direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C23.7Q8, C12⋊9(C22⋊C4), (C22×C4)⋊10C12, (C22×C12)⋊19C4, (C2×C12).511D4, C23.8(C3×Q8), C22.9(C6×Q8), (C23×C4).12C6, C24.30(C2×C6), C22.30(C6×D4), C23.26(C3×D4), (C22×C6).19Q8, C2.C42⋊4C6, C23.30(C2×C12), (C23×C12).22C2, (C22×C6).126D4, C6.82(C22⋊Q8), C6.132(C4⋊D4), C23.57(C22×C6), (C23×C6).84C22, C6.53(C42⋊C2), (C22×C6).444C23, C22.29(C22×C12), (C22×C12).573C22, (C2×C4⋊C4)⋊1C6, C2.4(C6×C4⋊C4), (C6×C4⋊C4)⋊28C2, (C2×C6)⋊4(C4⋊C4), C6.59(C2×C4⋊C4), C22⋊2(C3×C4⋊C4), C4⋊2(C3×C22⋊C4), C2.1(C3×C4⋊D4), C2.5(C6×C22⋊C4), (C2×C4).55(C2×C12), C2.1(C3×C22⋊Q8), (C2×C6).597(C2×D4), (C2×C4).116(C3×D4), (C2×C22⋊C4).3C6, (C6×C22⋊C4).9C2, C6.92(C2×C22⋊C4), (C2×C6).101(C2×Q8), (C2×C12).330(C2×C4), (C22×C4).91(C2×C6), C2.5(C3×C42⋊C2), C22.15(C3×C4○D4), (C2×C6).205(C4○D4), (C3×C2.C42)⋊3C2, (C2×C6).216(C22×C4), (C22×C6).111(C2×C4), SmallGroup(192,813)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.7Q8
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=1, f2=ce2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 370 in 234 conjugacy classes, 114 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C22×C12, C23×C6, C23.7Q8, C3×C2.C42, C6×C22⋊C4, C6×C4⋊C4, C23×C12, C3×C23.7Q8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C23.7Q8, C6×C22⋊C4, C6×C4⋊C4, C3×C42⋊C2, C3×C4⋊D4, C3×C22⋊Q8, C3×C23.7Q8
►On 96 points
Generators in S96
(1 37 33)(2 38 34)(3 39 35)(4 40 36)(5 96 92)(6 93 89)(7 94 90)(8 95 91)(9 17 13)(10 18 14)(11 19 15)(12 20 16)(21 49 53)(22 50 54)(23 51 55)(24 52 56)(25 68 29)(26 65 30)(27 66 31)(28 67 32)(41 73 45)(42 74 46)(43 75 47)(44 76 48)(57 61 69)(58 62 70)(59 63 71)(60 64 72)(77 85 81)(78 86 82)(79 87 83)(80 88 84)
(1 43)(2 44)(3 41)(4 42)(5 86)(6 87)(7 88)(8 85)(9 21)(10 22)(11 23)(12 24)(13 53)(14 54)(15 55)(16 56)(17 49)(18 50)(19 51)(20 52)(25 60)(26 57)(27 58)(28 59)(29 72)(30 69)(31 70)(32 71)(33 47)(34 48)(35 45)(36 46)(37 75)(38 76)(39 73)(40 74)(61 65)(62 66)(63 67)(64 68)(77 91)(78 92)(79 89)(80 90)(81 95)(82 96)(83 93)(84 94)
(1 43)(2 44)(3 41)(4 42)(5 70)(6 71)(7 72)(8 69)(9 21)(10 22)(11 23)(12 24)(13 53)(14 54)(15 55)(16 56)(17 49)(18 50)(19 51)(20 52)(25 84)(26 81)(27 82)(28 83)(29 88)(30 85)(31 86)(32 87)(33 47)(34 48)(35 45)(36 46)(37 75)(38 76)(39 73)(40 74)(57 95)(58 96)(59 93)(60 94)(61 91)(62 92)(63 89)(64 90)(65 77)(66 78)(67 79)(68 80)
(1 9)(2 10)(3 11)(4 12)(5 31)(6 32)(7 29)(8 30)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 43)(22 44)(23 41)(24 42)(25 94)(26 95)(27 96)(28 93)(45 55)(46 56)(47 53)(48 54)(49 75)(50 76)(51 73)(52 74)(57 81)(58 82)(59 83)(60 84)(61 77)(62 78)(63 79)(64 80)(65 91)(66 92)(67 89)(68 90)(69 85)(70 86)(71 87)(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 91 41 63)(2 90 42 62)(3 89 43 61)(4 92 44 64)(5 76 72 40)(6 75 69 39)(7 74 70 38)(8 73 71 37)(9 65 23 79)(10 68 24 78)(11 67 21 77)(12 66 22 80)(13 26 55 83)(14 25 56 82)(15 28 53 81)(16 27 54 84)(17 30 51 87)(18 29 52 86)(19 32 49 85)(20 31 50 88)(33 95 45 59)(34 94 46 58)(35 93 47 57)(36 96 48 60)
G:=sub<Sym(96)| (1,37,33)(2,38,34)(3,39,35)(4,40,36)(5,96,92)(6,93,89)(7,94,90)(8,95,91)(9,17,13)(10,18,14)(11,19,15)(12,20,16)(21,49,53)(22,50,54)(23,51,55)(24,52,56)(25,68,29)(26,65,30)(27,66,31)(28,67,32)(41,73,45)(42,74,46)(43,75,47)(44,76,48)(57,61,69)(58,62,70)(59,63,71)(60,64,72)(77,85,81)(78,86,82)(79,87,83)(80,88,84), (1,43)(2,44)(3,41)(4,42)(5,86)(6,87)(7,88)(8,85)(9,21)(10,22)(11,23)(12,24)(13,53)(14,54)(15,55)(16,56)(17,49)(18,50)(19,51)(20,52)(25,60)(26,57)(27,58)(28,59)(29,72)(30,69)(31,70)(32,71)(33,47)(34,48)(35,45)(36,46)(37,75)(38,76)(39,73)(40,74)(61,65)(62,66)(63,67)(64,68)(77,91)(78,92)(79,89)(80,90)(81,95)(82,96)(83,93)(84,94), (1,43)(2,44)(3,41)(4,42)(5,70)(6,71)(7,72)(8,69)(9,21)(10,22)(11,23)(12,24)(13,53)(14,54)(15,55)(16,56)(17,49)(18,50)(19,51)(20,52)(25,84)(26,81)(27,82)(28,83)(29,88)(30,85)(31,86)(32,87)(33,47)(34,48)(35,45)(36,46)(37,75)(38,76)(39,73)(40,74)(57,95)(58,96)(59,93)(60,94)(61,91)(62,92)(63,89)(64,90)(65,77)(66,78)(67,79)(68,80), (1,9)(2,10)(3,11)(4,12)(5,31)(6,32)(7,29)(8,30)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,43)(22,44)(23,41)(24,42)(25,94)(26,95)(27,96)(28,93)(45,55)(46,56)(47,53)(48,54)(49,75)(50,76)(51,73)(52,74)(57,81)(58,82)(59,83)(60,84)(61,77)(62,78)(63,79)(64,80)(65,91)(66,92)(67,89)(68,90)(69,85)(70,86)(71,87)(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), (1,91,41,63)(2,90,42,62)(3,89,43,61)(4,92,44,64)(5,76,72,40)(6,75,69,39)(7,74,70,38)(8,73,71,37)(9,65,23,79)(10,68,24,78)(11,67,21,77)(12,66,22,80)(13,26,55,83)(14,25,56,82)(15,28,53,81)(16,27,54,84)(17,30,51,87)(18,29,52,86)(19,32,49,85)(20,31,50,88)(33,95,45,59)(34,94,46,58)(35,93,47,57)(36,96,48,60)>;
G:=Group( (1,37,33)(2,38,34)(3,39,35)(4,40,36)(5,96,92)(6,93,89)(7,94,90)(8,95,91)(9,17,13)(10,18,14)(11,19,15)(12,20,16)(21,49,53)(22,50,54)(23,51,55)(24,52,56)(25,68,29)(26,65,30)(27,66,31)(28,67,32)(41,73,45)(42,74,46)(43,75,47)(44,76,48)(57,61,69)(58,62,70)(59,63,71)(60,64,72)(77,85,81)(78,86,82)(79,87,83)(80,88,84), (1,43)(2,44)(3,41)(4,42)(5,86)(6,87)(7,88)(8,85)(9,21)(10,22)(11,23)(12,24)(13,53)(14,54)(15,55)(16,56)(17,49)(18,50)(19,51)(20,52)(25,60)(26,57)(27,58)(28,59)(29,72)(30,69)(31,70)(32,71)(33,47)(34,48)(35,45)(36,46)(37,75)(38,76)(39,73)(40,74)(61,65)(62,66)(63,67)(64,68)(77,91)(78,92)(79,89)(80,90)(81,95)(82,96)(83,93)(84,94), (1,43)(2,44)(3,41)(4,42)(5,70)(6,71)(7,72)(8,69)(9,21)(10,22)(11,23)(12,24)(13,53)(14,54)(15,55)(16,56)(17,49)(18,50)(19,51)(20,52)(25,84)(26,81)(27,82)(28,83)(29,88)(30,85)(31,86)(32,87)(33,47)(34,48)(35,45)(36,46)(37,75)(38,76)(39,73)(40,74)(57,95)(58,96)(59,93)(60,94)(61,91)(62,92)(63,89)(64,90)(65,77)(66,78)(67,79)(68,80), (1,9)(2,10)(3,11)(4,12)(5,31)(6,32)(7,29)(8,30)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,43)(22,44)(23,41)(24,42)(25,94)(26,95)(27,96)(28,93)(45,55)(46,56)(47,53)(48,54)(49,75)(50,76)(51,73)(52,74)(57,81)(58,82)(59,83)(60,84)(61,77)(62,78)(63,79)(64,80)(65,91)(66,92)(67,89)(68,90)(69,85)(70,86)(71,87)(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), (1,91,41,63)(2,90,42,62)(3,89,43,61)(4,92,44,64)(5,76,72,40)(6,75,69,39)(7,74,70,38)(8,73,71,37)(9,65,23,79)(10,68,24,78)(11,67,21,77)(12,66,22,80)(13,26,55,83)(14,25,56,82)(15,28,53,81)(16,27,54,84)(17,30,51,87)(18,29,52,86)(19,32,49,85)(20,31,50,88)(33,95,45,59)(34,94,46,58)(35,93,47,57)(36,96,48,60) );
G=PermutationGroup([[(1,37,33),(2,38,34),(3,39,35),(4,40,36),(5,96,92),(6,93,89),(7,94,90),(8,95,91),(9,17,13),(10,18,14),(11,19,15),(12,20,16),(21,49,53),(22,50,54),(23,51,55),(24,52,56),(25,68,29),(26,65,30),(27,66,31),(28,67,32),(41,73,45),(42,74,46),(43,75,47),(44,76,48),(57,61,69),(58,62,70),(59,63,71),(60,64,72),(77,85,81),(78,86,82),(79,87,83),(80,88,84)], [(1,43),(2,44),(3,41),(4,42),(5,86),(6,87),(7,88),(8,85),(9,21),(10,22),(11,23),(12,24),(13,53),(14,54),(15,55),(16,56),(17,49),(18,50),(19,51),(20,52),(25,60),(26,57),(27,58),(28,59),(29,72),(30,69),(31,70),(32,71),(33,47),(34,48),(35,45),(36,46),(37,75),(38,76),(39,73),(40,74),(61,65),(62,66),(63,67),(64,68),(77,91),(78,92),(79,89),(80,90),(81,95),(82,96),(83,93),(84,94)], [(1,43),(2,44),(3,41),(4,42),(5,70),(6,71),(7,72),(8,69),(9,21),(10,22),(11,23),(12,24),(13,53),(14,54),(15,55),(16,56),(17,49),(18,50),(19,51),(20,52),(25,84),(26,81),(27,82),(28,83),(29,88),(30,85),(31,86),(32,87),(33,47),(34,48),(35,45),(36,46),(37,75),(38,76),(39,73),(40,74),(57,95),(58,96),(59,93),(60,94),(61,91),(62,92),(63,89),(64,90),(65,77),(66,78),(67,79),(68,80)], [(1,9),(2,10),(3,11),(4,12),(5,31),(6,32),(7,29),(8,30),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,43),(22,44),(23,41),(24,42),(25,94),(26,95),(27,96),(28,93),(45,55),(46,56),(47,53),(48,54),(49,75),(50,76),(51,73),(52,74),(57,81),(58,82),(59,83),(60,84),(61,77),(62,78),(63,79),(64,80),(65,91),(66,92),(67,89),(68,90),(69,85),(70,86),(71,87),(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)], [(1,91,41,63),(2,90,42,62),(3,89,43,61),(4,92,44,64),(5,76,72,40),(6,75,69,39),(7,74,70,38),(8,73,71,37),(9,65,23,79),(10,68,24,78),(11,67,21,77),(12,66,22,80),(13,26,55,83),(14,25,56,82),(15,28,53,81),(16,27,54,84),(17,30,51,87),(18,29,52,86),(19,32,49,85),(20,31,50,88),(33,95,45,59),(34,94,46,58),(35,93,47,57),(36,96,48,60)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4P | 6A | ··· | 6N | 6O | ··· | 6V | 12A | ··· | 12P | 12Q | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | D4 | D4 | Q8 | C4○D4 | C3×D4 | C3×D4 | C3×Q8 | C3×C4○D4 |
| kernel | C3×C23.7Q8 | C3×C2.C42 | C6×C22⋊C4 | C6×C4⋊C4 | C23×C12 | C23.7Q8 | C22×C12 | C2.C42 | C2×C22⋊C4 | C2×C4⋊C4 | C23×C4 | C22×C4 | C2×C12 | C22×C6 | C22×C6 | C2×C6 | C2×C4 | C23 | C23 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 8 | 4 | 4 | 4 | 2 | 16 | 4 | 2 | 2 | 4 | 8 | 4 | 4 | 8 |
Matrix representation of C3×C23.7Q8 ►in GL6(𝔽13)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 10 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 11 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 2 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 11 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 |
| 0 | 0 | 0 | 0 | 11 | 10 |
G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,10,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,2,8],[5,11,0,0,0,0,12,8,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,3,11,0,0,0,0,4,10] >;
C3×C23.7Q8 in GAP, Magma, Sage, TeX
C_3\times C_2^3._7Q_8
% in TeX
G:=Group("C3xC2^3.7Q8"); // GroupNames label
G:=SmallGroup(192,813);
// by ID
G=gap.SmallGroup(192,813);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,176,1094]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^4=1,f^2=c*e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations