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G = (C2×C6).Q16order 192 = 26·3

5th non-split extension by C2×C6 of Q16 acting via Q16/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.64D6, (C2×C6).5Q16, (C2×C12).76D4, (C2×Q8).50D6, C6.37(C2×Q16), C22⋊Q8.3S3, C6.Q1638C2, (C22×C6).90D4, Q82Dic313C2, (C22×C4).141D6, C12.188(C4○D4), (C6×Q8).44C22, C2.14(D4⋊D6), C4.94(D42S3), C6.115(C8⋊C22), (C2×C12).363C23, C12.55D4.7C2, C23.68(C3⋊D4), C35(C23.48D4), C22.3(C3⋊Q16), C4⋊Dic3.338C22, (C22×C12).167C22, C6.81(C22.D4), C2.15(C23.23D6), C2.8(C2×C3⋊Q16), (C2×C6).494(C2×D4), (C3×C22⋊Q8).2C2, (C2×C4).54(C3⋊D4), (C2×C3⋊C8).113C22, (C2×C4⋊Dic3).38C2, (C3×C4⋊C4).111C22, (C2×C4).463(C22×S3), C22.169(C2×C3⋊D4), SmallGroup(192,603)

Series: Derived Chief Lower central Upper central

C1C2×C12 — (C2×C6).Q16
C1C3C6C12C2×C12C4⋊Dic3C2×C4⋊Dic3 — (C2×C6).Q16
C3C6C2×C12 — (C2×C6).Q16
C1C22C22×C4C22⋊Q8

Generators and relations for (C2×C6).Q16
 G = < a,b,c,d | a2=b6=c8=1, d2=b3c4, ab=ba, cac-1=dad-1=ab3, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 256 in 104 conjugacy classes, 43 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×8], Q8 [×2], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C22×C4, C22×C4, C2×Q8, C3⋊C8 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], C22×C6, C22⋊C8, Q8⋊C4 [×2], C2.D8 [×2], C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C6×Q8, C23.48D4, C6.Q16 [×2], C12.55D4, Q82Dic3 [×2], C2×C4⋊Dic3, C3×C22⋊Q8, (C2×C6).Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], Q16 [×2], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C22.D4, C2×Q16, C8⋊C22, C3⋊Q16 [×2], D42S3 [×2], C2×C3⋊D4, C23.48D4, C23.23D6, C2×C3⋊Q16, D4⋊D6, (C2×C6).Q16

Smallest permutation representation of (C2×C6).Q16
On 96 points
Generators in S96
(1 5)(2 9)(3 7)(4 11)(6 13)(8 15)(10 14)(12 16)(17 60)(18 22)(19 62)(20 24)(21 64)(23 58)(25 71)(26 30)(27 65)(28 32)(29 67)(31 69)(33 91)(34 38)(35 93)(36 40)(37 95)(39 89)(41 45)(42 85)(43 47)(44 87)(46 81)(48 83)(49 53)(50 80)(51 55)(52 74)(54 76)(56 78)(57 61)(59 63)(66 70)(68 72)(73 77)(75 79)(82 86)(84 88)(90 94)(92 96)
(1 77 41 12 51 88)(2 81 52 13 42 78)(3 79 43 14 53 82)(4 83 54 15 44 80)(5 73 45 16 55 84)(6 85 56 9 46 74)(7 75 47 10 49 86)(8 87 50 11 48 76)(17 67 35 64 25 89)(18 90 26 57 36 68)(19 69 37 58 27 91)(20 92 28 59 38 70)(21 71 39 60 29 93)(22 94 30 61 40 72)(23 65 33 62 31 95)(24 96 32 63 34 66)
(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 23 16 58)(2 22 9 57)(3 21 10 64)(4 20 11 63)(5 19 12 62)(6 18 13 61)(7 17 14 60)(8 24 15 59)(25 79 71 49)(26 78 72 56)(27 77 65 55)(28 76 66 54)(29 75 67 53)(30 74 68 52)(31 73 69 51)(32 80 70 50)(33 84 91 41)(34 83 92 48)(35 82 93 47)(36 81 94 46)(37 88 95 45)(38 87 96 44)(39 86 89 43)(40 85 90 42)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F12G12H
order12222234444444446666688881212121212121212
size11112222248812121212222441212121244448888

33 irreducible representations

dim11111122222222224444
type++++++++++++-+--+
imageC1C2C2C2C2C2S3D4D4D6D6D6C4○D4Q16C3⋊D4C3⋊D4C8⋊C22D42S3C3⋊Q16D4⋊D6
kernel(C2×C6).Q16C6.Q16C12.55D4Q82Dic3C2×C4⋊Dic3C3×C22⋊Q8C22⋊Q8C2×C12C22×C6C4⋊C4C22×C4C2×Q8C12C2×C6C2×C4C23C6C4C22C2
# reps12121111111144221222

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

100000
010000
001000
00137200
0000720
0000072
,
72720000
100000
0072000
0007200
000010
000001
,
25620000
37480000
00526500
00552100
0000032
00005732
,
30600000
13430000
0017300
00255600
00001371
00001260

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,13,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,37,0,0,0,0,62,48,0,0,0,0,0,0,52,55,0,0,0,0,65,21,0,0,0,0,0,0,0,57,0,0,0,0,32,32],[30,13,0,0,0,0,60,43,0,0,0,0,0,0,17,25,0,0,0,0,3,56,0,0,0,0,0,0,13,12,0,0,0,0,71,60] >;

(C2×C6).Q16 in GAP, Magma, Sage, TeX

(C_2\times C_6).Q_{16}
% in TeX

G:=Group("(C2xC6).Q16");
// GroupNames label

G:=SmallGroup(192,603);
// by ID

G=gap.SmallGroup(192,603);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,336,254,219,268,1123,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^8=1,d^2=b^3*c^4,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^3,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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