metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.30D28, C22.2Dic28, C14.8C4≀C2, C4⋊Dic7⋊3C4, C22⋊C8.1D7, (C2×C14).1Q16, (C2×Dic14)⋊2C4, (C2×C28).437D4, (C2×C14).1SD16, C14.3(C23⋊C4), (C22×C4).54D14, (C22×C14).39D4, C2.6(D28⋊4C4), C14.6(Q8⋊C4), C28.48D4.1C2, C22.4(C56⋊C2), C7⋊2(C23.31D4), C22.58(D14⋊C4), C2.3(C28.44D4), (C22×C28).40C22, C14.C42.21C2, C2.6(C23.1D14), (C2×C4).12(C4×D7), (C2×C28).24(C2×C4), (C7×C22⋊C8).1C2, (C2×C4).208(C7⋊D4), (C2×C14).39(C22⋊C4), SmallGroup(448,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.30D28
G = < a,b,c,d,e | a2=b2=c2=1, d28=cb=bc, e2=b, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=acd27 >
Subgroups: 444 in 80 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C2.C42, C22⋊C8, C22⋊Q8, C56, Dic14, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.31D4, Dic7⋊C4, C4⋊Dic7, C23.D7, C2×C56, C2×Dic14, C22×Dic7, C22×C28, C14.C42, C7×C22⋊C8, C28.48D4, C23.30D28
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, SD16, Q16, D14, C23⋊C4, Q8⋊C4, C4≀C2, C4×D7, D28, C7⋊D4, C23.31D4, C56⋊C2, Dic28, D14⋊C4, C23.1D14, C28.44D4, D28⋊4C4, C23.30D28
►On 112 points
Generators in S112
(2 93)(4 95)(6 97)(8 99)(10 101)(12 103)(14 105)(16 107)(18 109)(20 111)(22 57)(24 59)(26 61)(28 63)(30 65)(32 67)(34 69)(36 71)(38 73)(40 75)(42 77)(44 79)(46 81)(48 83)(50 85)(52 87)(54 89)(56 91)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 105)(15 106)(16 107)(17 108)(18 109)(19 110)(20 111)(21 112)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)
(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 64 91)(2 55 65 62)(3 61 66 54)(4 88 67 25)(5 24 68 87)(6 51 69 58)(7 57 70 50)(8 84 71 21)(9 20 72 83)(10 47 73 110)(11 109 74 46)(12 80 75 17)(13 16 76 79)(14 43 77 106)(15 105 78 42)(18 39 81 102)(19 101 82 38)(22 35 85 98)(23 97 86 34)(26 31 89 94)(27 93 90 30)(29 56 92 63)(32 60 95 53)(33 52 96 59)(36 112 99 49)(37 48 100 111)(40 108 103 45)(41 44 104 107)
G:=sub<Sym(112)| (2,93)(4,95)(6,97)(8,99)(10,101)(12,103)(14,105)(16,107)(18,109)(20,111)(22,57)(24,59)(26,61)(28,63)(30,65)(32,67)(34,69)(36,71)(38,73)(40,75)(42,77)(44,79)(46,81)(48,83)(50,85)(52,87)(54,89)(56,91), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63), (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,100)(10,101)(11,102)(12,103)(13,104)(14,105)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91), (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), (1,28,64,91)(2,55,65,62)(3,61,66,54)(4,88,67,25)(5,24,68,87)(6,51,69,58)(7,57,70,50)(8,84,71,21)(9,20,72,83)(10,47,73,110)(11,109,74,46)(12,80,75,17)(13,16,76,79)(14,43,77,106)(15,105,78,42)(18,39,81,102)(19,101,82,38)(22,35,85,98)(23,97,86,34)(26,31,89,94)(27,93,90,30)(29,56,92,63)(32,60,95,53)(33,52,96,59)(36,112,99,49)(37,48,100,111)(40,108,103,45)(41,44,104,107)>;
G:=Group( (2,93)(4,95)(6,97)(8,99)(10,101)(12,103)(14,105)(16,107)(18,109)(20,111)(22,57)(24,59)(26,61)(28,63)(30,65)(32,67)(34,69)(36,71)(38,73)(40,75)(42,77)(44,79)(46,81)(48,83)(50,85)(52,87)(54,89)(56,91), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63), (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,100)(10,101)(11,102)(12,103)(13,104)(14,105)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91), (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), (1,28,64,91)(2,55,65,62)(3,61,66,54)(4,88,67,25)(5,24,68,87)(6,51,69,58)(7,57,70,50)(8,84,71,21)(9,20,72,83)(10,47,73,110)(11,109,74,46)(12,80,75,17)(13,16,76,79)(14,43,77,106)(15,105,78,42)(18,39,81,102)(19,101,82,38)(22,35,85,98)(23,97,86,34)(26,31,89,94)(27,93,90,30)(29,56,92,63)(32,60,95,53)(33,52,96,59)(36,112,99,49)(37,48,100,111)(40,108,103,45)(41,44,104,107) );
G=PermutationGroup([[(2,93),(4,95),(6,97),(8,99),(10,101),(12,103),(14,105),(16,107),(18,109),(20,111),(22,57),(24,59),(26,61),(28,63),(30,65),(32,67),(34,69),(36,71),(38,73),(40,75),(42,77),(44,79),(46,81),(48,83),(50,85),(52,87),(54,89),(56,91)], [(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,85),(23,86),(24,87),(25,88),(26,89),(27,90),(28,91),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,99),(37,100),(38,101),(39,102),(40,103),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63)], [(1,92),(2,93),(3,94),(4,95),(5,96),(6,97),(7,98),(8,99),(9,100),(10,101),(11,102),(12,103),(13,104),(14,105),(15,106),(16,107),(17,108),(18,109),(19,110),(20,111),(21,112),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71),(37,72),(38,73),(39,74),(40,75),(41,76),(42,77),(43,78),(44,79),(45,80),(46,81),(47,82),(48,83),(49,84),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91)], [(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)], [(1,28,64,91),(2,55,65,62),(3,61,66,54),(4,88,67,25),(5,24,68,87),(6,51,69,58),(7,57,70,50),(8,84,71,21),(9,20,72,83),(10,47,73,110),(11,109,74,46),(12,80,75,17),(13,16,76,79),(14,43,77,106),(15,105,78,42),(18,39,81,102),(19,101,82,38),(22,35,85,98),(23,97,86,34),(26,31,89,94),(27,93,90,30),(29,56,92,63),(32,60,95,53),(33,52,96,59),(36,112,99,49),(37,48,100,111),(40,108,103,45),(41,44,104,107)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 28 | 28 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D7 | SD16 | Q16 | D14 | C4≀C2 | C4×D7 | C7⋊D4 | D28 | C56⋊C2 | Dic28 | C23⋊C4 | C23.1D14 | D28⋊4C4 |
| kernel | C23.30D28 | C14.C42 | C7×C22⋊C8 | C28.48D4 | C4⋊Dic7 | C2×Dic14 | C2×C28 | C22×C14 | C22⋊C8 | C2×C14 | C2×C14 | C22×C4 | C14 | C2×C4 | C2×C4 | C23 | C22 | C22 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 12 | 12 | 1 | 6 | 6 |
Matrix representation of C23.30D28 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 32 | 112 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 99 | 112 | 0 | 0 | 0 | 0 |
| 57 | 101 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 9 | 0 | 0 |
| 0 | 0 | 104 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 102 | 36 |
| 0 | 0 | 0 | 0 | 59 | 11 |
| 99 | 112 | 0 | 0 | 0 | 0 |
| 84 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 9 | 0 | 0 |
| 0 | 0 | 80 | 80 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 77 |
| 0 | 0 | 0 | 0 | 41 | 102 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,32,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[99,57,0,0,0,0,112,101,0,0,0,0,0,0,33,104,0,0,0,0,9,1,0,0,0,0,0,0,102,59,0,0,0,0,36,11],[99,84,0,0,0,0,112,14,0,0,0,0,0,0,33,80,0,0,0,0,9,80,0,0,0,0,0,0,11,41,0,0,0,0,77,102] >;
C23.30D28 in GAP, Magma, Sage, TeX
C_2^3._{30}D_{28} % in TeX
G:=Group("C2^3.30D28"); // GroupNames label
G:=SmallGroup(448,24);
// by ID
G=gap.SmallGroup(448,24);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,85,92,422,387,268,570,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=c*b=b*c,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*c*d^27>;
// generators/relations