direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C22.49C24, C6.1662+ (1+4), C4⋊Q8⋊16C6, (C4×D4)⋊21C6, (D4×C12)⋊50C2, C4⋊D4⋊16C6, C4.4D4⋊13C6, C42.49(C2×C6), C42⋊C2⋊17C6, (C2×C6).375C24, C12.326(C4○D4), (C2×C12).964C23, (C4×C12).290C22, (C6×D4).222C22, C22.49(C23×C6), C23.20(C22×C6), (C6×Q8).185C22, (C22×C6).103C23, C2.18(C3×2+ (1+4)), (C22×C12).460C22, (C3×C4⋊Q8)⋊37C2, C4⋊C4.75(C2×C6), C4.38(C3×C4○D4), C2.28(C6×C4○D4), (C3×C4⋊D4)⋊43C2, (C2×D4).35(C2×C6), C6.247(C2×C4○D4), (C2×Q8).29(C2×C6), (C3×C4.4D4)⋊33C2, C22⋊C4.25(C2×C6), (C22×C4).76(C2×C6), (C2×C4).37(C22×C6), (C3×C42⋊C2)⋊38C2, (C3×C4⋊C4).409C22, (C3×C22⋊C4).157C22, SmallGroup(192,1444)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 362 in 236 conjugacy classes, 150 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×4], C4 [×9], C22, C22 [×12], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×10], C2×C4 [×8], D4 [×8], Q8 [×2], C23 [×4], C12 [×4], C12 [×9], C2×C6, C2×C6 [×12], C42, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], C2×C12, C2×C12 [×10], C2×C12 [×8], C3×D4 [×8], C3×Q8 [×2], C22×C6 [×4], C42⋊C2 [×4], C4×D4 [×2], C4⋊D4 [×4], C4.4D4 [×4], C4⋊Q8, C4×C12, C4×C12 [×4], C3×C22⋊C4 [×12], C3×C4⋊C4 [×6], C22×C12 [×4], C6×D4 [×6], C6×Q8 [×2], C22.49C24, C3×C42⋊C2 [×4], D4×C12 [×2], C3×C4⋊D4 [×4], C3×C4.4D4 [×4], C3×C4⋊Q8, C3×C22.49C24
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], C2×C6 [×35], C4○D4 [×4], C24, C22×C6 [×15], C2×C4○D4 [×2], 2+ (1+4), C3×C4○D4 [×4], C23×C6, C22.49C24, C6×C4○D4 [×2], C3×2+ (1+4), C3×C22.49C24
Generators and relations
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=1, e2=c, f2=g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg-1=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
►On 96 points
Generators in S96
(1 59 11)(2 60 12)(3 57 9)(4 58 10)(5 54 26)(6 55 27)(7 56 28)(8 53 25)(13 17 61)(14 18 62)(15 19 63)(16 20 64)(21 65 69)(22 66 70)(23 67 71)(24 68 72)(29 73 77)(30 74 78)(31 75 79)(32 76 80)(33 37 81)(34 38 82)(35 39 83)(36 40 84)(41 85 89)(42 86 90)(43 87 91)(44 88 92)(45 51 93)(46 52 94)(47 49 95)(48 50 96)
(1 75)(2 76)(3 73)(4 74)(5 52)(6 49)(7 50)(8 51)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(53 93)(54 94)(55 95)(56 96)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)(65 67)(66 68)(69 71)(70 72)(73 75)(74 76)(77 79)(78 80)(81 83)(82 84)(85 87)(86 88)(89 91)(90 92)(93 95)(94 96)
(1 26)(2 47)(3 28)(4 45)(5 59)(6 80)(7 57)(8 78)(9 56)(10 93)(11 54)(12 95)(13 68)(14 85)(15 66)(16 87)(17 72)(18 89)(19 70)(20 91)(21 82)(22 63)(23 84)(24 61)(25 74)(27 76)(29 96)(30 53)(31 94)(32 55)(33 88)(34 65)(35 86)(36 67)(37 92)(38 69)(39 90)(40 71)(41 62)(42 83)(43 64)(44 81)(46 75)(48 73)(49 60)(50 77)(51 58)(52 79)
(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 87 75 67)(2 88 76 68)(3 85 73 65)(4 86 74 66)(5 18 52 38)(6 19 49 39)(7 20 50 40)(8 17 51 37)(9 41 29 21)(10 42 30 22)(11 43 31 23)(12 44 32 24)(13 45 33 25)(14 46 34 26)(15 47 35 27)(16 48 36 28)(53 61 93 81)(54 62 94 82)(55 63 95 83)(56 64 96 84)(57 89 77 69)(58 90 78 70)(59 91 79 71)(60 92 80 72)
(1 35 75 15)(2 16 76 36)(3 33 73 13)(4 14 74 34)(5 90 52 70)(6 71 49 91)(7 92 50 72)(8 69 51 89)(9 81 29 61)(10 62 30 82)(11 83 31 63)(12 64 32 84)(17 57 37 77)(18 78 38 58)(19 59 39 79)(20 80 40 60)(21 93 41 53)(22 54 42 94)(23 95 43 55)(24 56 44 96)(25 65 45 85)(26 86 46 66)(27 67 47 87)(28 88 48 68)
G:=sub<Sym(96)| (1,59,11)(2,60,12)(3,57,9)(4,58,10)(5,54,26)(6,55,27)(7,56,28)(8,53,25)(13,17,61)(14,18,62)(15,19,63)(16,20,64)(21,65,69)(22,66,70)(23,67,71)(24,68,72)(29,73,77)(30,74,78)(31,75,79)(32,76,80)(33,37,81)(34,38,82)(35,39,83)(36,40,84)(41,85,89)(42,86,90)(43,87,91)(44,88,92)(45,51,93)(46,52,94)(47,49,95)(48,50,96), (1,75)(2,76)(3,73)(4,74)(5,52)(6,49)(7,50)(8,51)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(53,93)(54,94)(55,95)(56,96)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84)(85,87)(86,88)(89,91)(90,92)(93,95)(94,96), (1,26)(2,47)(3,28)(4,45)(5,59)(6,80)(7,57)(8,78)(9,56)(10,93)(11,54)(12,95)(13,68)(14,85)(15,66)(16,87)(17,72)(18,89)(19,70)(20,91)(21,82)(22,63)(23,84)(24,61)(25,74)(27,76)(29,96)(30,53)(31,94)(32,55)(33,88)(34,65)(35,86)(36,67)(37,92)(38,69)(39,90)(40,71)(41,62)(42,83)(43,64)(44,81)(46,75)(48,73)(49,60)(50,77)(51,58)(52,79), (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), (1,87,75,67)(2,88,76,68)(3,85,73,65)(4,86,74,66)(5,18,52,38)(6,19,49,39)(7,20,50,40)(8,17,51,37)(9,41,29,21)(10,42,30,22)(11,43,31,23)(12,44,32,24)(13,45,33,25)(14,46,34,26)(15,47,35,27)(16,48,36,28)(53,61,93,81)(54,62,94,82)(55,63,95,83)(56,64,96,84)(57,89,77,69)(58,90,78,70)(59,91,79,71)(60,92,80,72), (1,35,75,15)(2,16,76,36)(3,33,73,13)(4,14,74,34)(5,90,52,70)(6,71,49,91)(7,92,50,72)(8,69,51,89)(9,81,29,61)(10,62,30,82)(11,83,31,63)(12,64,32,84)(17,57,37,77)(18,78,38,58)(19,59,39,79)(20,80,40,60)(21,93,41,53)(22,54,42,94)(23,95,43,55)(24,56,44,96)(25,65,45,85)(26,86,46,66)(27,67,47,87)(28,88,48,68)>;
G:=Group( (1,59,11)(2,60,12)(3,57,9)(4,58,10)(5,54,26)(6,55,27)(7,56,28)(8,53,25)(13,17,61)(14,18,62)(15,19,63)(16,20,64)(21,65,69)(22,66,70)(23,67,71)(24,68,72)(29,73,77)(30,74,78)(31,75,79)(32,76,80)(33,37,81)(34,38,82)(35,39,83)(36,40,84)(41,85,89)(42,86,90)(43,87,91)(44,88,92)(45,51,93)(46,52,94)(47,49,95)(48,50,96), (1,75)(2,76)(3,73)(4,74)(5,52)(6,49)(7,50)(8,51)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(53,93)(54,94)(55,95)(56,96)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84)(85,87)(86,88)(89,91)(90,92)(93,95)(94,96), (1,26)(2,47)(3,28)(4,45)(5,59)(6,80)(7,57)(8,78)(9,56)(10,93)(11,54)(12,95)(13,68)(14,85)(15,66)(16,87)(17,72)(18,89)(19,70)(20,91)(21,82)(22,63)(23,84)(24,61)(25,74)(27,76)(29,96)(30,53)(31,94)(32,55)(33,88)(34,65)(35,86)(36,67)(37,92)(38,69)(39,90)(40,71)(41,62)(42,83)(43,64)(44,81)(46,75)(48,73)(49,60)(50,77)(51,58)(52,79), (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), (1,87,75,67)(2,88,76,68)(3,85,73,65)(4,86,74,66)(5,18,52,38)(6,19,49,39)(7,20,50,40)(8,17,51,37)(9,41,29,21)(10,42,30,22)(11,43,31,23)(12,44,32,24)(13,45,33,25)(14,46,34,26)(15,47,35,27)(16,48,36,28)(53,61,93,81)(54,62,94,82)(55,63,95,83)(56,64,96,84)(57,89,77,69)(58,90,78,70)(59,91,79,71)(60,92,80,72), (1,35,75,15)(2,16,76,36)(3,33,73,13)(4,14,74,34)(5,90,52,70)(6,71,49,91)(7,92,50,72)(8,69,51,89)(9,81,29,61)(10,62,30,82)(11,83,31,63)(12,64,32,84)(17,57,37,77)(18,78,38,58)(19,59,39,79)(20,80,40,60)(21,93,41,53)(22,54,42,94)(23,95,43,55)(24,56,44,96)(25,65,45,85)(26,86,46,66)(27,67,47,87)(28,88,48,68) );
G=PermutationGroup([(1,59,11),(2,60,12),(3,57,9),(4,58,10),(5,54,26),(6,55,27),(7,56,28),(8,53,25),(13,17,61),(14,18,62),(15,19,63),(16,20,64),(21,65,69),(22,66,70),(23,67,71),(24,68,72),(29,73,77),(30,74,78),(31,75,79),(32,76,80),(33,37,81),(34,38,82),(35,39,83),(36,40,84),(41,85,89),(42,86,90),(43,87,91),(44,88,92),(45,51,93),(46,52,94),(47,49,95),(48,50,96)], [(1,75),(2,76),(3,73),(4,74),(5,52),(6,49),(7,50),(8,51),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(53,93),(54,94),(55,95),(56,96),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,91),(72,92)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64),(65,67),(66,68),(69,71),(70,72),(73,75),(74,76),(77,79),(78,80),(81,83),(82,84),(85,87),(86,88),(89,91),(90,92),(93,95),(94,96)], [(1,26),(2,47),(3,28),(4,45),(5,59),(6,80),(7,57),(8,78),(9,56),(10,93),(11,54),(12,95),(13,68),(14,85),(15,66),(16,87),(17,72),(18,89),(19,70),(20,91),(21,82),(22,63),(23,84),(24,61),(25,74),(27,76),(29,96),(30,53),(31,94),(32,55),(33,88),(34,65),(35,86),(36,67),(37,92),(38,69),(39,90),(40,71),(41,62),(42,83),(43,64),(44,81),(46,75),(48,73),(49,60),(50,77),(51,58),(52,79)], [(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)], [(1,87,75,67),(2,88,76,68),(3,85,73,65),(4,86,74,66),(5,18,52,38),(6,19,49,39),(7,20,50,40),(8,17,51,37),(9,41,29,21),(10,42,30,22),(11,43,31,23),(12,44,32,24),(13,45,33,25),(14,46,34,26),(15,47,35,27),(16,48,36,28),(53,61,93,81),(54,62,94,82),(55,63,95,83),(56,64,96,84),(57,89,77,69),(58,90,78,70),(59,91,79,71),(60,92,80,72)], [(1,35,75,15),(2,16,76,36),(3,33,73,13),(4,14,74,34),(5,90,52,70),(6,71,49,91),(7,92,50,72),(8,69,51,89),(9,81,29,61),(10,62,30,82),(11,83,31,63),(12,64,32,84),(17,57,37,77),(18,78,38,58),(19,59,39,79),(20,80,40,60),(21,93,41,53),(22,54,42,94),(23,95,43,55),(24,56,44,96),(25,65,45,85),(26,86,46,66),(27,67,47,87),(28,88,48,68)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 8 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 8 | 10 |
| 0 | 0 | 8 | 5 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 12 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 2 |
| 0 | 0 | 12 | 1 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[0,5,0,0,8,0,0,0,0,0,8,8,0,0,10,5],[8,0,0,0,0,8,0,0,0,0,12,12,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,2,1] >;
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | ··· | 4L | 4M | ··· | 4Q | 6A | ··· | 6F | 6G | ··· | 6N | 12A | ··· | 12X | 12Y | ··· | 12AH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C4○D4 | C3×C4○D4 | 2+ (1+4) | C3×2+ (1+4) |
| kernel | C3×C22.49C24 | C3×C42⋊C2 | D4×C12 | C3×C4⋊D4 | C3×C4.4D4 | C3×C4⋊Q8 | C22.49C24 | C42⋊C2 | C4×D4 | C4⋊D4 | C4.4D4 | C4⋊Q8 | C12 | C4 | C6 | C2 |
| # reps | 1 | 4 | 2 | 4 | 4 | 1 | 2 | 8 | 4 | 8 | 8 | 2 | 8 | 16 | 1 | 2 |
In GAP, Magma, Sage, TeX
C_3\times C_2^2._{49}C_2^4 % in TeX
G:=Group("C3xC2^2.49C2^4"); // GroupNames label
G:=SmallGroup(192,1444);
// by ID
G=gap.SmallGroup(192,1444);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,680,2102,268,794,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=1,e^2=c,f^2=g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀