metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.20D28, M4(2)⋊24D14, C4○D28⋊8C4, (C2×D28)⋊16C4, (C2×C4).55D28, D28.27(C2×C4), C28.446(C2×D4), (C2×C28).178D4, D28⋊4C4⋊13C2, C4.30(D14⋊C4), (C2×Dic14)⋊16C4, (C2×M4(2))⋊15D7, C28.70(C22×C4), (C4×Dic7)⋊4C22, C22.17(C2×D28), C28.29(C22⋊C4), (C14×M4(2))⋊23C2, (C2×C28).420C23, Dic14.28(C2×C4), C7⋊3(C42⋊C22), C4○D28.43C22, (C22×C4).146D14, (C22×C14).107D4, C22.29(D14⋊C4), (C7×M4(2))⋊36C22, C23.21D14⋊17C2, (C22×C28).193C22, C4.55(C2×C4×D7), (C2×C4).56(C4×D7), C2.35(C2×D14⋊C4), (C2×C14).33(C2×D4), C4.137(C2×C7⋊D4), (C2×C28).113(C2×C4), (C2×C4○D28).15C2, C14.63(C2×C22⋊C4), (C2×C4).258(C7⋊D4), (C2×C4).513(C22×D7), (C2×C14).23(C22⋊C4), SmallGroup(448,673)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.20D28
G = < a,b,c,d,e | a2=b2=c2=1, d28=c, e2=b, ab=ba, dad-1=eae-1=ac=ca, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd27 >
Subgroups: 804 in 154 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×C14, C42⋊C22, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×M4(2), C7×M4(2), C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C4○D28, C2×C7⋊D4, C22×C28, D28⋊4C4, C23.21D14, C14×M4(2), C2×C4○D28, C23.20D28
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C4×D7, D28, C7⋊D4, C22×D7, C42⋊C22, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, C2×D14⋊C4, C23.20D28
►On 112 points
Generators in S112
(1 67)(2 96)(3 69)(4 98)(5 71)(6 100)(7 73)(8 102)(9 75)(10 104)(11 77)(12 106)(13 79)(14 108)(15 81)(16 110)(17 83)(18 112)(19 85)(20 58)(21 87)(22 60)(23 89)(24 62)(25 91)(26 64)(27 93)(28 66)(29 95)(30 68)(31 97)(32 70)(33 99)(34 72)(35 101)(36 74)(37 103)(38 76)(39 105)(40 78)(41 107)(42 80)(43 109)(44 82)(45 111)(46 84)(47 57)(48 86)(49 59)(50 88)(51 61)(52 90)(53 63)(54 92)(55 65)(56 94)
(2 30)(4 32)(6 34)(8 36)(10 38)(12 40)(14 42)(16 44)(18 46)(20 48)(22 50)(24 52)(26 54)(28 56)(58 86)(60 88)(62 90)(64 92)(66 94)(68 96)(70 98)(72 100)(74 102)(76 104)(78 106)(80 108)(82 110)(84 112)
(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)
(1 101)(2 72 30 100)(3 71)(4 98 32 70)(5 97)(6 68 34 96)(7 67)(8 94 36 66)(9 93)(10 64 38 92)(11 63)(12 90 40 62)(13 89)(14 60 42 88)(15 59)(16 86 44 58)(17 85)(18 112 46 84)(19 111)(20 82 48 110)(21 81)(22 108 50 80)(23 107)(24 78 52 106)(25 77)(26 104 54 76)(27 103)(28 74 56 102)(29 73)(31 99)(33 69)(35 95)(37 65)(39 91)(41 61)(43 87)(45 57)(47 83)(49 109)(51 79)(53 105)(55 75)
G:=sub<Sym(112)| (1,67)(2,96)(3,69)(4,98)(5,71)(6,100)(7,73)(8,102)(9,75)(10,104)(11,77)(12,106)(13,79)(14,108)(15,81)(16,110)(17,83)(18,112)(19,85)(20,58)(21,87)(22,60)(23,89)(24,62)(25,91)(26,64)(27,93)(28,66)(29,95)(30,68)(31,97)(32,70)(33,99)(34,72)(35,101)(36,74)(37,103)(38,76)(39,105)(40,78)(41,107)(42,80)(43,109)(44,82)(45,111)(46,84)(47,57)(48,86)(49,59)(50,88)(51,61)(52,90)(53,63)(54,92)(55,65)(56,94), (2,30)(4,32)(6,34)(8,36)(10,38)(12,40)(14,42)(16,44)(18,46)(20,48)(22,50)(24,52)(26,54)(28,56)(58,86)(60,88)(62,90)(64,92)(66,94)(68,96)(70,98)(72,100)(74,102)(76,104)(78,106)(80,108)(82,110)(84,112), (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), (1,101)(2,72,30,100)(3,71)(4,98,32,70)(5,97)(6,68,34,96)(7,67)(8,94,36,66)(9,93)(10,64,38,92)(11,63)(12,90,40,62)(13,89)(14,60,42,88)(15,59)(16,86,44,58)(17,85)(18,112,46,84)(19,111)(20,82,48,110)(21,81)(22,108,50,80)(23,107)(24,78,52,106)(25,77)(26,104,54,76)(27,103)(28,74,56,102)(29,73)(31,99)(33,69)(35,95)(37,65)(39,91)(41,61)(43,87)(45,57)(47,83)(49,109)(51,79)(53,105)(55,75)>;
G:=Group( (1,67)(2,96)(3,69)(4,98)(5,71)(6,100)(7,73)(8,102)(9,75)(10,104)(11,77)(12,106)(13,79)(14,108)(15,81)(16,110)(17,83)(18,112)(19,85)(20,58)(21,87)(22,60)(23,89)(24,62)(25,91)(26,64)(27,93)(28,66)(29,95)(30,68)(31,97)(32,70)(33,99)(34,72)(35,101)(36,74)(37,103)(38,76)(39,105)(40,78)(41,107)(42,80)(43,109)(44,82)(45,111)(46,84)(47,57)(48,86)(49,59)(50,88)(51,61)(52,90)(53,63)(54,92)(55,65)(56,94), (2,30)(4,32)(6,34)(8,36)(10,38)(12,40)(14,42)(16,44)(18,46)(20,48)(22,50)(24,52)(26,54)(28,56)(58,86)(60,88)(62,90)(64,92)(66,94)(68,96)(70,98)(72,100)(74,102)(76,104)(78,106)(80,108)(82,110)(84,112), (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), (1,101)(2,72,30,100)(3,71)(4,98,32,70)(5,97)(6,68,34,96)(7,67)(8,94,36,66)(9,93)(10,64,38,92)(11,63)(12,90,40,62)(13,89)(14,60,42,88)(15,59)(16,86,44,58)(17,85)(18,112,46,84)(19,111)(20,82,48,110)(21,81)(22,108,50,80)(23,107)(24,78,52,106)(25,77)(26,104,54,76)(27,103)(28,74,56,102)(29,73)(31,99)(33,69)(35,95)(37,65)(39,91)(41,61)(43,87)(45,57)(47,83)(49,109)(51,79)(53,105)(55,75) );
G=PermutationGroup([[(1,67),(2,96),(3,69),(4,98),(5,71),(6,100),(7,73),(8,102),(9,75),(10,104),(11,77),(12,106),(13,79),(14,108),(15,81),(16,110),(17,83),(18,112),(19,85),(20,58),(21,87),(22,60),(23,89),(24,62),(25,91),(26,64),(27,93),(28,66),(29,95),(30,68),(31,97),(32,70),(33,99),(34,72),(35,101),(36,74),(37,103),(38,76),(39,105),(40,78),(41,107),(42,80),(43,109),(44,82),(45,111),(46,84),(47,57),(48,86),(49,59),(50,88),(51,61),(52,90),(53,63),(54,92),(55,65),(56,94)], [(2,30),(4,32),(6,34),(8,36),(10,38),(12,40),(14,42),(16,44),(18,46),(20,48),(22,50),(24,52),(26,54),(28,56),(58,86),(60,88),(62,90),(64,92),(66,94),(68,96),(70,98),(72,100),(74,102),(76,104),(78,106),(80,108),(82,110),(84,112)], [(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)], [(1,101),(2,72,30,100),(3,71),(4,98,32,70),(5,97),(6,68,34,96),(7,67),(8,94,36,66),(9,93),(10,64,38,92),(11,63),(12,90,40,62),(13,89),(14,60,42,88),(15,59),(16,86,44,58),(17,85),(18,112,46,84),(19,111),(20,82,48,110),(21,81),(22,108,50,80),(23,107),(24,78,52,106),(25,77),(26,104,54,76),(27,103),(28,74,56,102),(29,73),(31,99),(33,69),(35,95),(37,65),(39,91),(41,61),(43,87),(45,57),(47,83),(49,109),(51,79),(53,105),(55,75)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4K | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 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 | 28 | 28 | 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 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D7 | D14 | D14 | C4×D7 | D28 | C7⋊D4 | D28 | C42⋊C22 | C23.20D28 |
| kernel | C23.20D28 | D28⋊4C4 | C23.21D14 | C14×M4(2) | C2×C4○D28 | C2×Dic14 | C2×D28 | C4○D28 | C2×C28 | C22×C14 | C2×M4(2) | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 1 | 3 | 6 | 3 | 12 | 6 | 12 | 6 | 2 | 12 |
Matrix representation of C23.20D28 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 91 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 41 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 41 | 0 | 0 | 112 |
| 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 |
| 107 | 25 | 0 | 0 | 0 | 0 |
| 88 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 91 | 0 | 0 |
| 0 | 0 | 61 | 1 | 0 | 0 |
| 0 | 0 | 41 | 112 | 0 | 112 |
| 0 | 0 | 11 | 1 | 98 | 0 |
| 37 | 92 | 0 | 0 | 0 | 0 |
| 76 | 76 | 0 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 | 104 | 0 |
| 0 | 0 | 52 | 0 | 98 | 112 |
| 0 | 0 | 50 | 0 | 15 | 0 |
| 0 | 0 | 11 | 1 | 98 | 0 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,91,1,112,1,0,0,0,1,0,0],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,41,0,41,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,112],[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],[107,88,0,0,0,0,25,29,0,0,0,0,0,0,112,61,41,11,0,0,91,1,112,1,0,0,0,0,0,98,0,0,0,0,112,0],[37,76,0,0,0,0,92,76,0,0,0,0,0,0,98,52,50,11,0,0,0,0,0,1,0,0,104,98,15,98,0,0,0,112,0,0] >;
C23.20D28 in GAP, Magma, Sage, TeX
C_2^3._{20}D_{28} % in TeX
G:=Group("C2^3.20D28"); // GroupNames label
G:=SmallGroup(448,673);
// by ID
G=gap.SmallGroup(448,673);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,387,58,136,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=c,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^27>;
// generators/relations