direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C22.11C24, C14.1492+ 1+4, (C4×D4)⋊5C14, D4⋊7(C2×C28), (C2×D4)⋊11C28, (D4×C28)⋊34C2, (D4×C14)⋊23C4, C23⋊4(C2×C28), C42⋊4(C2×C14), (C4×C28)⋊38C22, C2.7(C23×C28), C42⋊C2⋊6C14, C4.19(C22×C28), C14.59(C23×C4), C24.14(C2×C14), (C22×C28)⋊5C22, (C22×D4).9C14, (C2×C28).709C23, (C2×C14).338C24, C28.164(C22×C4), C22.2(C22×C28), C2.1(C7×2+ 1+4), (D4×C14).332C22, (C23×C14).11C22, C23.30(C22×C14), C22.11(C23×C14), (C22×C14).254C23, (C2×C4)⋊4(C2×C28), C4⋊C4⋊20(C2×C14), (C2×C28)⋊25(C2×C4), (C7×D4)⋊27(C2×C4), (D4×C2×C14).22C2, (C7×C4⋊C4)⋊77C22, (C2×C22⋊C4)⋊5C14, (C22×C14)⋊5(C2×C4), (C22×C4)⋊3(C2×C14), C22⋊C4⋊18(C2×C14), (C14×C22⋊C4)⋊10C2, (C2×D4).78(C2×C14), (C7×C42⋊C2)⋊27C2, (C7×C22⋊C4)⋊72C22, (C2×C14).33(C22×C4), (C2×C4).56(C22×C14), SmallGroup(448,1301)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C22.11C24
G = < a,b,c,d,e,f,g | a7=b2=c2=e2=f2=g2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=gdg=bd=db, fef=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, df=fd, eg=ge, fg=gf >
Subgroups: 514 in 338 conjugacy classes, 242 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C28, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C42⋊C2, C4×D4, C22×D4, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C22.11C24, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, D4×C14, C23×C14, C14×C22⋊C4, C7×C42⋊C2, D4×C28, D4×C2×C14, C7×C22.11C24
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, C22×C4, C24, C28, C2×C14, C23×C4, 2+ 1+4, C2×C28, C22×C14, C22.11C24, C22×C28, C23×C14, C23×C28, C7×2+ 1+4, C7×C22.11C24
►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 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 106)(16 107)(17 108)(18 109)(19 110)(20 111)(21 112)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 43)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 71)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)
(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 21)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(36 47)(37 48)(38 49)(39 43)(40 44)(41 45)(42 46)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)
(1 91 35 94)(2 85 29 95)(3 86 30 96)(4 87 31 97)(5 88 32 98)(6 89 33 92)(7 90 34 93)(8 62 21 65)(9 63 15 66)(10 57 16 67)(11 58 17 68)(12 59 18 69)(13 60 19 70)(14 61 20 64)(22 76 111 55)(23 77 112 56)(24 71 106 50)(25 72 107 51)(26 73 108 52)(27 74 109 53)(28 75 110 54)(36 103 47 82)(37 104 48 83)(38 105 49 84)(39 99 43 78)(40 100 44 79)(41 101 45 80)(42 102 46 81)
(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 71)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 64)(7 65)(8 90)(9 91)(10 85)(11 86)(12 87)(13 88)(14 89)(15 94)(16 95)(17 96)(18 97)(19 98)(20 92)(21 93)(22 104)(23 105)(24 99)(25 100)(26 101)(27 102)(28 103)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 54)(37 55)(38 56)(39 50)(40 51)(41 52)(42 53)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 106)(16 107)(17 108)(18 109)(19 110)(20 111)(21 112)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)
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), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(36,47)(37,48)(38,49)(39,43)(40,44)(41,45)(42,46)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94), (1,91,35,94)(2,85,29,95)(3,86,30,96)(4,87,31,97)(5,88,32,98)(6,89,33,92)(7,90,34,93)(8,62,21,65)(9,63,15,66)(10,57,16,67)(11,58,17,68)(12,59,18,69)(13,60,19,70)(14,61,20,64)(22,76,111,55)(23,77,112,56)(24,71,106,50)(25,72,107,51)(26,73,108,52)(27,74,109,53)(28,75,110,54)(36,103,47,82)(37,104,48,83)(38,105,49,84)(39,99,43,78)(40,100,44,79)(41,101,45,80)(42,102,46,81), (50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99), (1,66)(2,67)(3,68)(4,69)(5,70)(6,64)(7,65)(8,90)(9,91)(10,85)(11,86)(12,87)(13,88)(14,89)(15,94)(16,95)(17,96)(18,97)(19,98)(20,92)(21,93)(22,104)(23,105)(24,99)(25,100)(26,101)(27,102)(28,103)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,54)(37,55)(38,56)(39,50)(40,51)(41,52)(42,53)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99)>;
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), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(36,47)(37,48)(38,49)(39,43)(40,44)(41,45)(42,46)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94), (1,91,35,94)(2,85,29,95)(3,86,30,96)(4,87,31,97)(5,88,32,98)(6,89,33,92)(7,90,34,93)(8,62,21,65)(9,63,15,66)(10,57,16,67)(11,58,17,68)(12,59,18,69)(13,60,19,70)(14,61,20,64)(22,76,111,55)(23,77,112,56)(24,71,106,50)(25,72,107,51)(26,73,108,52)(27,74,109,53)(28,75,110,54)(36,103,47,82)(37,104,48,83)(38,105,49,84)(39,99,43,78)(40,100,44,79)(41,101,45,80)(42,102,46,81), (50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99), (1,66)(2,67)(3,68)(4,69)(5,70)(6,64)(7,65)(8,90)(9,91)(10,85)(11,86)(12,87)(13,88)(14,89)(15,94)(16,95)(17,96)(18,97)(19,98)(20,92)(21,93)(22,104)(23,105)(24,99)(25,100)(26,101)(27,102)(28,103)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,54)(37,55)(38,56)(39,50)(40,51)(41,52)(42,53)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99) );
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)], [(1,39),(2,40),(3,41),(4,42),(5,36),(6,37),(7,38),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,22),(15,106),(16,107),(17,108),(18,109),(19,110),(20,111),(21,112),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,43),(50,66),(51,67),(52,68),(53,69),(54,70),(55,64),(56,65),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,71),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,99)], [(1,35),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,21),(9,15),(10,16),(11,17),(12,18),(13,19),(14,20),(22,111),(23,112),(24,106),(25,107),(26,108),(27,109),(28,110),(36,47),(37,48),(38,49),(39,43),(40,44),(41,45),(42,46),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,67),(58,68),(59,69),(60,70),(61,64),(62,65),(63,66),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105),(85,95),(86,96),(87,97),(88,98),(89,92),(90,93),(91,94)], [(1,91,35,94),(2,85,29,95),(3,86,30,96),(4,87,31,97),(5,88,32,98),(6,89,33,92),(7,90,34,93),(8,62,21,65),(9,63,15,66),(10,57,16,67),(11,58,17,68),(12,59,18,69),(13,60,19,70),(14,61,20,64),(22,76,111,55),(23,77,112,56),(24,71,106,50),(25,72,107,51),(26,73,108,52),(27,74,109,53),(28,75,110,54),(36,103,47,82),(37,104,48,83),(38,105,49,84),(39,99,43,78),(40,100,44,79),(41,101,45,80),(42,102,46,81)], [(50,66),(51,67),(52,68),(53,69),(54,70),(55,64),(56,65),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,71),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,99)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,64),(7,65),(8,90),(9,91),(10,85),(11,86),(12,87),(13,88),(14,89),(15,94),(16,95),(17,96),(18,97),(19,98),(20,92),(21,93),(22,104),(23,105),(24,99),(25,100),(26,101),(27,102),(28,103),(29,57),(30,58),(31,59),(32,60),(33,61),(34,62),(35,63),(36,54),(37,55),(38,56),(39,50),(40,51),(41,52),(42,53),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,22),(15,106),(16,107),(17,108),(18,109),(19,110),(20,111),(21,112),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,99)]])
238 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 4A | ··· | 4T | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14BZ | 28A | ··· | 28DP |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
238 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | 2+ 1+4 | C7×2+ 1+4 |
| kernel | C7×C22.11C24 | C14×C22⋊C4 | C7×C42⋊C2 | D4×C28 | D4×C2×C14 | D4×C14 | C22.11C24 | C2×C22⋊C4 | C42⋊C2 | C4×D4 | C22×D4 | C2×D4 | C14 | C2 |
| # reps | 1 | 4 | 2 | 8 | 1 | 16 | 6 | 24 | 12 | 48 | 6 | 96 | 2 | 12 |
Matrix representation of C7×C22.11C24 ►in GL5(𝔽29)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 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 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 2 | 0 |
| 0 | 0 | 0 | 28 | 1 |
| 0 | 1 | 0 | 12 | 0 |
| 0 | 1 | 1 | 12 | 0 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 28 | 28 | 0 | 0 |
| 0 | 12 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 12 | 0 | 1 |
| 0 | 0 | 12 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 0 | 28 | 0 |
| 0 | 12 | 0 | 0 | 28 |
G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[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],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,17,0,1,1,0,0,0,0,1,0,2,28,12,12,0,0,1,0,0],[28,0,0,0,0,0,1,28,12,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1],[28,0,0,0,0,0,1,0,0,0,0,2,28,12,12,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,12,12,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28] >;
C7×C22.11C24 in GAP, Magma, Sage, TeX
C_7\times C_2^2._{11}C_2^4 % in TeX
G:=Group("C7xC2^2.11C2^4"); // GroupNames label
G:=SmallGroup(448,1301);
// by ID
G=gap.SmallGroup(448,1301);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,1227,3363]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=e^2=f^2=g^2=1,d^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=g*d*g=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*f=f*d,e*g=g*e,f*g=g*f>;
// generators/relations