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G = C6.Q16⋊C2order 192 = 26·3

35th semidirect product of C6.Q16 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.56D6, (C2×D4).36D6, C4⋊D4.4S3, C6.95(C4○D8), C6.Q1635C2, (C2×C12).261D4, (C22×C6).81D4, C12.55D49C2, D4⋊Dic314C2, C6.89(C8⋊C22), C12.Q834C2, (C6×D4).52C22, (C22×C4).134D6, C12.182(C4○D4), C4.92(D42S3), (C2×C12).354C23, C37(C23.19D4), C23.29(C3⋊D4), C2.14(Q8.13D6), C2.10(D126C22), C23.26D615C2, C4⋊Dic3.336C22, (C22×C12).158C22, C6.79(C22.D4), C2.13(C23.23D6), (C3×C4⋊D4).3C2, (C2×C6).485(C2×D4), (C2×C3⋊C8).107C22, (C2×C4).170(C3⋊D4), (C3×C4⋊C4).103C22, (C2×C4).454(C22×S3), C22.160(C2×C3⋊D4), SmallGroup(192,594)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C6.Q16⋊C2
C1C3C6C12C2×C12C4⋊Dic3C23.26D6 — C6.Q16⋊C2
C3C6C2×C12 — C6.Q16⋊C2
C1C22C22×C4C4⋊D4

Generators and relations for C6.Q16⋊C2
 G = < a,b,c,d | a12=b4=d2=1, c2=a9b2, bab-1=dad=a7, cac-1=a5, cbc-1=a9b-1, dbd=a6b-1, dcd=a9c >

Subgroups: 272 in 106 conjugacy classes, 39 normal (all characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×4], C23, C23, Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×6], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], C22×C4, C2×D4, C2×D4, C3⋊C8 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C22×C6, C22×C6, C22⋊C8, D4⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C2×C3⋊C8 [×2], C4×Dic3, C4⋊Dic3 [×2], C6.D4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×D4, C23.19D4, C6.Q16, C12.Q8, C12.55D4, D4⋊Dic3 [×2], C23.26D6, C3×C4⋊D4, C6.Q16⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C22.D4, C4○D8, C8⋊C22, D42S3 [×2], C2×C3⋊D4, C23.19D4, D126C22, C23.23D6, Q8.13D6, C6.Q16⋊C2

Smallest permutation representation of C6.Q16⋊C2
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 57 37 61)(2 52 38 68)(3 59 39 63)(4 54 40 70)(5 49 41 65)(6 56 42 72)(7 51 43 67)(8 58 44 62)(9 53 45 69)(10 60 46 64)(11 55 47 71)(12 50 48 66)(13 81 88 29)(14 76 89 36)(15 83 90 31)(16 78 91 26)(17 73 92 33)(18 80 93 28)(19 75 94 35)(20 82 95 30)(21 77 96 25)(22 84 85 32)(23 79 86 27)(24 74 87 34)
(1 15 46 87 7 21 40 93)(2 20 47 92 8 14 41 86)(3 13 48 85 9 19 42 91)(4 18 37 90 10 24 43 96)(5 23 38 95 11 17 44 89)(6 16 39 88 12 22 45 94)(25 67 74 60 31 61 80 54)(26 72 75 53 32 66 81 59)(27 65 76 58 33 71 82 52)(28 70 77 51 34 64 83 57)(29 63 78 56 35 69 84 50)(30 68 79 49 36 62 73 55)
(2 8)(4 10)(6 12)(13 22)(14 17)(15 24)(16 19)(18 21)(20 23)(25 74)(26 81)(27 76)(28 83)(29 78)(30 73)(31 80)(32 75)(33 82)(34 77)(35 84)(36 79)(38 44)(40 46)(42 48)(49 71)(50 66)(51 61)(52 68)(53 63)(54 70)(55 65)(56 72)(57 67)(58 62)(59 69)(60 64)(85 88)(86 95)(87 90)(89 92)(91 94)(93 96)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E6F6G8A8B8C8D12A12B12C12D12E12F
order122222344444444466666668888121212121212
size11114822222812121212222448812121212444488

33 irreducible representations

dim111111122222222224444
type++++++++++++++-
imageC1C2C2C2C2C2C2S3D4D4D6D6D6C4○D4C3⋊D4C3⋊D4C4○D8C8⋊C22D42S3D126C22Q8.13D6
kernelC6.Q16⋊C2C6.Q16C12.Q8C12.55D4D4⋊Dic3C23.26D6C3×C4⋊D4C4⋊D4C2×C12C22×C6C4⋊C4C22×C4C2×D4C12C2×C4C23C6C6C4C2C2
# reps111121111111142241222

Matrix representation of C6.Q16⋊C2 in GL6(𝔽73)

100000
010000
000100
0072000
0000072
0000172
,
5630000
25170000
0046000
0002700
000010
000001
,
4600000
59270000
00571600
00575700
0000283
00003145
,
100000
60720000
001000
0007200
000010
000001

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[56,25,0,0,0,0,3,17,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[46,59,0,0,0,0,0,27,0,0,0,0,0,0,57,57,0,0,0,0,16,57,0,0,0,0,0,0,28,31,0,0,0,0,3,45],[1,60,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C6.Q16⋊C2 in GAP, Magma, Sage, TeX

C_6.Q_{16}\rtimes C_2
% in TeX

G:=Group("C6.Q16:C2");
// GroupNames label

G:=SmallGroup(192,594);
// by ID

G=gap.SmallGroup(192,594);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,254,219,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^4=d^2=1,c^2=a^9*b^2,b*a*b^-1=d*a*d=a^7,c*a*c^-1=a^5,c*b*c^-1=a^9*b^-1,d*b*d=a^6*b^-1,d*c*d=a^9*c>;
// generators/relations

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