direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C22.45C24, C14.1632+ (1+4), (D4×C28)⋊48C2, (C4×D4)⋊19C14, C42⋊10(C2×C14), (C4×C28)⋊44C22, C22⋊Q8⋊15C14, C42⋊2C2⋊4C14, C22≀C2.2C14, C4.4D4⋊12C14, C24.23(C2×C14), (C22×C28)⋊6C22, (Q8×C14)⋊30C22, C42⋊C2⋊14C14, (C2×C28).678C23, (C2×C14).371C24, (D4×C14).323C22, C22.D4⋊10C14, (C23×C14).20C22, C23.18(C22×C14), C22.45(C23×C14), C2.15(C7×2+ (1+4)), (C22×C14).266C23, C4⋊C4⋊17(C2×C14), (C2×Q8)⋊5(C2×C14), C22⋊C4⋊6(C2×C14), (C7×C4⋊C4)⋊74C22, (C22×C4)⋊4(C2×C14), C2.24(C14×C4○D4), (C7×C22⋊Q8)⋊42C2, C22.9(C7×C4○D4), (C14×C22⋊C4)⋊35C2, (C2×C22⋊C4)⋊15C14, (C7×C22≀C2).4C2, (C2×D4).69(C2×C14), C14.243(C2×C4○D4), (C7×C4.4D4)⋊32C2, (C7×C42⋊C2)⋊35C2, (C7×C42⋊2C2)⋊15C2, (C7×C22⋊C4)⋊41C22, (C2×C4).61(C22×C14), (C2×C14).118(C4○D4), (C7×C22.D4)⋊29C2, SmallGroup(448,1334)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 394 in 248 conjugacy classes, 150 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×11], C22, C22 [×4], C22 [×14], C7, C2×C4, C2×C4 [×10], C2×C4 [×7], D4 [×5], Q8, C23 [×2], C23 [×2], C23 [×5], C14, C14 [×2], C14 [×6], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, C24, C28 [×11], C2×C14, C2×C14 [×4], C2×C14 [×14], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C22.D4 [×2], C4.4D4, C42⋊2C2 [×2], C2×C28, C2×C28 [×10], C2×C28 [×7], C7×D4 [×5], C7×Q8, C22×C14 [×2], C22×C14 [×2], C22×C14 [×5], C22.45C24, C4×C28, C4×C28 [×2], C7×C22⋊C4 [×2], C7×C22⋊C4 [×12], C7×C4⋊C4 [×8], C22×C28, C22×C28 [×4], D4×C14, D4×C14 [×2], Q8×C14, C23×C14, C14×C22⋊C4 [×2], C7×C42⋊C2 [×2], D4×C28 [×2], C7×C22≀C2, C7×C22⋊Q8 [×2], C7×C22.D4, C7×C22.D4 [×2], C7×C4.4D4, C7×C42⋊2C2 [×2], C7×C22.45C24
Quotients:
C1, C2 [×15], C22 [×35], C7, C23 [×15], C14 [×15], C4○D4 [×4], C24, C2×C14 [×35], C2×C4○D4 [×2], 2+ (1+4), C22×C14 [×15], C22.45C24, C7×C4○D4 [×4], C23×C14, C14×C4○D4 [×2], C7×2+ (1+4), C7×C22.45C24
Generators and relations
G = < a,b,c,d,e,f,g | a7=b2=c2=f2=g2=1, d2=b, e2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >
►On 112 points
Generators in S112
(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 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 36)(7 37)(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)(15 109)(16 110)(17 111)(18 112)(19 106)(20 107)(21 108)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 43)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 64)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(91 92)
(1 46)(2 47)(3 48)(4 49)(5 43)(6 44)(7 45)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)
(1 78 38 102)(2 79 39 103)(3 80 40 104)(4 81 41 105)(5 82 42 99)(6 83 36 100)(7 84 37 101)(8 70 27 62)(9 64 28 63)(10 65 22 57)(11 66 23 58)(12 67 24 59)(13 68 25 60)(14 69 26 61)(15 77 109 53)(16 71 110 54)(17 72 111 55)(18 73 112 56)(19 74 106 50)(20 75 107 51)(21 76 108 52)(29 93 44 85)(30 94 45 86)(31 95 46 87)(32 96 47 88)(33 97 48 89)(34 98 49 90)(35 92 43 91)
(1 67 46 74)(2 68 47 75)(3 69 48 76)(4 70 49 77)(5 64 43 71)(6 65 44 72)(7 66 45 73)(8 90 15 81)(9 91 16 82)(10 85 17 83)(11 86 18 84)(12 87 19 78)(13 88 20 79)(14 89 21 80)(22 93 111 100)(23 94 112 101)(24 95 106 102)(25 96 107 103)(26 97 108 104)(27 98 109 105)(28 92 110 99)(29 55 36 57)(30 56 37 58)(31 50 38 59)(32 51 39 60)(33 52 40 61)(34 53 41 62)(35 54 42 63)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 36)(7 37)(8 109)(9 110)(10 111)(11 112)(12 106)(13 107)(14 108)(15 27)(16 28)(17 22)(18 23)(19 24)(20 25)(21 26)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 43)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 64)(78 95)(79 96)(80 97)(81 98)(82 92)(83 93)(84 94)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)
(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)(15 109)(16 110)(17 111)(18 112)(19 106)(20 107)(21 108)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 64)
G:=sub<Sym(112)| (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)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (1,78,38,102)(2,79,39,103)(3,80,40,104)(4,81,41,105)(5,82,42,99)(6,83,36,100)(7,84,37,101)(8,70,27,62)(9,64,28,63)(10,65,22,57)(11,66,23,58)(12,67,24,59)(13,68,25,60)(14,69,26,61)(15,77,109,53)(16,71,110,54)(17,72,111,55)(18,73,112,56)(19,74,106,50)(20,75,107,51)(21,76,108,52)(29,93,44,85)(30,94,45,86)(31,95,46,87)(32,96,47,88)(33,97,48,89)(34,98,49,90)(35,92,43,91), (1,67,46,74)(2,68,47,75)(3,69,48,76)(4,70,49,77)(5,64,43,71)(6,65,44,72)(7,66,45,73)(8,90,15,81)(9,91,16,82)(10,85,17,83)(11,86,18,84)(12,87,19,78)(13,88,20,79)(14,89,21,80)(22,93,111,100)(23,94,112,101)(24,95,106,102)(25,96,107,103)(26,97,108,104)(27,98,109,105)(28,92,110,99)(29,55,36,57)(30,56,37,58)(31,50,38,59)(32,51,39,60)(33,52,40,61)(34,53,41,62)(35,54,42,63), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,109)(9,110)(10,111)(11,112)(12,106)(13,107)(14,108)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,95)(79,96)(80,97)(81,98)(82,92)(83,93)(84,94)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99), (8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)>;
G:=Group( (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)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105), (1,78,38,102)(2,79,39,103)(3,80,40,104)(4,81,41,105)(5,82,42,99)(6,83,36,100)(7,84,37,101)(8,70,27,62)(9,64,28,63)(10,65,22,57)(11,66,23,58)(12,67,24,59)(13,68,25,60)(14,69,26,61)(15,77,109,53)(16,71,110,54)(17,72,111,55)(18,73,112,56)(19,74,106,50)(20,75,107,51)(21,76,108,52)(29,93,44,85)(30,94,45,86)(31,95,46,87)(32,96,47,88)(33,97,48,89)(34,98,49,90)(35,92,43,91), (1,67,46,74)(2,68,47,75)(3,69,48,76)(4,70,49,77)(5,64,43,71)(6,65,44,72)(7,66,45,73)(8,90,15,81)(9,91,16,82)(10,85,17,83)(11,86,18,84)(12,87,19,78)(13,88,20,79)(14,89,21,80)(22,93,111,100)(23,94,112,101)(24,95,106,102)(25,96,107,103)(26,97,108,104)(27,98,109,105)(28,92,110,99)(29,55,36,57)(30,56,37,58)(31,50,38,59)(32,51,39,60)(33,52,40,61)(34,53,41,62)(35,54,42,63), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,109)(9,110)(10,111)(11,112)(12,106)(13,107)(14,108)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,95)(79,96)(80,97)(81,98)(82,92)(83,93)(84,94)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99), (8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64) );
G=PermutationGroup([(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),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,36),(7,37),(8,27),(9,28),(10,22),(11,23),(12,24),(13,25),(14,26),(15,109),(16,110),(17,111),(18,112),(19,106),(20,107),(21,108),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,43),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,64),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,93),(86,94),(87,95),(88,96),(89,97),(90,98),(91,92)], [(1,46),(2,47),(3,48),(4,49),(5,43),(6,44),(7,45),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(22,111),(23,112),(24,106),(25,107),(26,108),(27,109),(28,110),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105)], [(1,78,38,102),(2,79,39,103),(3,80,40,104),(4,81,41,105),(5,82,42,99),(6,83,36,100),(7,84,37,101),(8,70,27,62),(9,64,28,63),(10,65,22,57),(11,66,23,58),(12,67,24,59),(13,68,25,60),(14,69,26,61),(15,77,109,53),(16,71,110,54),(17,72,111,55),(18,73,112,56),(19,74,106,50),(20,75,107,51),(21,76,108,52),(29,93,44,85),(30,94,45,86),(31,95,46,87),(32,96,47,88),(33,97,48,89),(34,98,49,90),(35,92,43,91)], [(1,67,46,74),(2,68,47,75),(3,69,48,76),(4,70,49,77),(5,64,43,71),(6,65,44,72),(7,66,45,73),(8,90,15,81),(9,91,16,82),(10,85,17,83),(11,86,18,84),(12,87,19,78),(13,88,20,79),(14,89,21,80),(22,93,111,100),(23,94,112,101),(24,95,106,102),(25,96,107,103),(26,97,108,104),(27,98,109,105),(28,92,110,99),(29,55,36,57),(30,56,37,58),(31,50,38,59),(32,51,39,60),(33,52,40,61),(34,53,41,62),(35,54,42,63)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,36),(7,37),(8,109),(9,110),(10,111),(11,112),(12,106),(13,107),(14,108),(15,27),(16,28),(17,22),(18,23),(19,24),(20,25),(21,26),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,43),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,64),(78,95),(79,96),(80,97),(81,98),(82,92),(83,93),(84,94),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,99)], [(8,27),(9,28),(10,22),(11,23),(12,24),(13,25),(14,26),(15,109),(16,110),(17,111),(18,112),(19,106),(20,107),(21,108),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,64)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 23 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 12 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 28 | 27 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 28 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(29))| [23,0,0,0,0,23,0,0,0,0,25,0,0,0,0,25],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[12,0,0,0,0,17,0,0,0,0,28,0,0,0,27,1],[0,1,0,0,1,0,0,0,0,0,12,0,0,0,0,12],[28,0,0,0,0,28,0,0,0,0,1,28,0,0,0,28],[1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1] >;
175 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4O | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AP | 14AQ | ··· | 14BB | 28A | ··· | 28AV | 28AW | ··· | 28CL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
175 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | C4○D4 | C7×C4○D4 | 2+ (1+4) | C7×2+ (1+4) |
| kernel | C7×C22.45C24 | C14×C22⋊C4 | C7×C42⋊C2 | D4×C28 | C7×C22≀C2 | C7×C22⋊Q8 | C7×C22.D4 | C7×C4.4D4 | C7×C42⋊2C2 | C22.45C24 | C2×C22⋊C4 | C42⋊C2 | C4×D4 | C22≀C2 | C22⋊Q8 | C22.D4 | C4.4D4 | C42⋊2C2 | C2×C14 | C22 | C14 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 3 | 1 | 2 | 6 | 12 | 12 | 12 | 6 | 12 | 18 | 6 | 12 | 8 | 48 | 1 | 6 |
In GAP, Magma, Sage, TeX
C_7\times C_2^2._{45}C_2^4 % in TeX
G:=Group("C7xC2^2.45C2^4"); // GroupNames label
G:=SmallGroup(448,1334);
// by ID
G=gap.SmallGroup(448,1334);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,1576,4790,1690]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=f^2=g^2=1,d^2=b,e^2=c,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=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=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
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