metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.44D28, C24.48D14, (C23×D7)⋊5C4, (C22×C4)⋊1D14, C7⋊1(C24⋊3C4), (D7×C24).1C2, C23.51(C4×D7), C22⋊3(D14⋊C4), C14.37C22≀C2, D14⋊3(C22⋊C4), (C22×C28)⋊1C22, (C22×D7).87D4, C22.100(D4×D7), (C22×C14).67D4, C22.43(C2×D28), C2.4(C22⋊D28), C2.2(C23⋊D14), C23.52(C7⋊D4), (C23×C14).38C22, (C22×Dic7)⋊2C22, (C23×D7).87C22, C23.282(C22×D7), (C22×C14).329C23, (C2×D14⋊C4)⋊3C2, C2.9(C2×D14⋊C4), (C2×C22⋊C4)⋊2D7, (C14×C22⋊C4)⋊2C2, (C2×C23.D7)⋊2C2, C2.28(D7×C22⋊C4), C22.126(C2×C4×D7), (C2×C14)⋊1(C22⋊C4), (C2×C14).321(C2×D4), C14.36(C2×C22⋊C4), C22.50(C2×C7⋊D4), (C22×C14).53(C2×C4), (C22×D7).60(C2×C4), (C2×C14).108(C22×C4), SmallGroup(448,489)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.44D28
G = < a,b,c,d,e | a2=b2=c2=d28=1, e2=b, ab=ba, dad-1=eae-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 3028 in 506 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C23, C23, C23, D7, C14, C14, C14, C22⋊C4, C22×C4, C22×C4, C24, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C22⋊C4, C25, C2×Dic7, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C24⋊3C4, D14⋊C4, C23.D7, C7×C22⋊C4, C22×Dic7, C22×C28, C23×D7, C23×D7, C23×C14, C2×D14⋊C4, C2×C23.D7, C14×C22⋊C4, D7×C24, C23.44D28
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C22≀C2, C4×D7, D28, C7⋊D4, C22×D7, C24⋊3C4, D14⋊C4, C2×C4×D7, C2×D28, D4×D7, C2×C7⋊D4, D7×C22⋊C4, C22⋊D28, C2×D14⋊C4, C23⋊D14, C23.44D28
►On 112 points
Generators in S112
(2 29)(4 31)(6 33)(8 35)(10 37)(12 39)(14 41)(16 43)(18 45)(20 47)(22 49)(24 51)(26 53)(28 55)(57 106)(59 108)(61 110)(63 112)(65 86)(67 88)(69 90)(71 92)(73 94)(75 96)(77 98)(79 100)(81 102)(83 104)
(1 82)(2 83)(3 84)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)(53 100)(54 101)(55 102)(56 103)
(1 56)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)(63 112)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)
(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 67 82 14)(2 13 83 66)(3 65 84 12)(4 11 57 64)(5 63 58 10)(6 9 59 62)(7 61 60 8)(15 81 68 28)(16 27 69 80)(17 79 70 26)(18 25 71 78)(19 77 72 24)(20 23 73 76)(21 75 74 22)(29 40 104 87)(30 86 105 39)(31 38 106 85)(32 112 107 37)(33 36 108 111)(34 110 109 35)(41 56 88 103)(42 102 89 55)(43 54 90 101)(44 100 91 53)(45 52 92 99)(46 98 93 51)(47 50 94 97)(48 96 95 49)
G:=sub<Sym(112)| (2,29)(4,31)(6,33)(8,35)(10,37)(12,39)(14,41)(16,43)(18,45)(20,47)(22,49)(24,51)(26,53)(28,55)(57,106)(59,108)(61,110)(63,112)(65,86)(67,88)(69,90)(71,92)(73,94)(75,96)(77,98)(79,100)(81,102)(83,104), (1,82)(2,83)(3,84)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103), (1,56)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (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), (1,67,82,14)(2,13,83,66)(3,65,84,12)(4,11,57,64)(5,63,58,10)(6,9,59,62)(7,61,60,8)(15,81,68,28)(16,27,69,80)(17,79,70,26)(18,25,71,78)(19,77,72,24)(20,23,73,76)(21,75,74,22)(29,40,104,87)(30,86,105,39)(31,38,106,85)(32,112,107,37)(33,36,108,111)(34,110,109,35)(41,56,88,103)(42,102,89,55)(43,54,90,101)(44,100,91,53)(45,52,92,99)(46,98,93,51)(47,50,94,97)(48,96,95,49)>;
G:=Group( (2,29)(4,31)(6,33)(8,35)(10,37)(12,39)(14,41)(16,43)(18,45)(20,47)(22,49)(24,51)(26,53)(28,55)(57,106)(59,108)(61,110)(63,112)(65,86)(67,88)(69,90)(71,92)(73,94)(75,96)(77,98)(79,100)(81,102)(83,104), (1,82)(2,83)(3,84)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,96)(50,97)(51,98)(52,99)(53,100)(54,101)(55,102)(56,103), (1,56)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105), (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), (1,67,82,14)(2,13,83,66)(3,65,84,12)(4,11,57,64)(5,63,58,10)(6,9,59,62)(7,61,60,8)(15,81,68,28)(16,27,69,80)(17,79,70,26)(18,25,71,78)(19,77,72,24)(20,23,73,76)(21,75,74,22)(29,40,104,87)(30,86,105,39)(31,38,106,85)(32,112,107,37)(33,36,108,111)(34,110,109,35)(41,56,88,103)(42,102,89,55)(43,54,90,101)(44,100,91,53)(45,52,92,99)(46,98,93,51)(47,50,94,97)(48,96,95,49) );
G=PermutationGroup([[(2,29),(4,31),(6,33),(8,35),(10,37),(12,39),(14,41),(16,43),(18,45),(20,47),(22,49),(24,51),(26,53),(28,55),(57,106),(59,108),(61,110),(63,112),(65,86),(67,88),(69,90),(71,92),(73,94),(75,96),(77,98),(79,100),(81,102),(83,104)], [(1,82),(2,83),(3,84),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,73),(21,74),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,81),(29,104),(30,105),(31,106),(32,107),(33,108),(34,109),(35,110),(36,111),(37,112),(38,85),(39,86),(40,87),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,96),(50,97),(51,98),(52,99),(53,100),(54,101),(55,102),(56,103)], [(1,56),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(57,106),(58,107),(59,108),(60,109),(61,110),(62,111),(63,112),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105)], [(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)], [(1,67,82,14),(2,13,83,66),(3,65,84,12),(4,11,57,64),(5,63,58,10),(6,9,59,62),(7,61,60,8),(15,81,68,28),(16,27,69,80),(17,79,70,26),(18,25,71,78),(19,77,72,24),(20,23,73,76),(21,75,74,22),(29,40,104,87),(30,86,105,39),(31,38,106,85),(32,112,107,37),(33,36,108,111),(34,110,109,35),(41,56,88,103),(42,102,89,55),(43,54,90,101),(44,100,91,53),(45,52,92,99),(46,98,93,51),(47,50,94,97),(48,96,95,49)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D14 | D14 | C4×D7 | D28 | C7⋊D4 | D4×D7 |
| kernel | C23.44D28 | C2×D14⋊C4 | C2×C23.D7 | C14×C22⋊C4 | D7×C24 | C23×D7 | C22×D7 | C22×C14 | C2×C22⋊C4 | C22×C4 | C24 | C23 | C23 | C23 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 8 | 4 | 3 | 6 | 3 | 12 | 12 | 12 | 12 |
Matrix representation of C23.44D28 ►in GL6(𝔽29)
| 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 | 18 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 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 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 11 | 16 | 0 | 0 | 0 | 0 |
| 7 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 22 | 0 | 0 |
| 0 | 0 | 7 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 13 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 16 | 0 | 0 | 0 | 0 |
| 25 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 22 | 0 | 0 |
| 0 | 0 | 26 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 13 |
| 0 | 0 | 0 | 0 | 11 | 1 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,18,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[11,7,0,0,0,0,16,18,0,0,0,0,0,0,26,7,0,0,0,0,22,16,0,0,0,0,0,0,28,0,0,0,0,0,13,1],[11,25,0,0,0,0,16,18,0,0,0,0,0,0,26,26,0,0,0,0,22,3,0,0,0,0,0,0,28,11,0,0,0,0,13,1] >;
C23.44D28 in GAP, Magma, Sage, TeX
C_2^3._{44}D_{28} % in TeX
G:=Group("C2^3.44D28"); // GroupNames label
G:=SmallGroup(448,489);
// by ID
G=gap.SmallGroup(448,489);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,387,58,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=1,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations