metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.9D28, C28.20C42, (C2×C56)⋊2C4, (C2×C8)⋊2Dic7, (C4×Dic7)⋊4C4, C28.12(C4⋊C4), (C2×C28).12Q8, C7⋊2(C4.9C42), (C2×C28).110D4, (C2×C4).6Dic14, C4.25(C4×Dic7), (C22×C4).63D14, (C22×C14).47D4, (C2×M4(2)).8D7, C28.87(C22⋊C4), C4.12(Dic7⋊C4), C4.18(C23.D7), C22.20(D14⋊C4), (C14×M4(2)).12C2, C22.10(C4⋊Dic7), (C22×C28).126C22, C14.16(C2.C42), C2.16(C14.C42), C23.21D14.10C2, (C2×C28).63(C2×C4), (C2×C4).140(C4×D7), (C2×C14).40(C4⋊C4), (C2×C4).22(C7⋊D4), (C2×C4).77(C2×Dic7), (C2×C14).11(C22⋊C4), SmallGroup(448,114)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.9D28
G = < a,b,c,d,e | a2=b2=c2=1, d28=c, e2=abc, ab=ba, dad-1=eae-1=ac=ca, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=bd27 >
Subgroups: 388 in 94 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C42⋊C2, C2×M4(2), C56, C2×Dic7, C2×C28, C2×C28, C22×C14, C4.9C42, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×M4(2), C22×C28, C23.21D14, C14×M4(2), C23.9D28
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C4.9C42, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C14.C42, C23.9D28
►On 112 points
Generators in S112
(1 110)(2 83)(3 112)(4 85)(5 58)(6 87)(7 60)(8 89)(9 62)(10 91)(11 64)(12 93)(13 66)(14 95)(15 68)(16 97)(17 70)(18 99)(19 72)(20 101)(21 74)(22 103)(23 76)(24 105)(25 78)(26 107)(27 80)(28 109)(29 82)(30 111)(31 84)(32 57)(33 86)(34 59)(35 88)(36 61)(37 90)(38 63)(39 92)(40 65)(41 94)(42 67)(43 96)(44 69)(45 98)(46 71)(47 100)(48 73)(49 102)(50 75)(51 104)(52 77)(53 106)(54 79)(55 108)(56 81)
(1 110)(2 111)(3 112)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 97)(45 98)(46 99)(47 100)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 109)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(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 60 29 88)(2 6)(3 58 31 86)(5 112 33 84)(7 110 35 82)(8 56)(9 108 37 80)(10 54)(11 106 39 78)(12 52)(13 104 41 76)(14 50)(15 102 43 74)(16 48)(17 100 45 72)(18 46)(19 98 47 70)(20 44)(21 96 49 68)(22 42)(23 94 51 66)(24 40)(25 92 53 64)(26 38)(27 90 55 62)(28 36)(30 34)(57 85)(59 83)(61 81)(63 79)(65 77)(67 75)(69 73)(87 111)(89 109)(91 107)(93 105)(95 103)(97 101)
G:=sub<Sym(112)| (1,110)(2,83)(3,112)(4,85)(5,58)(6,87)(7,60)(8,89)(9,62)(10,91)(11,64)(12,93)(13,66)(14,95)(15,68)(16,97)(17,70)(18,99)(19,72)(20,101)(21,74)(22,103)(23,76)(24,105)(25,78)(26,107)(27,80)(28,109)(29,82)(30,111)(31,84)(32,57)(33,86)(34,59)(35,88)(36,61)(37,90)(38,63)(39,92)(40,65)(41,94)(42,67)(43,96)(44,69)(45,98)(46,71)(47,100)(48,73)(49,102)(50,75)(51,104)(52,77)(53,106)(54,79)(55,108)(56,81), (1,110)(2,111)(3,112)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (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), (1,60,29,88)(2,6)(3,58,31,86)(5,112,33,84)(7,110,35,82)(8,56)(9,108,37,80)(10,54)(11,106,39,78)(12,52)(13,104,41,76)(14,50)(15,102,43,74)(16,48)(17,100,45,72)(18,46)(19,98,47,70)(20,44)(21,96,49,68)(22,42)(23,94,51,66)(24,40)(25,92,53,64)(26,38)(27,90,55,62)(28,36)(30,34)(57,85)(59,83)(61,81)(63,79)(65,77)(67,75)(69,73)(87,111)(89,109)(91,107)(93,105)(95,103)(97,101)>;
G:=Group( (1,110)(2,83)(3,112)(4,85)(5,58)(6,87)(7,60)(8,89)(9,62)(10,91)(11,64)(12,93)(13,66)(14,95)(15,68)(16,97)(17,70)(18,99)(19,72)(20,101)(21,74)(22,103)(23,76)(24,105)(25,78)(26,107)(27,80)(28,109)(29,82)(30,111)(31,84)(32,57)(33,86)(34,59)(35,88)(36,61)(37,90)(38,63)(39,92)(40,65)(41,94)(42,67)(43,96)(44,69)(45,98)(46,71)(47,100)(48,73)(49,102)(50,75)(51,104)(52,77)(53,106)(54,79)(55,108)(56,81), (1,110)(2,111)(3,112)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (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), (1,60,29,88)(2,6)(3,58,31,86)(5,112,33,84)(7,110,35,82)(8,56)(9,108,37,80)(10,54)(11,106,39,78)(12,52)(13,104,41,76)(14,50)(15,102,43,74)(16,48)(17,100,45,72)(18,46)(19,98,47,70)(20,44)(21,96,49,68)(22,42)(23,94,51,66)(24,40)(25,92,53,64)(26,38)(27,90,55,62)(28,36)(30,34)(57,85)(59,83)(61,81)(63,79)(65,77)(67,75)(69,73)(87,111)(89,109)(91,107)(93,105)(95,103)(97,101) );
G=PermutationGroup([[(1,110),(2,83),(3,112),(4,85),(5,58),(6,87),(7,60),(8,89),(9,62),(10,91),(11,64),(12,93),(13,66),(14,95),(15,68),(16,97),(17,70),(18,99),(19,72),(20,101),(21,74),(22,103),(23,76),(24,105),(25,78),(26,107),(27,80),(28,109),(29,82),(30,111),(31,84),(32,57),(33,86),(34,59),(35,88),(36,61),(37,90),(38,63),(39,92),(40,65),(41,94),(42,67),(43,96),(44,69),(45,98),(46,71),(47,100),(48,73),(49,102),(50,75),(51,104),(52,77),(53,106),(54,79),(55,108),(56,81)], [(1,110),(2,111),(3,112),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,73),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,81),(29,82),(30,83),(31,84),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,91),(39,92),(40,93),(41,94),(42,95),(43,96),(44,97),(45,98),(46,99),(47,100),(48,101),(49,102),(50,103),(51,104),(52,105),(53,106),(54,107),(55,108),(56,109)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(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)], [(1,60,29,88),(2,6),(3,58,31,86),(5,112,33,84),(7,110,35,82),(8,56),(9,108,37,80),(10,54),(11,106,39,78),(12,52),(13,104,41,76),(14,50),(15,102,43,74),(16,48),(17,100,45,72),(18,46),(19,98,47,70),(20,44),(21,96,49,68),(22,42),(23,94,51,66),(24,40),(25,92,53,64),(26,38),(27,90,55,62),(28,36),(30,34),(57,85),(59,83),(61,81),(63,79),(65,77),(67,75),(69,73),(87,111),(89,109),(91,107),(93,105),(95,103),(97,101)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | + | - | + | - | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | D7 | Dic7 | D14 | Dic14 | C4×D7 | C7⋊D4 | D28 | C4.9C42 | C23.9D28 |
| kernel | C23.9D28 | C23.21D14 | C14×M4(2) | C4×Dic7 | C2×C56 | C2×C28 | C2×C28 | C22×C14 | C2×M4(2) | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 2 | 1 | 8 | 4 | 2 | 1 | 1 | 3 | 6 | 3 | 6 | 12 | 12 | 6 | 2 | 12 |
Matrix representation of C23.9D28 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 98 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 |
| 0 | 0 | 0 | 0 | 98 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 82 | 25 | 0 | 0 | 0 | 0 |
| 28 | 54 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 76 | 0 | 0 | 0 | 0 |
| 55 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,98,0,0,0,0,15,0,0,0,0,0,0,0,0,15,0,0,0,0,98,0],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,98,0,0,0,0,15,0,0,0,0,0,0,0,0,98,0,0,0,0,15,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[82,28,0,0,0,0,25,54,0,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,55,0,0,0,0,76,0,0,0,0,0,0,0,15,0,0,0,0,0,0,98,0,0,0,0,0,0,1,0,0,0,0,0,0,112] >;
C23.9D28 in GAP, Magma, Sage, TeX
C_2^3._9D_{28} % in TeX
G:=Group("C2^3.9D28"); // GroupNames label
G:=SmallGroup(448,114);
// by ID
G=gap.SmallGroup(448,114);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,387,1123,136,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=c,e^2=a*b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^27>;
// generators/relations