direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8.11D6, C12.31C24, D12.28C23, Dic6.27C23, (C2×Q8)⋊30D6, C3⋊C8.13C23, C12.255(C2×D4), (C2×C12).211D4, C6⋊4(C8.C22), C4.31(S3×C23), (C6×Q8)⋊34C22, (C22×Q8)⋊11S3, C3⋊Q16⋊15C22, (C22×C6).210D4, C6.150(C22×D4), (C22×C4).290D6, Q8.30(C22×S3), (C3×Q8).20C23, (C2×C12).548C23, Q8⋊2S3⋊16C22, C4○D12.57C22, C4.Dic3⋊33C22, (C2×D12).277C22, C23.100(C3⋊D4), (C22×C12).280C22, (C2×Dic6).305C22, (Q8×C2×C6)⋊3C2, C3⋊5(C2×C8.C22), C4.25(C2×C3⋊D4), (C2×C3⋊Q16)⋊30C2, (C2×C6).585(C2×D4), (C2×Q8⋊2S3)⋊30C2, (C2×C4○D12).24C2, (C2×C4).93(C3⋊D4), (C2×C3⋊C8).183C22, (C2×C4.Dic3)⋊27C2, C2.23(C22×C3⋊D4), (C2×C4).240(C22×S3), C22.113(C2×C3⋊D4), SmallGroup(192,1367)
Series: Derived ►Chief ►Lower central ►Upper central
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, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C3⋊C8, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C22×C6, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C2×C3⋊C8, C4.Dic3, Q8⋊2S3, C3⋊Q16, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, C22×C12, C6×Q8, C6×Q8, C2×C8.C22, C2×C4.Dic3, C2×Q8⋊2S3, Q8.11D6, C2×C3⋊Q16, C2×C4○D12, Q8×C2×C6, C2×Q8.11D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C8.C22, C22×D4, C2×C3⋊D4, S3×C23, C2×C8.C22, Q8.11D6, C22×C3⋊D4, C2×Q8.11D6
►On 96 points
Generators in S96
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 25)(10 26)(11 27)(12 28)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 73)(23 74)(24 75)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 61)(46 62)(47 63)(48 64)(49 93)(50 94)(51 95)(52 96)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 91)(60 92)
(1 41 7 47)(2 42 8 48)(3 43 9 37)(4 44 10 38)(5 45 11 39)(6 46 12 40)(13 89 19 95)(14 90 20 96)(15 91 21 85)(16 92 22 86)(17 93 23 87)(18 94 24 88)(25 65 31 71)(26 66 32 72)(27 67 33 61)(28 68 34 62)(29 69 35 63)(30 70 36 64)(49 74 55 80)(50 75 56 81)(51 76 57 82)(52 77 58 83)(53 78 59 84)(54 79 60 73)
(1 50 7 56)(2 57 8 51)(3 52 9 58)(4 59 10 53)(5 54 11 60)(6 49 12 55)(13 64 19 70)(14 71 20 65)(15 66 21 72)(16 61 22 67)(17 68 23 62)(18 63 24 69)(25 90 31 96)(26 85 32 91)(27 92 33 86)(28 87 34 93)(29 94 35 88)(30 89 36 95)(37 77 43 83)(38 84 44 78)(39 79 45 73)(40 74 46 80)(41 81 47 75)(42 76 48 82)
(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 92 19 86)(14 85 20 91)(15 90 21 96)(16 95 22 89)(17 88 23 94)(18 93 24 87)(25 26 31 32)(27 36 33 30)(28 29 34 35)(37 44 43 38)(39 42 45 48)(40 47 46 41)(49 75 55 81)(50 80 56 74)(51 73 57 79)(52 78 58 84)(53 83 59 77)(54 76 60 82)(61 64 67 70)(62 69 68 63)(65 72 71 66)
G:=sub<Sym(96)| (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,25)(10,26)(11,27)(12,28)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,73)(23,74)(24,75)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,61)(46,62)(47,63)(48,64)(49,93)(50,94)(51,95)(52,96)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92), (1,41,7,47)(2,42,8,48)(3,43,9,37)(4,44,10,38)(5,45,11,39)(6,46,12,40)(13,89,19,95)(14,90,20,96)(15,91,21,85)(16,92,22,86)(17,93,23,87)(18,94,24,88)(25,65,31,71)(26,66,32,72)(27,67,33,61)(28,68,34,62)(29,69,35,63)(30,70,36,64)(49,74,55,80)(50,75,56,81)(51,76,57,82)(52,77,58,83)(53,78,59,84)(54,79,60,73), (1,50,7,56)(2,57,8,51)(3,52,9,58)(4,59,10,53)(5,54,11,60)(6,49,12,55)(13,64,19,70)(14,71,20,65)(15,66,21,72)(16,61,22,67)(17,68,23,62)(18,63,24,69)(25,90,31,96)(26,85,32,91)(27,92,33,86)(28,87,34,93)(29,94,35,88)(30,89,36,95)(37,77,43,83)(38,84,44,78)(39,79,45,73)(40,74,46,80)(41,81,47,75)(42,76,48,82), (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)(13,92,19,86)(14,85,20,91)(15,90,21,96)(16,95,22,89)(17,88,23,94)(18,93,24,87)(25,26,31,32)(27,36,33,30)(28,29,34,35)(37,44,43,38)(39,42,45,48)(40,47,46,41)(49,75,55,81)(50,80,56,74)(51,73,57,79)(52,78,58,84)(53,83,59,77)(54,76,60,82)(61,64,67,70)(62,69,68,63)(65,72,71,66)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,25)(10,26)(11,27)(12,28)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,73)(23,74)(24,75)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,61)(46,62)(47,63)(48,64)(49,93)(50,94)(51,95)(52,96)(53,85)(54,86)(55,87)(56,88)(57,89)(58,90)(59,91)(60,92), (1,41,7,47)(2,42,8,48)(3,43,9,37)(4,44,10,38)(5,45,11,39)(6,46,12,40)(13,89,19,95)(14,90,20,96)(15,91,21,85)(16,92,22,86)(17,93,23,87)(18,94,24,88)(25,65,31,71)(26,66,32,72)(27,67,33,61)(28,68,34,62)(29,69,35,63)(30,70,36,64)(49,74,55,80)(50,75,56,81)(51,76,57,82)(52,77,58,83)(53,78,59,84)(54,79,60,73), (1,50,7,56)(2,57,8,51)(3,52,9,58)(4,59,10,53)(5,54,11,60)(6,49,12,55)(13,64,19,70)(14,71,20,65)(15,66,21,72)(16,61,22,67)(17,68,23,62)(18,63,24,69)(25,90,31,96)(26,85,32,91)(27,92,33,86)(28,87,34,93)(29,94,35,88)(30,89,36,95)(37,77,43,83)(38,84,44,78)(39,79,45,73)(40,74,46,80)(41,81,47,75)(42,76,48,82), (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)(13,92,19,86)(14,85,20,91)(15,90,21,96)(16,95,22,89)(17,88,23,94)(18,93,24,87)(25,26,31,32)(27,36,33,30)(28,29,34,35)(37,44,43,38)(39,42,45,48)(40,47,46,41)(49,75,55,81)(50,80,56,74)(51,73,57,79)(52,78,58,84)(53,83,59,77)(54,76,60,82)(61,64,67,70)(62,69,68,63)(65,72,71,66) );
G=PermutationGroup([[(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,25),(10,26),(11,27),(12,28),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,73),(23,74),(24,75),(37,65),(38,66),(39,67),(40,68),(41,69),(42,70),(43,71),(44,72),(45,61),(46,62),(47,63),(48,64),(49,93),(50,94),(51,95),(52,96),(53,85),(54,86),(55,87),(56,88),(57,89),(58,90),(59,91),(60,92)], [(1,41,7,47),(2,42,8,48),(3,43,9,37),(4,44,10,38),(5,45,11,39),(6,46,12,40),(13,89,19,95),(14,90,20,96),(15,91,21,85),(16,92,22,86),(17,93,23,87),(18,94,24,88),(25,65,31,71),(26,66,32,72),(27,67,33,61),(28,68,34,62),(29,69,35,63),(30,70,36,64),(49,74,55,80),(50,75,56,81),(51,76,57,82),(52,77,58,83),(53,78,59,84),(54,79,60,73)], [(1,50,7,56),(2,57,8,51),(3,52,9,58),(4,59,10,53),(5,54,11,60),(6,49,12,55),(13,64,19,70),(14,71,20,65),(15,66,21,72),(16,61,22,67),(17,68,23,62),(18,63,24,69),(25,90,31,96),(26,85,32,91),(27,92,33,86),(28,87,34,93),(29,94,35,88),(30,89,36,95),(37,77,43,83),(38,84,44,78),(39,79,45,73),(40,74,46,80),(41,81,47,75),(42,76,48,82)], [(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),(13,92,19,86),(14,85,20,91),(15,90,21,96),(16,95,22,89),(17,88,23,94),(18,93,24,87),(25,26,31,32),(27,36,33,30),(28,29,34,35),(37,44,43,38),(39,42,45,48),(40,47,46,41),(49,75,55,81),(50,80,56,74),(51,73,57,79),(52,78,58,84),(53,83,59,77),(54,76,60,82),(61,64,67,70),(62,69,68,63),(65,72,71,66)]])
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