊C4 [×2], S3×C2×C4, C2×D12, C6×Q8 [×2], D4.D4, C12⋊C8, Q82Dic3 [×2], C4×D12, C2×Q82S3 [×2], C3×C4⋊Q8, C125SD16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], SD16 [×2], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C2×SD16, C8.C22, Q82S3 [×2], 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 32 66 76 19 92 58 39)(2 31 67 75 20 91 59 38)(3 30 68 74 21 90 60 37)(4 29 69 73 22 89 49 48)(5 28 70 84 23 88 50 47)(6 27 71 83 24 87 51 46)(7 26 72 82 13 86 52 45)(8 25 61 81 14 85 53 44)(9 36 62 80 15 96 54 43)(10 35 63 79 16 95 55 42)(11 34 64 78 17 94 56 41)(12 33 65 77 18 93 57 40)
(2 6)(3 11)(5 9)(8 12)(14 18)(15 23)(17 21)(20 24)(25 77)(26 82)(27 75)(28 80)(29 73)(30 78)(31 83)(32 76)(33 81)(34 74)(35 79)(36 84)(37 94)(38 87)(39 92)(40 85)(41 90)(42 95)(43 88)(44 93)(45 86)(46 91)(47 96)(48 89)(49 69)(50 62)(51 67)(52 72)(53 65)(54 70)(55 63)(56 68)(57 61)(58 66)(59 71)(60 64)

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

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

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

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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