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G = (C2×C6)⋊8Q16order 192 = 26·3

2nd semidirect product of C2×C6 and Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C2×C6)⋊8Q16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C12.48D4 — (C2×C6)⋊8Q16
 Lower central C3 — C6 — C2×C12 — (C2×C6)⋊8Q16
 Upper central C1 — C22 — C22×C4 — C22×Q8

Generators and relations for (C2×C6)⋊8Q16
G = < a,b,c,d | a2=b6=c8=1, d2=c4, ab=ba, cac-1=ab3, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 328 in 148 conjugacy classes, 51 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×C6, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C3⋊Q16, C6.D4, C2×Dic6, C22×C12, C22×C12, C6×Q8, C6×Q8, C22⋊Q16, C12.55D4, Q82Dic3, C12.48D4, C2×C3⋊Q16, Q8×C2×C6, (C2×C6)⋊8Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C3⋊D4, C22×S3, C22≀C2, C2×Q16, C8.C22, C3⋊Q16, C2×C3⋊D4, C22⋊Q16, Q8.11D6, C2×C3⋊Q16, C244S3, (C2×C6)⋊8Q16

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

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C ··· 4G 4H 4I 6A ··· 6G 8A 8B 8C 8D 12A ··· 12L order 1 2 2 2 2 2 3 4 4 4 ··· 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 1 1 2 2 2 2 2 4 ··· 4 24 24 2 ··· 2 12 12 12 12 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 Q16 C3⋊D4 C3⋊D4 C3⋊D4 C8.C22 C3⋊Q16 Q8.11D6 kernel (C2×C6)⋊8Q16 C12.55D4 Q8⋊2Dic3 C12.48D4 C2×C3⋊Q16 Q8×C2×C6 C22×Q8 C2×C12 C3×Q8 C22×C6 C22×C4 C2×Q8 C2×C6 C2×C4 Q8 C23 C6 C22 C2 # reps 1 1 2 1 2 1 1 1 4 1 1 2 4 2 8 2 1 2 2

Matrix representation of (C2×C6)⋊8Q16 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 72
,
 1 0 0 0 0 1 0 0 0 0 65 0 0 0 0 9
,
 57 16 0 0 57 57 0 0 0 0 0 1 0 0 72 0
,
 61 1 0 0 1 12 0 0 0 0 1 0 0 0 0 72
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,65,0,0,0,0,9],[57,57,0,0,16,57,0,0,0,0,0,72,0,0,1,0],[61,1,0,0,1,12,0,0,0,0,1,0,0,0,0,72] >;

(C2×C6)⋊8Q16 in GAP, Magma, Sage, TeX

(C_2\times C_6)\rtimes_8Q_{16}
% in TeX

G:=Group("(C2xC6):8Q16");
// GroupNames label

G:=SmallGroup(192,787);
// by ID

G=gap.SmallGroup(192,787);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,184,1123,297,136,6278]);
// Polycyclic

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

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