direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C22.19C24, (C4×D4)⋊7C14, (C2×C28)⋊41D4, (D4×C28)⋊36C2, C42⋊5(C2×C14), (C23×C4)⋊8C14, C4.64(D4×C14), C22≀C2⋊9C14, C4⋊D4⋊19C14, (C4×C28)⋊39C22, (C23×C28)⋊15C2, C28.471(C2×D4), C22⋊Q8⋊21C14, C42⋊C2⋊8C14, (D4×C14)⋊62C22, C24.36(C2×C14), (Q8×C14)⋊49C22, C22.20(D4×C14), (C2×C14).345C24, (C2×C28).658C23, (C22×C28)⋊46C22, C14.184(C22×D4), C22.D4⋊15C14, (C22×C14).84C23, C23.33(C22×C14), (C23×C14).93C22, C22.19(C23×C14), C2.8(D4×C2×C14), (C2×C4)⋊11(C7×D4), C4⋊C4⋊12(C2×C14), (C2×C4○D4)⋊2C14, (C2×Q8)⋊9(C2×C14), C2.8(C14×C4○D4), C22⋊1(C7×C4○D4), (C14×C4○D4)⋊18C2, (C2×D4)⋊10(C2×C14), (C7×C4⋊D4)⋊46C2, (C7×C4⋊C4)⋊68C22, (C22×C4)⋊6(C2×C14), (C7×C22⋊Q8)⋊48C2, (C7×C22≀C2)⋊19C2, (C2×C14)⋊11(C4○D4), C22⋊C4⋊13(C2×C14), C14.227(C2×C4○D4), (C2×C14).416(C2×D4), (C7×C42⋊C2)⋊29C2, (C7×C22⋊C4)⋊67C22, (C2×C4).14(C22×C14), (C7×C22.D4)⋊34C2, SmallGroup(448,1308)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 498 in 330 conjugacy classes, 170 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C7, C2×C4 [×2], C2×C4 [×12], C2×C4 [×14], D4 [×14], Q8 [×2], C23, C23 [×4], C23 [×6], C14, C14 [×2], C14 [×8], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×6], C2×Q8, C4○D4 [×4], C24, C28 [×4], C28 [×8], C2×C14, C2×C14 [×6], C2×C14 [×20], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, C2×C28 [×2], C2×C28 [×12], C2×C28 [×14], C7×D4 [×14], C7×Q8 [×2], C22×C14, C22×C14 [×4], C22×C14 [×6], C22.19C24, C4×C28 [×2], C7×C22⋊C4 [×10], C7×C4⋊C4 [×6], C22×C28 [×2], C22×C28 [×6], C22×C28 [×4], D4×C14, D4×C14 [×6], Q8×C14, C7×C4○D4 [×4], C23×C14, C7×C42⋊C2, D4×C28 [×4], C7×C22≀C2 [×2], C7×C4⋊D4 [×2], C7×C22⋊Q8 [×2], C7×C22.D4 [×2], C23×C28, C14×C4○D4, C7×C22.19C24
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C4○D4 [×4], C24, C2×C14 [×35], C22×D4, C2×C4○D4 [×2], C7×D4 [×4], C22×C14 [×15], C22.19C24, D4×C14 [×6], C7×C4○D4 [×4], C23×C14, D4×C2×C14, C14×C4○D4 [×2], C7×C22.19C24
Generators and relations
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=e2=f2=1, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, 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)
(1 102)(2 103)(3 104)(4 105)(5 99)(6 100)(7 101)(8 70)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 77)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 83)(37 84)(38 78)(39 79)(40 80)(41 81)(42 82)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)
(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)(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)
(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)(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 50 38 74)(2 51 39 75)(3 52 40 76)(4 53 41 77)(5 54 42 71)(6 55 36 72)(7 56 37 73)(8 98 27 90)(9 92 28 91)(10 93 22 85)(11 94 23 86)(12 95 24 87)(13 96 25 88)(14 97 26 89)(15 105 109 81)(16 99 110 82)(17 100 111 83)(18 101 112 84)(19 102 106 78)(20 103 107 79)(21 104 108 80)(29 65 44 57)(30 66 45 58)(31 67 46 59)(32 68 47 60)(33 69 48 61)(34 70 49 62)(35 64 43 63)
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), (1,102)(2,103)(3,104)(4,105)(5,99)(6,100)(7,101)(8,70)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,77)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,83)(37,84)(38,78)(39,79)(40,80)(41,81)(42,82)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112), (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)(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), (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)(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), (1,50,38,74)(2,51,39,75)(3,52,40,76)(4,53,41,77)(5,54,42,71)(6,55,36,72)(7,56,37,73)(8,98,27,90)(9,92,28,91)(10,93,22,85)(11,94,23,86)(12,95,24,87)(13,96,25,88)(14,97,26,89)(15,105,109,81)(16,99,110,82)(17,100,111,83)(18,101,112,84)(19,102,106,78)(20,103,107,79)(21,104,108,80)(29,65,44,57)(30,66,45,58)(31,67,46,59)(32,68,47,60)(33,69,48,61)(34,70,49,62)(35,64,43,63)>;
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), (1,102)(2,103)(3,104)(4,105)(5,99)(6,100)(7,101)(8,70)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,77)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,83)(37,84)(38,78)(39,79)(40,80)(41,81)(42,82)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112), (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)(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), (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)(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), (1,50,38,74)(2,51,39,75)(3,52,40,76)(4,53,41,77)(5,54,42,71)(6,55,36,72)(7,56,37,73)(8,98,27,90)(9,92,28,91)(10,93,22,85)(11,94,23,86)(12,95,24,87)(13,96,25,88)(14,97,26,89)(15,105,109,81)(16,99,110,82)(17,100,111,83)(18,101,112,84)(19,102,106,78)(20,103,107,79)(21,104,108,80)(29,65,44,57)(30,66,45,58)(31,67,46,59)(32,68,47,60)(33,69,48,61)(34,70,49,62)(35,64,43,63) );
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)], [(1,102),(2,103),(3,104),(4,105),(5,99),(6,100),(7,101),(8,70),(9,64),(10,65),(11,66),(12,67),(13,68),(14,69),(15,77),(16,71),(17,72),(18,73),(19,74),(20,75),(21,76),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,83),(37,84),(38,78),(39,79),(40,80),(41,81),(42,82),(43,92),(44,93),(45,94),(46,95),(47,96),(48,97),(49,98),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112)], [(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),(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)], [(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),(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)], [(1,50,38,74),(2,51,39,75),(3,52,40,76),(4,53,41,77),(5,54,42,71),(6,55,36,72),(7,56,37,73),(8,98,27,90),(9,92,28,91),(10,93,22,85),(11,94,23,86),(12,95,24,87),(13,96,25,88),(14,97,26,89),(15,105,109,81),(16,99,110,82),(17,100,111,83),(18,101,112,84),(19,102,106,78),(20,103,107,79),(21,104,108,80),(29,65,44,57),(30,66,45,58),(31,67,46,59),(32,68,47,60),(33,69,48,61),(34,70,49,62),(35,64,43,63)])
Matrix representation ►G ⊆ GL5(𝔽29)
| 7 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(5,GF(29))| [7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,17,0,0,0,0,0,17,0,0,0,0,0,17,0,0,0,0,0,17] >;
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4P | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14BB | 14BC | ··· | 14BN | 28A | ··· | 28X | 28Y | ··· | 28BH | 28BI | ··· | 28CR |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | D4 | C4○D4 | C7×D4 | C7×C4○D4 |
| kernel | C7×C22.19C24 | C7×C42⋊C2 | D4×C28 | C7×C22≀C2 | C7×C4⋊D4 | C7×C22⋊Q8 | C7×C22.D4 | C23×C28 | C14×C4○D4 | C22.19C24 | C42⋊C2 | C4×D4 | C22≀C2 | C4⋊D4 | C22⋊Q8 | C22.D4 | C23×C4 | C2×C4○D4 | C2×C28 | C2×C14 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 6 | 6 | 24 | 12 | 12 | 12 | 12 | 6 | 6 | 4 | 8 | 24 | 48 |
In GAP, Magma, Sage, TeX
C_7\times C_2^2._{19}C_2^4 % in TeX
G:=Group("C7xC2^2.19C2^4"); // GroupNames label
G:=SmallGroup(448,1308);
// by ID
G=gap.SmallGroup(448,1308);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,416]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=e^2=f^2=1,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=b*d=d*b,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,e*g=g*e,f*g=g*f>;
// generators/relations
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