metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.72D14, (C2×C28)⋊38D4, (C23×C4)⋊6D7, C28⋊7D4⋊51C2, (C23×C28)⋊10C2, C28.425(C2×D4), D14⋊C4⋊43C22, (C2×D28)⋊51C22, C22⋊5(C4○D28), C24⋊D7⋊15C2, C4⋊Dic7⋊65C22, C28.48D4⋊51C2, (C2×C14).289C24, (C2×C28).887C23, Dic7⋊C4⋊45C22, C7⋊7(C22.19C24), (C4×Dic7)⋊59C22, C14.135(C22×D4), (C22×C4).449D14, (C2×Dic14)⋊59C22, C23.235(C22×D7), C22.304(C23×D7), C23.23D14⋊33C2, C23.21D14⋊13C2, (C23×C14).111C22, (C22×C14).418C23, (C22×C28).530C22, (C2×Dic7).151C23, (C22×D7).127C23, C23.D7.130C22, (C4×C7⋊D4)⋊51C2, (C2×C4×D7)⋊54C22, (C2×C4○D28)⋊14C2, (C2×C4)⋊17(C7⋊D4), C2.72(C2×C4○D28), C14.64(C2×C4○D4), C4.145(C2×C7⋊D4), (C2×C14)⋊12(C4○D4), C2.8(C22×C7⋊D4), (C2×C14).575(C2×D4), C22.35(C2×C7⋊D4), (C2×C4).740(C22×D7), (C2×C7⋊D4).137C22, SmallGroup(448,1244)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1284 in 330 conjugacy classes, 119 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C7, C2×C4 [×2], C2×C4 [×6], C2×C4 [×20], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×8], D7 [×2], C14, C14 [×2], C14 [×6], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×7], C2×Q8, C4○D4 [×4], C24, Dic7 [×6], C28 [×4], C28 [×2], D14 [×6], C2×C14, C2×C14 [×6], C2×C14 [×14], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, Dic14 [×2], C4×D7 [×4], D28 [×2], C2×Dic7 [×6], C7⋊D4 [×12], C2×C28 [×2], C2×C28 [×6], C2×C28 [×10], C22×D7 [×2], C22×C14, C22×C14 [×2], C22×C14 [×6], C22.19C24, C4×Dic7 [×2], Dic7⋊C4 [×4], C4⋊Dic7 [×2], D14⋊C4 [×4], C23.D7 [×6], C2×Dic14, C2×C4×D7 [×2], C2×D28, C4○D28 [×4], C2×C7⋊D4 [×6], C22×C28 [×2], C22×C28 [×4], C22×C28 [×4], C23×C14, C28.48D4 [×2], C23.21D14, C4×C7⋊D4 [×4], C23.23D14 [×2], C28⋊7D4 [×2], C24⋊D7 [×2], C2×C4○D28, C23×C28, C24.72D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×4], C24, D14 [×7], C22×D4, C2×C4○D4 [×2], C7⋊D4 [×4], C22×D7 [×7], C22.19C24, C4○D28 [×4], C2×C7⋊D4 [×6], C23×D7, C2×C4○D28 [×2], C22×C7⋊D4, C24.72D14
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=f2=d, ab=ba, ac=ca, faf-1=ad=da, ae=ea, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >
►On 112 points
(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)
(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)(63 112)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)(63 112)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)
(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 58 15 72)(2 71 16 57)(3 84 17 70)(4 69 18 83)(5 82 19 68)(6 67 20 81)(7 80 21 66)(8 65 22 79)(9 78 23 64)(10 63 24 77)(11 76 25 62)(12 61 26 75)(13 74 27 60)(14 59 28 73)(29 108 43 94)(30 93 44 107)(31 106 45 92)(32 91 46 105)(33 104 47 90)(34 89 48 103)(35 102 49 88)(36 87 50 101)(37 100 51 86)(38 85 52 99)(39 98 53 112)(40 111 54 97)(41 96 55 110)(42 109 56 95)
G:=sub<Sym(112)| (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), (57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (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), (1,58,15,72)(2,71,16,57)(3,84,17,70)(4,69,18,83)(5,82,19,68)(6,67,20,81)(7,80,21,66)(8,65,22,79)(9,78,23,64)(10,63,24,77)(11,76,25,62)(12,61,26,75)(13,74,27,60)(14,59,28,73)(29,108,43,94)(30,93,44,107)(31,106,45,92)(32,91,46,105)(33,104,47,90)(34,89,48,103)(35,102,49,88)(36,87,50,101)(37,100,51,86)(38,85,52,99)(39,98,53,112)(40,111,54,97)(41,96,55,110)(42,109,56,95)>;
G:=Group( (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), (57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (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), (1,58,15,72)(2,71,16,57)(3,84,17,70)(4,69,18,83)(5,82,19,68)(6,67,20,81)(7,80,21,66)(8,65,22,79)(9,78,23,64)(10,63,24,77)(11,76,25,62)(12,61,26,75)(13,74,27,60)(14,59,28,73)(29,108,43,94)(30,93,44,107)(31,106,45,92)(32,91,46,105)(33,104,47,90)(34,89,48,103)(35,102,49,88)(36,87,50,101)(37,100,51,86)(38,85,52,99)(39,98,53,112)(40,111,54,97)(41,96,55,110)(42,109,56,95) );
G=PermutationGroup([(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)], [(57,106),(58,107),(59,108),(60,109),(61,110),(62,111),(63,112),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,43),(57,106),(58,107),(59,108),(60,109),(61,110),(62,111),(63,112),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105)], [(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)], [(1,58,15,72),(2,71,16,57),(3,84,17,70),(4,69,18,83),(5,82,19,68),(6,67,20,81),(7,80,21,66),(8,65,22,79),(9,78,23,64),(10,63,24,77),(11,76,25,62),(12,61,26,75),(13,74,27,60),(14,59,28,73),(29,108,43,94),(30,93,44,107),(31,106,45,92),(32,91,46,105),(33,104,47,90),(34,89,48,103),(35,102,49,88),(36,87,50,101),(37,100,51,86),(38,85,52,99),(39,98,53,112),(40,111,54,97),(41,96,55,110),(42,109,56,95)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 26 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 10 | 0 | 0 |
| 26 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[26,0,0,0,0,10,0,0,0,0,9,0,0,0,0,13],[0,26,0,0,10,0,0,0,0,0,0,9,0,0,13,0] >;
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4P | 7A | 7B | 7C | 14A | ··· | 14AS | 28A | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 28 | 28 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | C4○D28 |
| kernel | C24.72D14 | C28.48D4 | C23.21D14 | C4×C7⋊D4 | C23.23D14 | C28⋊7D4 | C24⋊D7 | C2×C4○D28 | C23×C28 | C2×C28 | C23×C4 | C2×C14 | C22×C4 | C24 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 1 | 1 | 4 | 3 | 8 | 18 | 3 | 24 | 48 |
In GAP, Magma, Sage, TeX
C_2^4._{72}D_{14} % in TeX
G:=Group("C2^4.72D14"); // GroupNames label
G:=SmallGroup(448,1244);
// by ID
G=gap.SmallGroup(448,1244);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,675,18822]);
// 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,a*c=c*a,f*a*f^-1=a*d=d*a,a*e=e*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*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