metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊7D12, C42.129D6, C6.1102+ (1+4), (C4×Q8)⋊14S3, (C3×Q8)⋊11D4, (C4×D12)⋊39C2, C4⋊C4.296D6, (Q8×C12)⋊12C2, C3⋊2(Q8⋊6D4), C4.26(C2×D12), C12.58(C2×D4), C12⋊D4⋊18C2, C12⋊17(C4○D4), C4⋊D12⋊13C2, C4⋊3(Q8⋊3S3), (C2×Q8).229D6, C6.20(C22×D4), C2.22(D4○D12), (C2×C6).121C24, C2.22(C22×D12), (C2×C12).170C23, (C4×C12).173C22, D6⋊C4.101C22, (C2×D12).28C22, (C6×Q8).221C22, (C22×S3).46C23, C4⋊Dic3.399C22, C22.142(S3×C23), (C2×Dic3).215C23, (C2×Q8⋊3S3)⋊4C2, C6.112(C2×C4○D4), (S3×C2×C4).73C22, C2.11(C2×Q8⋊3S3), (C3×C4⋊C4).349C22, (C2×C4).734(C22×S3), SmallGroup(192,1136)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 936 in 312 conjugacy classes, 115 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×8], C4 [×5], C22, C22 [×18], S3 [×6], C6 [×3], C2×C4, C2×C4 [×6], C2×C4 [×14], D4 [×24], Q8 [×4], C23 [×6], Dic3 [×2], C12 [×8], C12 [×3], D6 [×18], C2×C6, C42 [×3], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4, C22×C4 [×6], C2×D4 [×15], C2×Q8, C4○D4 [×8], C4×S3 [×12], D12 [×24], C2×Dic3 [×2], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3 [×6], C4×D4 [×3], C4×Q8, C4⋊D4 [×6], C4⋊1D4 [×3], C2×C4○D4 [×2], C4⋊Dic3, D6⋊C4 [×6], C4×C12 [×3], C3×C4⋊C4 [×3], S3×C2×C4 [×6], C2×D12 [×15], Q8⋊3S3 [×8], C6×Q8, Q8⋊6D4, C4×D12 [×3], C4⋊D12 [×3], C12⋊D4 [×6], Q8×C12, C2×Q8⋊3S3 [×2], Q8⋊7D12
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C2×D12 [×6], Q8⋊3S3 [×2], S3×C23, Q8⋊6D4, C22×D12, C2×Q8⋊3S3, D4○D12, Q8⋊7D12
Generators and relations
G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >
►On 96 points
(1 81 17 31)(2 82 18 32)(3 83 19 33)(4 84 20 34)(5 73 21 35)(6 74 22 36)(7 75 23 25)(8 76 24 26)(9 77 13 27)(10 78 14 28)(11 79 15 29)(12 80 16 30)(37 91 69 55)(38 92 70 56)(39 93 71 57)(40 94 72 58)(41 95 61 59)(42 96 62 60)(43 85 63 49)(44 86 64 50)(45 87 65 51)(46 88 66 52)(47 89 67 53)(48 90 68 54)
(1 59 17 95)(2 60 18 96)(3 49 19 85)(4 50 20 86)(5 51 21 87)(6 52 22 88)(7 53 23 89)(8 54 24 90)(9 55 13 91)(10 56 14 92)(11 57 15 93)(12 58 16 94)(25 47 75 67)(26 48 76 68)(27 37 77 69)(28 38 78 70)(29 39 79 71)(30 40 80 72)(31 41 81 61)(32 42 82 62)(33 43 83 63)(34 44 84 64)(35 45 73 65)(36 46 74 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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 84)(20 83)(21 82)(22 81)(23 80)(24 79)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 60)(46 59)(47 58)(48 57)(61 88)(62 87)(63 86)(64 85)(65 96)(66 95)(67 94)(68 93)(69 92)(70 91)(71 90)(72 89)
G:=sub<Sym(96)| (1,81,17,31)(2,82,18,32)(3,83,19,33)(4,84,20,34)(5,73,21,35)(6,74,22,36)(7,75,23,25)(8,76,24,26)(9,77,13,27)(10,78,14,28)(11,79,15,29)(12,80,16,30)(37,91,69,55)(38,92,70,56)(39,93,71,57)(40,94,72,58)(41,95,61,59)(42,96,62,60)(43,85,63,49)(44,86,64,50)(45,87,65,51)(46,88,66,52)(47,89,67,53)(48,90,68,54), (1,59,17,95)(2,60,18,96)(3,49,19,85)(4,50,20,86)(5,51,21,87)(6,52,22,88)(7,53,23,89)(8,54,24,90)(9,55,13,91)(10,56,14,92)(11,57,15,93)(12,58,16,94)(25,47,75,67)(26,48,76,68)(27,37,77,69)(28,38,78,70)(29,39,79,71)(30,40,80,72)(31,41,81,61)(32,42,82,62)(33,43,83,63)(34,44,84,64)(35,45,73,65)(36,46,74,66), (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,84)(20,83)(21,82)(22,81)(23,80)(24,79)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,60)(46,59)(47,58)(48,57)(61,88)(62,87)(63,86)(64,85)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)>;
G:=Group( (1,81,17,31)(2,82,18,32)(3,83,19,33)(4,84,20,34)(5,73,21,35)(6,74,22,36)(7,75,23,25)(8,76,24,26)(9,77,13,27)(10,78,14,28)(11,79,15,29)(12,80,16,30)(37,91,69,55)(38,92,70,56)(39,93,71,57)(40,94,72,58)(41,95,61,59)(42,96,62,60)(43,85,63,49)(44,86,64,50)(45,87,65,51)(46,88,66,52)(47,89,67,53)(48,90,68,54), (1,59,17,95)(2,60,18,96)(3,49,19,85)(4,50,20,86)(5,51,21,87)(6,52,22,88)(7,53,23,89)(8,54,24,90)(9,55,13,91)(10,56,14,92)(11,57,15,93)(12,58,16,94)(25,47,75,67)(26,48,76,68)(27,37,77,69)(28,38,78,70)(29,39,79,71)(30,40,80,72)(31,41,81,61)(32,42,82,62)(33,43,83,63)(34,44,84,64)(35,45,73,65)(36,46,74,66), (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,84)(20,83)(21,82)(22,81)(23,80)(24,79)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,60)(46,59)(47,58)(48,57)(61,88)(62,87)(63,86)(64,85)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89) );
G=PermutationGroup([(1,81,17,31),(2,82,18,32),(3,83,19,33),(4,84,20,34),(5,73,21,35),(6,74,22,36),(7,75,23,25),(8,76,24,26),(9,77,13,27),(10,78,14,28),(11,79,15,29),(12,80,16,30),(37,91,69,55),(38,92,70,56),(39,93,71,57),(40,94,72,58),(41,95,61,59),(42,96,62,60),(43,85,63,49),(44,86,64,50),(45,87,65,51),(46,88,66,52),(47,89,67,53),(48,90,68,54)], [(1,59,17,95),(2,60,18,96),(3,49,19,85),(4,50,20,86),(5,51,21,87),(6,52,22,88),(7,53,23,89),(8,54,24,90),(9,55,13,91),(10,56,14,92),(11,57,15,93),(12,58,16,94),(25,47,75,67),(26,48,76,68),(27,37,77,69),(28,38,78,70),(29,39,79,71),(30,40,80,72),(31,41,81,61),(32,42,82,62),(33,43,83,63),(34,44,84,64),(35,45,73,65),(36,46,74,66)], [(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)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,78),(14,77),(15,76),(16,75),(17,74),(18,73),(19,84),(20,83),(21,82),(22,81),(23,80),(24,79),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,60),(46,59),(47,58),(48,57),(61,88),(62,87),(63,86),(64,85),(65,96),(66,95),(67,94),(68,93),(69,92),(70,91),(71,90),(72,89)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 8 |
| 0 | 0 | 3 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 8 |
| 3 | 10 | 0 | 0 |
| 3 | 6 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 10 | 3 | 0 | 0 |
| 6 | 3 | 0 | 0 |
| 0 | 0 | 1 | 5 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,12,3,0,0,8,1],[12,0,0,0,0,12,0,0,0,0,5,0,0,0,12,8],[3,3,0,0,10,6,0,0,0,0,12,0,0,0,0,12],[10,6,0,0,3,3,0,0,0,0,1,0,0,0,5,12] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 12 | ··· | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | D12 | 2+ (1+4) | Q8⋊3S3 | D4○D12 |
| kernel | Q8⋊7D12 | C4×D12 | C4⋊D12 | C12⋊D4 | Q8×C12 | C2×Q8⋊3S3 | C4×Q8 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | Q8 | C6 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 1 | 4 | 3 | 3 | 1 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_7D_{12} % in TeX
G:=Group("Q8:7D12"); // GroupNames label
G:=SmallGroup(192,1136);
// by ID
G=gap.SmallGroup(192,1136);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,184,675,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations