metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.1D28, (C4×Dic7)⋊1C4, (C2×D4).3D14, C23⋊C4.2D7, C7⋊1(C42⋊3C4), (C2×Dic14)⋊3C4, C14.9(C23⋊C4), (D4×C14).3C22, (C22×C14).10D4, C23.3(C7⋊D4), C23⋊Dic7.1C2, C28.17D4.1C2, C22.10(D14⋊C4), C2.10(C23.1D14), (C2×C4).1(C4×D7), (C2×C28).1(C2×C4), (C7×C23⋊C4).2C2, (C2×C14).3(C22⋊C4), SmallGroup(448,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.D28
G = < a,b,c,d,e | a2=b2=c2=d28=1, e2=ca=ac, dad-1=ab=ba, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=ad-1 >
Subgroups: 412 in 70 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C2×D4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C23⋊C4, C23⋊C4, C4.4D4, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C42⋊3C4, C4×Dic7, C23.D7, C7×C22⋊C4, C2×Dic14, D4×C14, C23⋊Dic7, C7×C23⋊C4, C28.17D4, C23.D28
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D14, C23⋊C4, C4×D7, D28, C7⋊D4, C42⋊3C4, D14⋊C4, C23.1D14, C23.D28
►On 112 points
Generators in S112
(1 84)(2 37)(3 38)(4 59)(5 60)(6 41)(7 42)(8 63)(9 64)(10 45)(11 46)(12 67)(13 68)(14 49)(15 50)(16 71)(17 72)(18 53)(19 54)(20 75)(21 76)(22 29)(23 30)(24 79)(25 80)(26 33)(27 34)(28 83)(31 110)(32 111)(35 86)(36 87)(39 90)(40 91)(43 94)(44 95)(47 98)(48 99)(51 102)(52 103)(55 106)(56 107)(57 88)(58 89)(61 92)(62 93)(65 96)(66 97)(69 100)(70 101)(73 104)(74 105)(77 108)(78 109)(81 112)(82 85)
(1 87)(3 89)(5 91)(7 93)(9 95)(11 97)(13 99)(15 101)(17 103)(19 105)(21 107)(23 109)(25 111)(27 85)(30 78)(32 80)(34 82)(36 84)(38 58)(40 60)(42 62)(44 64)(46 66)(48 68)(50 70)(52 72)(54 74)(56 76)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 85)(28 86)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)
(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 69 36 14)(2 13 57 48)(3 67 58 98)(4 97 39 46)(5 65 40 10)(6 9 61 44)(7 63 62 94)(8 93 43 42)(11 59 66 90)(12 89 47 38)(15 83 70 86)(16 85 51 34)(17 81 52 26)(18 25 73 32)(19 79 74 110)(20 109 55 30)(21 77 56 22)(23 75 78 106)(24 105 31 54)(27 71 82 102)(28 101 35 50)(29 76 108 107)(33 72 112 103)(37 68 88 99)(41 64 92 95)(45 60 96 91)(49 84 100 87)(53 80 104 111)
G:=sub<Sym(112)| (1,84)(2,37)(3,38)(4,59)(5,60)(6,41)(7,42)(8,63)(9,64)(10,45)(11,46)(12,67)(13,68)(14,49)(15,50)(16,71)(17,72)(18,53)(19,54)(20,75)(21,76)(22,29)(23,30)(24,79)(25,80)(26,33)(27,34)(28,83)(31,110)(32,111)(35,86)(36,87)(39,90)(40,91)(43,94)(44,95)(47,98)(48,99)(51,102)(52,103)(55,106)(56,107)(57,88)(58,89)(61,92)(62,93)(65,96)(66,97)(69,100)(70,101)(73,104)(74,105)(77,108)(78,109)(81,112)(82,85), (1,87)(3,89)(5,91)(7,93)(9,95)(11,97)(13,99)(15,101)(17,103)(19,105)(21,107)(23,109)(25,111)(27,85)(30,78)(32,80)(34,82)(36,84)(38,58)(40,60)(42,62)(44,64)(46,66)(48,68)(50,70)(52,72)(54,74)(56,76), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,85)(28,86)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76), (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), (1,69,36,14)(2,13,57,48)(3,67,58,98)(4,97,39,46)(5,65,40,10)(6,9,61,44)(7,63,62,94)(8,93,43,42)(11,59,66,90)(12,89,47,38)(15,83,70,86)(16,85,51,34)(17,81,52,26)(18,25,73,32)(19,79,74,110)(20,109,55,30)(21,77,56,22)(23,75,78,106)(24,105,31,54)(27,71,82,102)(28,101,35,50)(29,76,108,107)(33,72,112,103)(37,68,88,99)(41,64,92,95)(45,60,96,91)(49,84,100,87)(53,80,104,111)>;
G:=Group( (1,84)(2,37)(3,38)(4,59)(5,60)(6,41)(7,42)(8,63)(9,64)(10,45)(11,46)(12,67)(13,68)(14,49)(15,50)(16,71)(17,72)(18,53)(19,54)(20,75)(21,76)(22,29)(23,30)(24,79)(25,80)(26,33)(27,34)(28,83)(31,110)(32,111)(35,86)(36,87)(39,90)(40,91)(43,94)(44,95)(47,98)(48,99)(51,102)(52,103)(55,106)(56,107)(57,88)(58,89)(61,92)(62,93)(65,96)(66,97)(69,100)(70,101)(73,104)(74,105)(77,108)(78,109)(81,112)(82,85), (1,87)(3,89)(5,91)(7,93)(9,95)(11,97)(13,99)(15,101)(17,103)(19,105)(21,107)(23,109)(25,111)(27,85)(30,78)(32,80)(34,82)(36,84)(38,58)(40,60)(42,62)(44,64)(46,66)(48,68)(50,70)(52,72)(54,74)(56,76), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,85)(28,86)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76), (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), (1,69,36,14)(2,13,57,48)(3,67,58,98)(4,97,39,46)(5,65,40,10)(6,9,61,44)(7,63,62,94)(8,93,43,42)(11,59,66,90)(12,89,47,38)(15,83,70,86)(16,85,51,34)(17,81,52,26)(18,25,73,32)(19,79,74,110)(20,109,55,30)(21,77,56,22)(23,75,78,106)(24,105,31,54)(27,71,82,102)(28,101,35,50)(29,76,108,107)(33,72,112,103)(37,68,88,99)(41,64,92,95)(45,60,96,91)(49,84,100,87)(53,80,104,111) );
G=PermutationGroup([[(1,84),(2,37),(3,38),(4,59),(5,60),(6,41),(7,42),(8,63),(9,64),(10,45),(11,46),(12,67),(13,68),(14,49),(15,50),(16,71),(17,72),(18,53),(19,54),(20,75),(21,76),(22,29),(23,30),(24,79),(25,80),(26,33),(27,34),(28,83),(31,110),(32,111),(35,86),(36,87),(39,90),(40,91),(43,94),(44,95),(47,98),(48,99),(51,102),(52,103),(55,106),(56,107),(57,88),(58,89),(61,92),(62,93),(65,96),(66,97),(69,100),(70,101),(73,104),(74,105),(77,108),(78,109),(81,112),(82,85)], [(1,87),(3,89),(5,91),(7,93),(9,95),(11,97),(13,99),(15,101),(17,103),(19,105),(21,107),(23,109),(25,111),(27,85),(30,78),(32,80),(34,82),(36,84),(38,58),(40,60),(42,62),(44,64),(46,66),(48,68),(50,70),(52,72),(54,74),(56,76)], [(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,103),(18,104),(19,105),(20,106),(21,107),(22,108),(23,109),(24,110),(25,111),(26,112),(27,85),(28,86),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76)], [(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)], [(1,69,36,14),(2,13,57,48),(3,67,58,98),(4,97,39,46),(5,65,40,10),(6,9,61,44),(7,63,62,94),(8,93,43,42),(11,59,66,90),(12,89,47,38),(15,83,70,86),(16,85,51,34),(17,81,52,26),(18,25,73,32),(19,79,74,110),(20,109,55,30),(21,77,56,22),(23,75,78,106),(24,105,31,54),(27,71,82,102),(28,101,35,50),(29,76,108,107),(33,72,112,103),(37,68,88,99),(41,64,92,95),(45,60,96,91),(49,84,100,87),(53,80,104,111)]])
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 14A | 14B | 14C | 14D | ··· | 14L | 14M | 14N | 14O | 28A | ··· | 28O |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 28 | 28 | 56 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D7 | D14 | C4×D7 | D28 | C7⋊D4 | C23⋊C4 | C42⋊3C4 | C23.1D14 | C23.D28 |
| kernel | C23.D28 | C23⋊Dic7 | C7×C23⋊C4 | C28.17D4 | C4×Dic7 | C2×Dic14 | C22×C14 | C23⋊C4 | C2×D4 | C2×C4 | C23 | C23 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 1 | 2 | 6 | 3 |
Matrix representation of C23.D28 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 24 | 0 | 0 |
| 0 | 0 | 17 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 24 |
| 0 | 0 | 0 | 0 | 17 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 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 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 5 | 4 | 0 | 0 | 0 | 0 |
| 22 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 27 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 18 | 20 | 0 | 0 | 0 | 0 |
| 23 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 27 |
| 0 | 0 | 0 | 0 | 1 | 28 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,17,0,0,0,0,24,12,0,0,0,0,0,0,17,17,0,0,0,0,24,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[5,22,0,0,0,0,4,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,27,28,0,0,1,0,0,0,0,0,0,1,0,0],[18,23,0,0,0,0,20,11,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,1,0,0,0,0,27,28,0,0] >;
C23.D28 in GAP, Magma, Sage, TeX
C_2^3.D_{28} % in TeX
G:=Group("C2^3.D28"); // GroupNames label
G:=SmallGroup(448,30);
// by ID
G=gap.SmallGroup(448,30);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,141,36,422,1123,794,297,136,851,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=1,e^2=c*a=a*c,d*a*d^-1=a*b=b*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*d^-1>;
// generators/relations