Copied to
clipboard

G = C2×Q8.11D6order 192 = 26·3

Direct product of C2 and Q8.11D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×Q8.11D6
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — C2×C4○D12 — C2×Q8.11D6
 Lower central C3 — C6 — C12 — C2×Q8.11D6
 Upper central C1 — C22 — C22×C4 — C22×Q8

Generators and relations for C2×Q8.11D6
G = < a,b,c,d,e | a2=b4=1, c2=d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d5 >

Subgroups: 584 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], S3 [×2], C6, C6 [×2], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×7], Q8 [×4], Q8 [×9], C23, C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×4], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8 [×6], C2×Q8 [×4], C4○D4 [×6], C3⋊C8 [×4], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×6], C3×Q8 [×4], C3×Q8 [×6], C22×S3, C22×C6, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C2×C3⋊C8 [×2], C4.Dic3 [×4], Q82S3 [×8], C3⋊Q16 [×8], C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C2×C3⋊D4, C22×C12, C22×C12, C6×Q8 [×6], C6×Q8 [×3], C2×C8.C22, C2×C4.Dic3, C2×Q82S3 [×2], Q8.11D6 [×8], C2×C3⋊Q16 [×2], C2×C4○D12, Q8×C2×C6, C2×Q8.11D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C8.C22 [×2], C22×D4, C2×C3⋊D4 [×6], S3×C23, C2×C8.C22, Q8.11D6 [×2], C22×C3⋊D4, C2×Q8.11D6

Smallest permutation representation of C2×Q8.11D6
On 96 points
Generators in S96
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 35)(14 36)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 61)(57 62)(58 63)(59 64)(60 65)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 85)(82 86)(83 87)(84 88)
(1 54 7 60)(2 55 8 49)(3 56 9 50)(4 57 10 51)(5 58 11 52)(6 59 12 53)(13 83 19 77)(14 84 20 78)(15 73 21 79)(16 74 22 80)(17 75 23 81)(18 76 24 82)(25 89 31 95)(26 90 32 96)(27 91 33 85)(28 92 34 86)(29 93 35 87)(30 94 36 88)(37 65 43 71)(38 66 44 72)(39 67 45 61)(40 68 46 62)(41 69 47 63)(42 70 48 64)
(1 13 7 19)(2 20 8 14)(3 15 9 21)(4 22 10 16)(5 17 11 23)(6 24 12 18)(25 39 31 45)(26 46 32 40)(27 41 33 47)(28 48 34 42)(29 43 35 37)(30 38 36 44)(49 78 55 84)(50 73 56 79)(51 80 57 74)(52 75 58 81)(53 82 59 76)(54 77 60 83)(61 95 67 89)(62 90 68 96)(63 85 69 91)(64 92 70 86)(65 87 71 93)(66 94 72 88)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 76 19 82)(14 81 20 75)(15 74 21 80)(16 79 22 73)(17 84 23 78)(18 77 24 83)(25 90 31 96)(26 95 32 89)(27 88 33 94)(28 93 34 87)(29 86 35 92)(30 91 36 85)(37 42 43 48)(38 47 44 41)(39 40 45 46)(49 52 55 58)(50 57 56 51)(53 60 59 54)(61 68 67 62)(63 66 69 72)(64 71 70 65)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C3⋊D4 C3⋊D4 C8.C22 Q8.11D6 kernel C2×Q8.11D6 C2×C4.Dic3 C2×Q8⋊2S3 Q8.11D6 C2×C3⋊Q16 C2×C4○D12 Q8×C2×C6 C22×Q8 C2×C12 C22×C6 C22×C4 C2×Q8 C2×C4 C23 C6 C2 # reps 1 1 2 8 2 1 1 1 3 1 1 6 6 2 2 4

Matrix representation of C2×Q8.11D6 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 72 0 0 0 0 0 0 72 0 0
,
 72 71 0 0 0 0 0 1 0 0 0 0 0 0 62 16 5 57 0 0 57 5 16 62 0 0 5 57 11 57 0 0 16 62 16 68
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 30 30 0 0 0 0 43 60 0 0 43 43 0 0 0 0 30 13 0 0
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 43 60 0 0 0 0 30 30 0 0 43 60 0 0 0 0 30 30 0 0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[72,0,0,0,0,0,71,1,0,0,0,0,0,0,62,57,5,16,0,0,16,5,57,62,0,0,5,16,11,16,0,0,57,62,57,68],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,43,30,0,0,0,0,43,13,0,0,30,43,0,0,0,0,30,60,0,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,43,30,0,0,0,0,60,30,0,0,43,30,0,0,0,0,60,30,0,0] >;

C2×Q8.11D6 in GAP, Magma, Sage, TeX

C_2\times Q_8._{11}D_6
% in TeX

G:=Group("C2xQ8.11D6");
// GroupNames label

G:=SmallGroup(192,1367);
// by ID

G=gap.SmallGroup(192,1367);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,136,1684,235,102,6278]);
// Polycyclic

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

׿
×
𝔽