metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.33D14, C14.272+ 1+4, C7⋊D4⋊5D4, (C2×D4)⋊19D14, C7⋊4(D4⋊5D4), C22≀C2⋊4D7, C22⋊C4⋊6D14, C23⋊D14⋊5C2, C28⋊2D4⋊13C2, (D4×Dic7)⋊12C2, D14.16(C2×D4), (D4×C14)⋊8C22, C22.11(D4×D7), D14⋊C4⋊12C22, Dic7⋊4D4⋊3C2, Dic7⋊D4⋊3C2, (C2×C28).29C23, C4⋊Dic7⋊26C22, Dic7.19(C2×D4), C14.57(C22×D4), D14.D4⋊13C2, C22⋊4(D4⋊2D7), C23.7(C22×D7), (C2×C14).135C24, Dic7⋊C4⋊10C22, (C4×Dic7)⋊15C22, (C22×C14).9C23, C2.29(D4⋊6D14), C23.D7⋊50C22, Dic7.D4⋊12C2, C22⋊Dic14⋊13C2, (C2×Dic14)⋊20C22, C22.D28⋊10C2, (C23×C14).68C22, (C23×D7).43C22, (C22×D7).54C23, C22.156(C23×D7), (C2×Dic7).222C23, (C22×Dic7)⋊14C22, C2.30(C2×D4×D7), (C2×C4×D7)⋊8C22, (D7×C22⋊C4)⋊3C2, (C7×C22≀C2)⋊6C2, (C2×D4⋊2D7)⋊6C2, C14.77(C2×C4○D4), (C2×C14).54(C2×D4), (C2×C7⋊D4)⋊8C22, (C22×C7⋊D4)⋊9C2, (C2×C14)⋊10(C4○D4), C2.28(C2×D4⋊2D7), (C7×C22⋊C4)⋊6C22, (C2×C23.D7)⋊20C2, (C2×C4).29(C22×D7), SmallGroup(448,1044)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.33D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >
Subgroups: 1644 in 334 conjugacy classes, 107 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22×C14, D4⋊5D4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C2×Dic14, C2×C4×D7, D4⋊2D7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, D4×C14, C23×D7, C23×C14, C22⋊Dic14, D7×C22⋊C4, Dic7⋊4D4, D14.D4, Dic7.D4, C22.D28, D4×Dic7, C23⋊D14, C28⋊2D4, Dic7⋊D4, C2×C23.D7, C7×C22≀C2, C2×D4⋊2D7, C22×C7⋊D4, C24.33D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2+ 1+4, C22×D7, D4⋊5D4, D4×D7, D4⋊2D7, C23×D7, C2×D4×D7, C2×D4⋊2D7, D4⋊6D14, C24.33D14
►On 112 points
Generators in S112
(2 103)(4 105)(6 107)(8 109)(10 111)(12 85)(14 87)(16 89)(18 91)(20 93)(22 95)(24 97)(26 99)(28 101)(30 72)(32 74)(34 76)(36 78)(38 80)(40 82)(42 84)(44 58)(46 60)(48 62)(50 64)(52 66)(54 68)(56 70)
(1 31)(2 46)(3 33)(4 48)(5 35)(6 50)(7 37)(8 52)(9 39)(10 54)(11 41)(12 56)(13 43)(14 30)(15 45)(16 32)(17 47)(18 34)(19 49)(20 36)(21 51)(22 38)(23 53)(24 40)(25 55)(26 42)(27 29)(28 44)(57 86)(58 101)(59 88)(60 103)(61 90)(62 105)(63 92)(64 107)(65 94)(66 109)(67 96)(68 111)(69 98)(70 85)(71 100)(72 87)(73 102)(74 89)(75 104)(76 91)(77 106)(78 93)(79 108)(80 95)(81 110)(82 97)(83 112)(84 99)
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 85)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 97)(25 98)(26 99)(27 100)(28 101)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(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 28 15 14)(2 13 16 27)(3 26 17 12)(4 11 18 25)(5 24 19 10)(6 9 20 23)(7 22 21 8)(29 60 43 74)(30 73 44 59)(31 58 45 72)(32 71 46 57)(33 84 47 70)(34 69 48 83)(35 82 49 68)(36 67 50 81)(37 80 51 66)(38 65 52 79)(39 78 53 64)(40 63 54 77)(41 76 55 62)(42 61 56 75)(85 104 99 90)(86 89 100 103)(87 102 101 88)(91 98 105 112)(92 111 106 97)(93 96 107 110)(94 109 108 95)
G:=sub<Sym(112)| (2,103)(4,105)(6,107)(8,109)(10,111)(12,85)(14,87)(16,89)(18,91)(20,93)(22,95)(24,97)(26,99)(28,101)(30,72)(32,74)(34,76)(36,78)(38,80)(40,82)(42,84)(44,58)(46,60)(48,62)(50,64)(52,66)(54,68)(56,70), (1,31)(2,46)(3,33)(4,48)(5,35)(6,50)(7,37)(8,52)(9,39)(10,54)(11,41)(12,56)(13,43)(14,30)(15,45)(16,32)(17,47)(18,34)(19,49)(20,36)(21,51)(22,38)(23,53)(24,40)(25,55)(26,42)(27,29)(28,44)(57,86)(58,101)(59,88)(60,103)(61,90)(62,105)(63,92)(64,107)(65,94)(66,109)(67,96)(68,111)(69,98)(70,85)(71,100)(72,87)(73,102)(74,89)(75,104)(76,91)(77,106)(78,93)(79,108)(80,95)(81,110)(82,97)(83,112)(84,99), (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,85)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,97)(25,98)(26,99)(27,100)(28,101)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,28,15,14)(2,13,16,27)(3,26,17,12)(4,11,18,25)(5,24,19,10)(6,9,20,23)(7,22,21,8)(29,60,43,74)(30,73,44,59)(31,58,45,72)(32,71,46,57)(33,84,47,70)(34,69,48,83)(35,82,49,68)(36,67,50,81)(37,80,51,66)(38,65,52,79)(39,78,53,64)(40,63,54,77)(41,76,55,62)(42,61,56,75)(85,104,99,90)(86,89,100,103)(87,102,101,88)(91,98,105,112)(92,111,106,97)(93,96,107,110)(94,109,108,95)>;
G:=Group( (2,103)(4,105)(6,107)(8,109)(10,111)(12,85)(14,87)(16,89)(18,91)(20,93)(22,95)(24,97)(26,99)(28,101)(30,72)(32,74)(34,76)(36,78)(38,80)(40,82)(42,84)(44,58)(46,60)(48,62)(50,64)(52,66)(54,68)(56,70), (1,31)(2,46)(3,33)(4,48)(5,35)(6,50)(7,37)(8,52)(9,39)(10,54)(11,41)(12,56)(13,43)(14,30)(15,45)(16,32)(17,47)(18,34)(19,49)(20,36)(21,51)(22,38)(23,53)(24,40)(25,55)(26,42)(27,29)(28,44)(57,86)(58,101)(59,88)(60,103)(61,90)(62,105)(63,92)(64,107)(65,94)(66,109)(67,96)(68,111)(69,98)(70,85)(71,100)(72,87)(73,102)(74,89)(75,104)(76,91)(77,106)(78,93)(79,108)(80,95)(81,110)(82,97)(83,112)(84,99), (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,85)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,97)(25,98)(26,99)(27,100)(28,101)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,28,15,14)(2,13,16,27)(3,26,17,12)(4,11,18,25)(5,24,19,10)(6,9,20,23)(7,22,21,8)(29,60,43,74)(30,73,44,59)(31,58,45,72)(32,71,46,57)(33,84,47,70)(34,69,48,83)(35,82,49,68)(36,67,50,81)(37,80,51,66)(38,65,52,79)(39,78,53,64)(40,63,54,77)(41,76,55,62)(42,61,56,75)(85,104,99,90)(86,89,100,103)(87,102,101,88)(91,98,105,112)(92,111,106,97)(93,96,107,110)(94,109,108,95) );
G=PermutationGroup([[(2,103),(4,105),(6,107),(8,109),(10,111),(12,85),(14,87),(16,89),(18,91),(20,93),(22,95),(24,97),(26,99),(28,101),(30,72),(32,74),(34,76),(36,78),(38,80),(40,82),(42,84),(44,58),(46,60),(48,62),(50,64),(52,66),(54,68),(56,70)], [(1,31),(2,46),(3,33),(4,48),(5,35),(6,50),(7,37),(8,52),(9,39),(10,54),(11,41),(12,56),(13,43),(14,30),(15,45),(16,32),(17,47),(18,34),(19,49),(20,36),(21,51),(22,38),(23,53),(24,40),(25,55),(26,42),(27,29),(28,44),(57,86),(58,101),(59,88),(60,103),(61,90),(62,105),(63,92),(64,107),(65,94),(66,109),(67,96),(68,111),(69,98),(70,85),(71,100),(72,87),(73,102),(74,89),(75,104),(76,91),(77,106),(78,93),(79,108),(80,95),(81,110),(82,97),(83,112),(84,99)], [(1,102),(2,103),(3,104),(4,105),(5,106),(6,107),(7,108),(8,109),(9,110),(10,111),(11,112),(12,85),(13,86),(14,87),(15,88),(16,89),(17,90),(18,91),(19,92),(20,93),(21,94),(22,95),(23,96),(24,97),(25,98),(26,99),(27,100),(28,101),(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(49,63),(50,64),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)], [(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)], [(1,28,15,14),(2,13,16,27),(3,26,17,12),(4,11,18,25),(5,24,19,10),(6,9,20,23),(7,22,21,8),(29,60,43,74),(30,73,44,59),(31,58,45,72),(32,71,46,57),(33,84,47,70),(34,69,48,83),(35,82,49,68),(36,67,50,81),(37,80,51,66),(38,65,52,79),(39,78,53,64),(40,63,54,77),(41,76,55,62),(42,61,56,75),(85,104,99,90),(86,89,100,103),(87,102,101,88),(91,98,105,112),(92,111,106,97),(93,96,107,110),(94,109,108,95)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | ··· | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 14AB | 14AC | 14AD | 28A | ··· | 28I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | 2+ 1+4 | D4×D7 | D4⋊2D7 | D4⋊6D14 |
| kernel | C24.33D14 | C22⋊Dic14 | D7×C22⋊C4 | Dic7⋊4D4 | D14.D4 | Dic7.D4 | C22.D28 | D4×Dic7 | C23⋊D14 | C28⋊2D4 | Dic7⋊D4 | C2×C23.D7 | C7×C22≀C2 | C2×D4⋊2D7 | C22×C7⋊D4 | C7⋊D4 | C22≀C2 | C2×C14 | C22⋊C4 | C2×D4 | C24 | C14 | C22 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 9 | 9 | 3 | 1 | 6 | 6 | 6 |
Matrix representation of C24.33D14 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 24 |
| 0 | 0 | 0 | 0 | 24 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 9 | 8 | 0 | 0 | 0 | 0 |
| 13 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 27 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 11 | 17 |
| 26 | 26 | 0 | 0 | 0 | 0 |
| 22 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 28 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,28,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,1,0,0,0,0,0,1,0,0,0,0,0,0,18,24,0,0,0,0,24,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[9,13,0,0,0,0,8,2,0,0,0,0,0,0,28,1,0,0,0,0,27,1,0,0,0,0,0,0,12,11,0,0,0,0,0,17],[26,22,0,0,0,0,26,3,0,0,0,0,0,0,1,28,0,0,0,0,2,28,0,0,0,0,0,0,17,0,0,0,0,0,0,17] >;
C24.33D14 in GAP, Magma, Sage, TeX
C_2^4._{33}D_{14} % in TeX
G:=Group("C2^4.33D14"); // GroupNames label
G:=SmallGroup(448,1044);
// by ID
G=gap.SmallGroup(448,1044);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,1571,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=f^2=d,a*b=b*a,e*a*e^-1=f*a*f^-1=a*c=c*a,a*d=d*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*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^13>;
// generators/relations