Copied to
clipboard

## G = C3×Q8○D8order 192 = 26·3

### Direct product of C3 and Q8○D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×Q8○D8
 Chief series C1 — C2 — C4 — C12 — C3×Q8 — C3×Q16 — C6×Q16 — C3×Q8○D8
 Lower central C1 — C2 — C4 — C3×Q8○D8
 Upper central C1 — C6 — C3×C4○D4 — C3×Q8○D8

Generators and relations for C3×Q8○D8
G = < a,b,c,d,e | a3=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >

Subgroups: 346 in 248 conjugacy classes, 158 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C12, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×Q8, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, C2×C24, C3×M4(2), C3×D8, C3×SD16, C3×Q16, C6×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, C3×C4○D4, Q8○D8, C3×C8○D4, C6×Q16, C3×C4○D8, C3×C8.C22, C3×2- 1+4, C3×Q8○D8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C22×D4, C6×D4, C23×C6, Q8○D8, D4×C2×C6, C3×Q8○D8

Smallest permutation representation of C3×Q8○D8
On 96 points
Generators in S96
(1 86 91)(2 87 92)(3 88 93)(4 81 94)(5 82 95)(6 83 96)(7 84 89)(8 85 90)(9 43 49)(10 44 50)(11 45 51)(12 46 52)(13 47 53)(14 48 54)(15 41 55)(16 42 56)(17 79 29)(18 80 30)(19 73 31)(20 74 32)(21 75 25)(22 76 26)(23 77 27)(24 78 28)(33 71 64)(34 72 57)(35 65 58)(36 66 59)(37 67 60)(38 68 61)(39 69 62)(40 70 63)
(1 42 5 46)(2 43 6 47)(3 44 7 48)(4 45 8 41)(9 96 13 92)(10 89 14 93)(11 90 15 94)(12 91 16 95)(17 58 21 62)(18 59 22 63)(19 60 23 64)(20 61 24 57)(25 69 29 65)(26 70 30 66)(27 71 31 67)(28 72 32 68)(33 73 37 77)(34 74 38 78)(35 75 39 79)(36 76 40 80)(49 83 53 87)(50 84 54 88)(51 85 55 81)(52 86 56 82)
(1 58 5 62)(2 59 6 63)(3 60 7 64)(4 61 8 57)(9 30 13 26)(10 31 14 27)(11 32 15 28)(12 25 16 29)(17 46 21 42)(18 47 22 43)(19 48 23 44)(20 41 24 45)(33 88 37 84)(34 81 38 85)(35 82 39 86)(36 83 40 87)(49 80 53 76)(50 73 54 77)(51 74 55 78)(52 75 56 79)(65 95 69 91)(66 96 70 92)(67 89 71 93)(68 90 72 94)
(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)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(18 24)(19 23)(20 22)(26 32)(27 31)(28 30)(33 37)(34 36)(38 40)(41 43)(44 48)(45 47)(49 55)(50 54)(51 53)(57 59)(60 64)(61 63)(66 72)(67 71)(68 70)(73 77)(74 76)(78 80)(81 83)(84 88)(85 87)(89 93)(90 92)(94 96)

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 C3×D4 C3×D4 Q8○D8 C3×Q8○D8 kernel C3×Q8○D8 C3×C8○D4 C6×Q16 C3×C4○D8 C3×C8.C22 C3×2- 1+4 Q8○D8 C8○D4 C2×Q16 C4○D8 C8.C22 2- 1+4 C3×D4 C3×Q8 D4 Q8 C3 C1 # reps 1 1 3 3 6 2 2 2 6 6 12 4 3 1 6 2 2 4

Matrix representation of C3×Q8○D8 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 0 6 5 1 3 0 5 6 3 3 6 1 1 6 3 1
,
 6 1 3 5 4 6 5 1 4 3 2 6 2 0 4 0
,
 5 0 2 6 6 5 5 6 6 1 1 2 2 2 6 2
,
 0 1 5 1 5 0 2 1 1 6 3 6 5 5 4 4
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[6,4,4,2,1,6,3,0,3,5,2,4,5,1,6,0],[5,6,6,2,0,5,1,2,2,5,1,6,6,6,2,2],[0,5,1,5,1,0,6,5,5,2,3,4,1,1,6,4] >;

C3×Q8○D8 in GAP, Magma, Sage, TeX

C_3\times Q_8\circ D_8
% in TeX

G:=Group("C3xQ8oD8");
// GroupNames label

G:=SmallGroup(192,1467);
// by ID

G=gap.SmallGroup(192,1467);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,680,745,6053,3036,124]);
// Polycyclic

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

׿
×
𝔽