metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23⋊3D28, C24.29D14, C14.42+ 1+4, C28⋊7D4⋊3C2, (C22×C4)⋊9D14, C22⋊D28⋊3C2, D14⋊C4⋊1C22, C22⋊C4⋊43D14, (C2×D28)⋊3C22, (C22×C14)⋊10D4, C7⋊1(C23⋊3D4), C4⋊Dic7⋊5C22, C14.8(C22×D4), (C2×C14).37C24, (C22×C28)⋊8C22, C22.18(C2×D28), C2.10(C22×D28), (C23×D7)⋊4C22, C2.8(D4⋊6D14), (C2×C28).130C23, C22.D28⋊2C2, (C22×D7).9C23, C22.76(C23×D7), (C23×C14).63C22, (C2×Dic7).10C23, (C22×Dic7)⋊7C22, C23.148(C22×D7), (C22×C14).127C23, (C2×C22⋊C4)⋊16D7, (C22×C7⋊D4)⋊6C2, (C14×C22⋊C4)⋊15C2, (C2×C14).173(C2×D4), (C2×C7⋊D4)⋊36C22, (C7×C22⋊C4)⋊48C22, (C2×C4).136(C22×D7), SmallGroup(448,946)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊3D28
G = < a,b,c,d,e | a2=b2=c2=d28=e2=1, ab=ba, dad-1=eae=ac=ca, ebe=bc=cb, bd=db, cd=dc, ce=ec, ede=d-1 >
Subgroups: 2004 in 346 conjugacy classes, 111 normal (13 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C22≀C2, C4⋊D4, C22.D4, C22×D4, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C23⋊3D4, C4⋊Dic7, D14⋊C4, C7×C22⋊C4, C2×D28, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C23×D7, C23×C14, C22⋊D28, C22.D28, C28⋊7D4, C14×C22⋊C4, C22×C7⋊D4, C23⋊3D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, D28, C22×D7, C23⋊3D4, C2×D28, C23×D7, C22×D28, D4⋊6D14, C23⋊3D28
►On 112 points
Generators in S112
(1 103)(2 72)(3 105)(4 74)(5 107)(6 76)(7 109)(8 78)(9 111)(10 80)(11 85)(12 82)(13 87)(14 84)(15 89)(16 58)(17 91)(18 60)(19 93)(20 62)(21 95)(22 64)(23 97)(24 66)(25 99)(26 68)(27 101)(28 70)(29 96)(30 65)(31 98)(32 67)(33 100)(34 69)(35 102)(36 71)(37 104)(38 73)(39 106)(40 75)(41 108)(42 77)(43 110)(44 79)(45 112)(46 81)(47 86)(48 83)(49 88)(50 57)(51 90)(52 59)(53 92)(54 61)(55 94)(56 63)
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 85)(26 86)(27 87)(28 88)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(28 35)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 85)(82 86)(83 87)(84 88)
(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 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(50 56)(51 55)(52 54)(57 95)(58 94)(59 93)(60 92)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 112)(69 111)(70 110)(71 109)(72 108)(73 107)(74 106)(75 105)(76 104)(77 103)(78 102)(79 101)(80 100)(81 99)(82 98)(83 97)(84 96)
G:=sub<Sym(112)| (1,103)(2,72)(3,105)(4,74)(5,107)(6,76)(7,109)(8,78)(9,111)(10,80)(11,85)(12,82)(13,87)(14,84)(15,89)(16,58)(17,91)(18,60)(19,93)(20,62)(21,95)(22,64)(23,97)(24,66)(25,99)(26,68)(27,101)(28,70)(29,96)(30,65)(31,98)(32,67)(33,100)(34,69)(35,102)(36,71)(37,104)(38,73)(39,106)(40,75)(41,108)(42,77)(43,110)(44,79)(45,112)(46,81)(47,86)(48,83)(49,88)(50,57)(51,90)(52,59)(53,92)(54,61)(55,94)(56,63), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,85)(26,86)(27,87)(28,88)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(28,35)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,85)(82,86)(83,87)(84,88), (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)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(50,56)(51,55)(52,54)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,112)(69,111)(70,110)(71,109)(72,108)(73,107)(74,106)(75,105)(76,104)(77,103)(78,102)(79,101)(80,100)(81,99)(82,98)(83,97)(84,96)>;
G:=Group( (1,103)(2,72)(3,105)(4,74)(5,107)(6,76)(7,109)(8,78)(9,111)(10,80)(11,85)(12,82)(13,87)(14,84)(15,89)(16,58)(17,91)(18,60)(19,93)(20,62)(21,95)(22,64)(23,97)(24,66)(25,99)(26,68)(27,101)(28,70)(29,96)(30,65)(31,98)(32,67)(33,100)(34,69)(35,102)(36,71)(37,104)(38,73)(39,106)(40,75)(41,108)(42,77)(43,110)(44,79)(45,112)(46,81)(47,86)(48,83)(49,88)(50,57)(51,90)(52,59)(53,92)(54,61)(55,94)(56,63), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,85)(26,86)(27,87)(28,88)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(28,35)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,85)(82,86)(83,87)(84,88), (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)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(50,56)(51,55)(52,54)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,112)(69,111)(70,110)(71,109)(72,108)(73,107)(74,106)(75,105)(76,104)(77,103)(78,102)(79,101)(80,100)(81,99)(82,98)(83,97)(84,96) );
G=PermutationGroup([[(1,103),(2,72),(3,105),(4,74),(5,107),(6,76),(7,109),(8,78),(9,111),(10,80),(11,85),(12,82),(13,87),(14,84),(15,89),(16,58),(17,91),(18,60),(19,93),(20,62),(21,95),(22,64),(23,97),(24,66),(25,99),(26,68),(27,101),(28,70),(29,96),(30,65),(31,98),(32,67),(33,100),(34,69),(35,102),(36,71),(37,104),(38,73),(39,106),(40,75),(41,108),(42,77),(43,110),(44,79),(45,112),(46,81),(47,86),(48,83),(49,88),(50,57),(51,90),(52,59),(53,92),(54,61),(55,94),(56,63)], [(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,97),(10,98),(11,99),(12,100),(13,101),(14,102),(15,103),(16,104),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,85),(26,86),(27,87),(28,88),(29,78),(30,79),(31,80),(32,81),(33,82),(34,83),(35,84),(36,57),(37,58),(38,59),(39,60),(40,61),(41,62),(42,63),(43,64),(44,65),(45,66),(46,67),(47,68),(48,69),(49,70),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,29),(23,30),(24,31),(25,32),(26,33),(27,34),(28,35),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,99),(68,100),(69,101),(70,102),(71,103),(72,104),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112),(81,85),(82,86),(83,87),(84,88)], [(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),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(50,56),(51,55),(52,54),(57,95),(58,94),(59,93),(60,92),(61,91),(62,90),(63,89),(64,88),(65,87),(66,86),(67,85),(68,112),(69,111),(70,110),(71,109),(72,108),(73,107),(74,106),(75,105),(76,104),(77,103),(78,102),(79,101),(80,100),(81,99),(82,98),(83,97),(84,96)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | D28 | 2+ 1+4 | D4⋊6D14 |
| kernel | C23⋊3D28 | C22⋊D28 | C22.D28 | C28⋊7D4 | C14×C22⋊C4 | C22×C7⋊D4 | C22×C14 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C24 | C23 | C14 | C2 |
| # reps | 1 | 4 | 4 | 4 | 1 | 2 | 4 | 3 | 12 | 6 | 3 | 24 | 2 | 12 |
Matrix representation of C23⋊3D28 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 27 | 0 |
| 0 | 0 | 1 | 0 | 1 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 28 | 28 | 28 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 28 | 28 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 | 1 | 0 |
| 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 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 22 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 27 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 28 | 28 | 28 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 25 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,1,0,28,0,0,0,0,0,28,0,0,27,1,1,28,0,0,0,28,0,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,2,28,28,1,0,0,0,28,0,0],[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],[3,22,0,0,0,0,7,13,0,0,0,0,0,0,28,1,0,28,0,0,27,1,1,28,0,0,0,0,0,28,0,0,0,0,1,0],[4,25,0,0,0,0,11,25,0,0,0,0,0,0,28,1,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28] >;
C23⋊3D28 in GAP, Magma, Sage, TeX
C_2^3\rtimes_3D_{28} % in TeX
G:=Group("C2^3:3D28"); // GroupNames label
G:=SmallGroup(448,946);
// by ID
G=gap.SmallGroup(448,946);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,675,570,80,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=e^2=1,a*b=b*a,d*a*d^-1=e*a*e=a*c=c*a,e*b*e=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations