metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.21D28, C56.19D4, D28.21D4, Dic14.21D4, M4(2).10D14, (C2×D56)⋊21C2, C8.C4⋊6D7, C4.57(C2×D28), (C2×C8).71D14, C4.136(D4×D7), C8⋊D14⋊10C2, C28.137(C2×D4), C7⋊2(D4.4D4), D28.2C4⋊6C2, C28.46D4⋊3C2, C14.50(C4⋊D4), C2.23(C4⋊D28), (C2×C28).313C23, (C2×C56).103C22, C4○D28.40C22, (C2×D28).88C22, C22.7(Q8⋊2D7), (C7×M4(2)).7C22, C4.Dic7.38C22, (C7×C8.C4)⋊7C2, (C2×C14).4(C4○D4), (C2×C4).114(C22×D7), SmallGroup(448,431)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.21D28
G = < a,b,c | a8=c2=1, b28=a4, bab-1=cac=a-1, cbc=a4b27 >
Subgroups: 796 in 108 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C7⋊C8, C56, C56, Dic14, C4×D7, D28, D28, C7⋊D4, C2×C28, C22×D7, D4.4D4, C8×D7, C8⋊D7, C56⋊C2, D56, C4.Dic7, C2×C56, C7×M4(2), C2×D28, C4○D28, C28.46D4, C7×C8.C4, D28.2C4, C2×D56, C8⋊D14, C8.21D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, D28, C22×D7, D4.4D4, C2×D28, D4×D7, Q8⋊2D7, C4⋊D28, C8.21D28
►On 112 points
Generators in S112
(1 101 43 87 29 73 15 59)(2 60 16 74 30 88 44 102)(3 103 45 89 31 75 17 61)(4 62 18 76 32 90 46 104)(5 105 47 91 33 77 19 63)(6 64 20 78 34 92 48 106)(7 107 49 93 35 79 21 65)(8 66 22 80 36 94 50 108)(9 109 51 95 37 81 23 67)(10 68 24 82 38 96 52 110)(11 111 53 97 39 83 25 69)(12 70 26 84 40 98 54 112)(13 57 55 99 41 85 27 71)(14 72 28 86 42 100 56 58)
(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 7)(2 6)(3 5)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(57 109)(58 108)(59 107)(60 106)(61 105)(62 104)(63 103)(64 102)(65 101)(66 100)(67 99)(68 98)(69 97)(70 96)(71 95)(72 94)(73 93)(74 92)(75 91)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(110 112)
G:=sub<Sym(112)| (1,101,43,87,29,73,15,59)(2,60,16,74,30,88,44,102)(3,103,45,89,31,75,17,61)(4,62,18,76,32,90,46,104)(5,105,47,91,33,77,19,63)(6,64,20,78,34,92,48,106)(7,107,49,93,35,79,21,65)(8,66,22,80,36,94,50,108)(9,109,51,95,37,81,23,67)(10,68,24,82,38,96,52,110)(11,111,53,97,39,83,25,69)(12,70,26,84,40,98,54,112)(13,57,55,99,41,85,27,71)(14,72,28,86,42,100,56,58), (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), (1,7)(2,6)(3,5)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(57,109)(58,108)(59,107)(60,106)(61,105)(62,104)(63,103)(64,102)(65,101)(66,100)(67,99)(68,98)(69,97)(70,96)(71,95)(72,94)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(110,112)>;
G:=Group( (1,101,43,87,29,73,15,59)(2,60,16,74,30,88,44,102)(3,103,45,89,31,75,17,61)(4,62,18,76,32,90,46,104)(5,105,47,91,33,77,19,63)(6,64,20,78,34,92,48,106)(7,107,49,93,35,79,21,65)(8,66,22,80,36,94,50,108)(9,109,51,95,37,81,23,67)(10,68,24,82,38,96,52,110)(11,111,53,97,39,83,25,69)(12,70,26,84,40,98,54,112)(13,57,55,99,41,85,27,71)(14,72,28,86,42,100,56,58), (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), (1,7)(2,6)(3,5)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(57,109)(58,108)(59,107)(60,106)(61,105)(62,104)(63,103)(64,102)(65,101)(66,100)(67,99)(68,98)(69,97)(70,96)(71,95)(72,94)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(110,112) );
G=PermutationGroup([[(1,101,43,87,29,73,15,59),(2,60,16,74,30,88,44,102),(3,103,45,89,31,75,17,61),(4,62,18,76,32,90,46,104),(5,105,47,91,33,77,19,63),(6,64,20,78,34,92,48,106),(7,107,49,93,35,79,21,65),(8,66,22,80,36,94,50,108),(9,109,51,95,37,81,23,67),(10,68,24,82,38,96,52,110),(11,111,53,97,39,83,25,69),(12,70,26,84,40,98,54,112),(13,57,55,99,41,85,27,71),(14,72,28,86,42,100,56,58)], [(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)], [(1,7),(2,6),(3,5),(8,56),(9,55),(10,54),(11,53),(12,52),(13,51),(14,50),(15,49),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(57,109),(58,108),(59,107),(60,106),(61,105),(62,104),(63,103),(64,102),(65,101),(66,100),(67,99),(68,98),(69,97),(70,96),(71,95),(72,94),(73,93),(74,92),(75,91),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(110,112)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 56 | 56 | 2 | 2 | 28 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D28 | D4.4D4 | D4×D7 | Q8⋊2D7 | C8.21D28 |
| kernel | C8.21D28 | C28.46D4 | C7×C8.C4 | D28.2C4 | C2×D56 | C8⋊D14 | C56 | Dic14 | D28 | C8.C4 | C2×C14 | C2×C8 | M4(2) | C8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 2 | 3 | 6 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of C8.21D28 ►in GL4(𝔽113) generated by
| 84 | 43 | 0 | 0 |
| 70 | 91 | 0 | 0 |
| 0 | 0 | 91 | 70 |
| 0 | 0 | 43 | 84 |
| 0 | 0 | 25 | 34 |
| 0 | 0 | 79 | 112 |
| 94 | 13 | 0 | 0 |
| 100 | 104 | 0 | 0 |
| 94 | 13 | 0 | 0 |
| 94 | 19 | 0 | 0 |
| 0 | 0 | 25 | 34 |
| 0 | 0 | 88 | 88 |
G:=sub<GL(4,GF(113))| [84,70,0,0,43,91,0,0,0,0,91,43,0,0,70,84],[0,0,94,100,0,0,13,104,25,79,0,0,34,112,0,0],[94,94,0,0,13,19,0,0,0,0,25,88,0,0,34,88] >;
C8.21D28 in GAP, Magma, Sage, TeX
C_8._{21}D_{28} % in TeX
G:=Group("C8.21D28"); // GroupNames label
G:=SmallGroup(448,431);
// by ID
G=gap.SmallGroup(448,431);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,219,226,1123,136,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=c^2=1,b^28=a^4,b*a*b^-1=c*a*c=a^-1,c*b*c=a^4*b^27>;
// generators/relations