metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.18C25, C28.53C24, D28.39C23, D14.10C24, 2- (1+4)⋊4D7, Dic7.13C24, Dic14.40C23, C4○D4⋊13D14, (C2×Q8)⋊25D14, (D4×D7)⋊15C22, (C2×C14).9C24, D4⋊8D14⋊12C2, (Q8×D7)⋊18C22, C2.19(D7×C24), C4.50(C23×D7), C7⋊D4.5C23, C4○D28⋊15C22, (C2×D28)⋊41C22, C7⋊3(C2.C25), (Q8×C14)⋊25C22, (C4×D7).22C23, D4.33(C22×D7), (C7×D4).33C23, Q8.34(C22×D7), (C7×Q8).34C23, D4⋊2D7⋊19C22, C22.6(C23×D7), (C2×C28).124C23, Q8.10D14⋊8C2, Q8⋊2D7⋊17C22, (C7×2- (1+4))⋊5C2, (C2×Dic7).300C23, (C22×D7).144C23, (D7×C4○D4)⋊10C2, (C2×C4×D7)⋊38C22, (C2×Q8⋊2D7)⋊22C2, (C7×C4○D4)⋊13C22, (C2×C4).108(C22×D7), SmallGroup(448,1382)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 3252 in 810 conjugacy classes, 443 normal (8 characteristic)
C1, C2, C2 [×15], C4 [×10], C4 [×6], C22 [×5], C22 [×25], C7, C2×C4 [×15], C2×C4 [×45], D4 [×10], D4 [×50], Q8 [×10], Q8 [×10], C23 [×15], D7 [×10], C14, C14 [×5], C22×C4 [×15], C2×D4 [×45], C2×Q8 [×5], C2×Q8 [×10], C4○D4 [×10], C4○D4 [×70], Dic7, Dic7 [×5], C28 [×10], D14 [×10], D14 [×15], C2×C14 [×5], C2×C4○D4 [×15], 2+ (1+4) [×10], 2- (1+4), 2- (1+4) [×5], Dic14 [×10], C4×D7 [×40], D28 [×30], C2×Dic7 [×5], C7⋊D4 [×20], C2×C28 [×15], C7×D4 [×10], C7×Q8 [×10], C22×D7 [×15], C2.C25, C2×C4×D7 [×15], C2×D28 [×15], C4○D28 [×30], D4×D7 [×30], D4⋊2D7 [×10], Q8×D7 [×10], Q8⋊2D7 [×30], Q8×C14 [×5], C7×C4○D4 [×10], C2×Q8⋊2D7 [×5], Q8.10D14 [×5], D7×C4○D4 [×10], D4⋊8D14 [×10], C7×2- (1+4), D28.39C23
Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], D7, C24 [×31], D14 [×15], C25, C22×D7 [×35], C2.C25, C23×D7 [×15], D7×C24, D28.39C23
Generators and relations
G = < a,b,c,d,e | a28=b2=c2=d2=e2=1, bab=a-1, ac=ca, ad=da, eae=a13, cbc=a14b, bd=db, ebe=a26b, dcd=ece=a14c, de=ed >
►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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(51 56)(52 55)(53 54)(57 76)(58 75)(59 74)(60 73)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(77 84)(78 83)(79 82)(80 81)(85 102)(86 101)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(103 112)(104 111)(105 110)(106 109)(107 108)
(1 101)(2 102)(3 103)(4 104)(5 105)(6 106)(7 107)(8 108)(9 109)(10 110)(11 111)(12 112)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 73)(40 74)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 81)(48 82)(49 83)(50 84)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)
(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)(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 74)(2 59)(3 72)(4 57)(5 70)(6 83)(7 68)(8 81)(9 66)(10 79)(11 64)(12 77)(13 62)(14 75)(15 60)(16 73)(17 58)(18 71)(19 84)(20 69)(21 82)(22 67)(23 80)(24 65)(25 78)(26 63)(27 76)(28 61)(29 112)(30 97)(31 110)(32 95)(33 108)(34 93)(35 106)(36 91)(37 104)(38 89)(39 102)(40 87)(41 100)(42 85)(43 98)(44 111)(45 96)(46 109)(47 94)(48 107)(49 92)(50 105)(51 90)(52 103)(53 88)(54 101)(55 86)(56 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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(51,56)(52,55)(53,54)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(77,84)(78,83)(79,82)(80,81)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(103,112)(104,111)(105,110)(106,109)(107,108), (1,101)(2,102)(3,103)(4,104)(5,105)(6,106)(7,107)(8,108)(9,109)(10,110)(11,111)(12,112)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62), (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)(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), (1,74)(2,59)(3,72)(4,57)(5,70)(6,83)(7,68)(8,81)(9,66)(10,79)(11,64)(12,77)(13,62)(14,75)(15,60)(16,73)(17,58)(18,71)(19,84)(20,69)(21,82)(22,67)(23,80)(24,65)(25,78)(26,63)(27,76)(28,61)(29,112)(30,97)(31,110)(32,95)(33,108)(34,93)(35,106)(36,91)(37,104)(38,89)(39,102)(40,87)(41,100)(42,85)(43,98)(44,111)(45,96)(46,109)(47,94)(48,107)(49,92)(50,105)(51,90)(52,103)(53,88)(54,101)(55,86)(56,99)>;
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(51,56)(52,55)(53,54)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(77,84)(78,83)(79,82)(80,81)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(103,112)(104,111)(105,110)(106,109)(107,108), (1,101)(2,102)(3,103)(4,104)(5,105)(6,106)(7,107)(8,108)(9,109)(10,110)(11,111)(12,112)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,57)(52,58)(53,59)(54,60)(55,61)(56,62), (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)(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), (1,74)(2,59)(3,72)(4,57)(5,70)(6,83)(7,68)(8,81)(9,66)(10,79)(11,64)(12,77)(13,62)(14,75)(15,60)(16,73)(17,58)(18,71)(19,84)(20,69)(21,82)(22,67)(23,80)(24,65)(25,78)(26,63)(27,76)(28,61)(29,112)(30,97)(31,110)(32,95)(33,108)(34,93)(35,106)(36,91)(37,104)(38,89)(39,102)(40,87)(41,100)(42,85)(43,98)(44,111)(45,96)(46,109)(47,94)(48,107)(49,92)(50,105)(51,90)(52,103)(53,88)(54,101)(55,86)(56,99) );
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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(29,50),(30,49),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(51,56),(52,55),(53,54),(57,76),(58,75),(59,74),(60,73),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(77,84),(78,83),(79,82),(80,81),(85,102),(86,101),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94),(103,112),(104,111),(105,110),(106,109),(107,108)], [(1,101),(2,102),(3,103),(4,104),(5,105),(6,106),(7,107),(8,108),(9,109),(10,110),(11,111),(12,112),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,99),(28,100),(29,63),(30,64),(31,65),(32,66),(33,67),(34,68),(35,69),(36,70),(37,71),(38,72),(39,73),(40,74),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,81),(48,82),(49,83),(50,84),(51,57),(52,58),(53,59),(54,60),(55,61),(56,62)], [(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),(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)], [(1,74),(2,59),(3,72),(4,57),(5,70),(6,83),(7,68),(8,81),(9,66),(10,79),(11,64),(12,77),(13,62),(14,75),(15,60),(16,73),(17,58),(18,71),(19,84),(20,69),(21,82),(22,67),(23,80),(24,65),(25,78),(26,63),(27,76),(28,61),(29,112),(30,97),(31,110),(32,95),(33,108),(34,93),(35,106),(36,91),(37,104),(38,89),(39,102),(40,87),(41,100),(42,85),(43,98),(44,111),(45,96),(46,109),(47,94),(48,107),(49,92),(50,105),(51,90),(52,103),(53,88),(54,101),(55,86),(56,99)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 10 | 7 | 0 | 0 | 0 | 0 |
| 22 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 5 | 17 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 |
| 19 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 17 | 12 | 0 |
| 0 | 0 | 24 | 12 | 17 | 17 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 27 | 28 | 1 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 28 | 28 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 22 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 28 | 28 |
| 0 | 0 | 27 | 28 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 28 | 0 |
G:=sub<GL(6,GF(29))| [10,22,0,0,0,0,7,1,0,0,0,0,0,0,12,5,0,0,0,0,0,17,0,0,0,0,0,12,0,12,0,0,0,12,12,0],[10,19,0,0,0,0,7,19,0,0,0,0,0,0,17,24,0,0,0,0,17,12,0,0,0,0,12,17,0,12,0,0,0,17,17,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,27,0,0,0,0,0,28,1,0,0,0,0,1,0,0,0,0,1,1,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,28,0,28,0,0,0,28,0,0,28],[28,22,0,0,0,0,0,1,0,0,0,0,0,0,1,27,0,0,0,0,0,28,0,0,0,0,28,1,0,28,0,0,28,1,28,0] >;
85 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2F | 2G | ··· | 2P | 4A | ··· | 4J | 4K | 4L | 4M | ··· | 4Q | 7A | 7B | 7C | 14A | 14B | 14C | 14D | ··· | 14R | 28A | ··· | 28AD |
| order | 1 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 7 | 7 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | C2.C25 | D28.39C23 |
| kernel | D28.39C23 | C2×Q8⋊2D7 | Q8.10D14 | D7×C4○D4 | D4⋊8D14 | C7×2- (1+4) | 2- (1+4) | C2×Q8 | C4○D4 | C7 | C1 |
| # reps | 1 | 5 | 5 | 10 | 10 | 1 | 3 | 15 | 30 | 2 | 3 |
In GAP, Magma, Sage, TeX
D_{28}._{39}C_2^3 % in TeX
G:=Group("D28.39C2^3"); // GroupNames label
G:=SmallGroup(448,1382);
// by ID
G=gap.SmallGroup(448,1382);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,570,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^28=b^2=c^2=d^2=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^13,c*b*c=a^14*b,b*d=d*b,e*b*e=a^26*b,d*c*d=e*c*e=a^14*c,d*e=e*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀