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G = C124SD16order 192 = 26·3

4th semidirect product of C12 and SD16 acting via SD16/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C124SD16, C42.219D6, C3⋊C812D4, C32(C85D4), C4.16(S3×D4), C41(D4.S3), (C2×D4).59D6, C41D4.7S3, C12.34(C2×D4), C122Q820C2, (C2×C12).150D4, C6.59(C2×SD16), C6.22(C41D4), (C6×D4).75C22, C2.13(C123D4), (C2×C12).394C23, (C4×C12).124C22, (C2×Dic6).112C22, (C4×C3⋊C8)⋊16C2, (C2×D4.S3)⋊15C2, (C3×C41D4).5C2, (C2×C6).525(C2×D4), C2.13(C2×D4.S3), (C2×C3⋊C8).258C22, (C2×C4).132(C3⋊D4), (C2×C4).492(C22×S3), C22.198(C2×C3⋊D4), SmallGroup(192,635)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C124SD16
C1C3C6C12C2×C12C2×Dic6C122Q8 — C124SD16
C3C6C2×C12 — C124SD16
C1C22C42C41D4

Generators and relations for C124SD16
 G = < a,b,c | a12=b8=c2=1, bab-1=a5, cac=a7, cbc=b3 >

Subgroups: 400 in 142 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×6], C4 [×2], C22, C22 [×6], C6, C6 [×2], C6 [×2], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×8], Q8 [×4], C23 [×2], Dic3 [×2], C12 [×6], C2×C6, C2×C6 [×6], C42, C4⋊C4 [×2], C2×C8 [×2], SD16 [×8], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C3⋊C8 [×4], Dic6 [×4], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×D4 [×8], C22×C6 [×2], C4×C8, C41D4, C4⋊Q8, C2×SD16 [×4], C2×C3⋊C8 [×2], C4⋊Dic3 [×2], D4.S3 [×8], C4×C12, C2×Dic6 [×2], C6×D4 [×2], C6×D4 [×2], C85D4, C4×C3⋊C8, C122Q8, C2×D4.S3 [×4], C3×C41D4, C124SD16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], SD16 [×4], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, C2×SD16 [×2], D4.S3 [×4], S3×D4 [×2], C2×C3⋊D4, C85D4, C2×D4.S3 [×2], C123D4, C124SD16

Smallest permutation representation of C124SD16
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 25 22 82 39 71 60 85)(2 30 23 75 40 64 49 90)(3 35 24 80 41 69 50 95)(4 28 13 73 42 62 51 88)(5 33 14 78 43 67 52 93)(6 26 15 83 44 72 53 86)(7 31 16 76 45 65 54 91)(8 36 17 81 46 70 55 96)(9 29 18 74 47 63 56 89)(10 34 19 79 48 68 57 94)(11 27 20 84 37 61 58 87)(12 32 21 77 38 66 59 92)
(2 8)(4 10)(6 12)(13 57)(14 52)(15 59)(16 54)(17 49)(18 56)(19 51)(20 58)(21 53)(22 60)(23 55)(24 50)(25 82)(26 77)(27 84)(28 79)(29 74)(30 81)(31 76)(32 83)(33 78)(34 73)(35 80)(36 75)(38 44)(40 46)(42 48)(61 87)(62 94)(63 89)(64 96)(65 91)(66 86)(67 93)(68 88)(69 95)(70 90)(71 85)(72 92)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A···4F4G4H6A6B6C6D6E6F6G8A···8H12A···12F
order12222234···44466666668···812···12
size11118822···2242422288886···64···4

36 irreducible representations

dim11111222222244
type++++++++++-+
imageC1C2C2C2C2S3D4D4D6D6SD16C3⋊D4D4.S3S3×D4
kernelC124SD16C4×C3⋊C8C122Q8C2×D4.S3C3×C41D4C41D4C3⋊C8C2×C12C42C2×D4C12C2×C4C4C4
# reps11141142128442

Matrix representation of C124SD16 in GL6(𝔽73)

0720000
100000
0017100
0017200
0000650
000099
,
6760000
67670000
0072200
0072100
0000154
00005358
,
7200000
010000
0072000
0072100
000010
00002972

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,0,0,0,0,65,9,0,0,0,0,0,9],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,72,72,0,0,0,0,2,1,0,0,0,0,0,0,15,53,0,0,0,0,4,58],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,29,0,0,0,0,0,72] >;

C124SD16 in GAP, Magma, Sage, TeX

C_{12}\rtimes_4{\rm SD}_{16}
% in TeX

G:=Group("C12:4SD16");
// GroupNames label

G:=SmallGroup(192,635);
// by ID

G=gap.SmallGroup(192,635);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,219,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^5,c*a*c=a^7,c*b*c=b^3>;
// generators/relations

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