metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.18D14, (C2×D4).5D7, (C2×C14).7D4, C14.47(C2×D4), (C2×C4).18D14, C23.D7⋊8C2, Dic7⋊C4⋊14C2, (D4×C14).10C2, C14.29(C4○D4), (C2×C14).50C23, (C2×C28).61C22, (C22×Dic7)⋊5C2, C22.4(C7⋊D4), C7⋊5(C22.D4), C2.15(D4⋊2D7), C22.57(C22×D7), (C22×C14).18C22, (C2×Dic7).17C22, C2.11(C2×C7⋊D4), SmallGroup(224,130)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.18D14
G = < a,b,c,d,e | a2=b2=c2=d14=1, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 254 in 78 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, C28, C2×C14, C2×C14, C2×C14, C22.D4, C2×Dic7, C2×Dic7, C2×C28, C7×D4, C22×C14, Dic7⋊C4, C23.D7, C23.D7, C22×Dic7, D4×C14, C23.18D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22.D4, C7⋊D4, C22×D7, D4⋊2D7, C2×C7⋊D4, C23.18D14
►On 112 points
Generators in S112
(1 38)(2 32)(3 40)(4 34)(5 42)(6 36)(7 30)(8 31)(9 39)(10 33)(11 41)(12 35)(13 29)(14 37)(15 93)(16 87)(17 95)(18 89)(19 97)(20 91)(21 85)(22 88)(23 96)(24 90)(25 98)(26 92)(27 86)(28 94)(43 75)(44 108)(45 77)(46 110)(47 79)(48 112)(49 81)(50 100)(51 83)(52 102)(53 71)(54 104)(55 73)(56 106)(57 72)(58 105)(59 74)(60 107)(61 76)(62 109)(63 78)(64 111)(65 80)(66 99)(67 82)(68 101)(69 84)(70 103)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 15)(7 16)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 97)(41 98)(42 85)(43 67)(44 68)(45 69)(46 70)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 27)(16 28)(17 22)(18 23)(19 24)(20 25)(21 26)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 60)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 68)(52 69)(53 70)(54 57)(55 58)(56 59)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 99)(82 100)(83 101)(84 102)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)
(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 81 22 106)(2 73 23 112)(3 79 24 104)(4 71 25 110)(5 77 26 102)(6 83 27 108)(7 75 28 100)(8 99 17 74)(9 105 18 80)(10 111 19 72)(11 103 20 78)(12 109 21 84)(13 101 15 76)(14 107 16 82)(29 51 93 44)(30 60 94 67)(31 49 95 56)(32 58 96 65)(33 47 97 54)(34 70 98 63)(35 45 85 52)(36 68 86 61)(37 43 87 50)(38 66 88 59)(39 55 89 48)(40 64 90 57)(41 53 91 46)(42 62 92 69)
G:=sub<Sym(112)| (1,38)(2,32)(3,40)(4,34)(5,42)(6,36)(7,30)(8,31)(9,39)(10,33)(11,41)(12,35)(13,29)(14,37)(15,93)(16,87)(17,95)(18,89)(19,97)(20,91)(21,85)(22,88)(23,96)(24,90)(25,98)(26,92)(27,86)(28,94)(43,75)(44,108)(45,77)(46,110)(47,79)(48,112)(49,81)(50,100)(51,83)(52,102)(53,71)(54,104)(55,73)(56,106)(57,72)(58,105)(59,74)(60,107)(61,76)(62,109)(63,78)(64,111)(65,80)(66,99)(67,82)(68,101)(69,84)(70,103), (1,17)(2,18)(3,19)(4,20)(5,21)(6,15)(7,16)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,98)(42,85)(43,67)(44,68)(45,69)(46,70)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,60)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,68)(52,69)(53,70)(54,57)(55,58)(56,59)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,99)(82,100)(83,101)(84,102)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98), (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), (1,81,22,106)(2,73,23,112)(3,79,24,104)(4,71,25,110)(5,77,26,102)(6,83,27,108)(7,75,28,100)(8,99,17,74)(9,105,18,80)(10,111,19,72)(11,103,20,78)(12,109,21,84)(13,101,15,76)(14,107,16,82)(29,51,93,44)(30,60,94,67)(31,49,95,56)(32,58,96,65)(33,47,97,54)(34,70,98,63)(35,45,85,52)(36,68,86,61)(37,43,87,50)(38,66,88,59)(39,55,89,48)(40,64,90,57)(41,53,91,46)(42,62,92,69)>;
G:=Group( (1,38)(2,32)(3,40)(4,34)(5,42)(6,36)(7,30)(8,31)(9,39)(10,33)(11,41)(12,35)(13,29)(14,37)(15,93)(16,87)(17,95)(18,89)(19,97)(20,91)(21,85)(22,88)(23,96)(24,90)(25,98)(26,92)(27,86)(28,94)(43,75)(44,108)(45,77)(46,110)(47,79)(48,112)(49,81)(50,100)(51,83)(52,102)(53,71)(54,104)(55,73)(56,106)(57,72)(58,105)(59,74)(60,107)(61,76)(62,109)(63,78)(64,111)(65,80)(66,99)(67,82)(68,101)(69,84)(70,103), (1,17)(2,18)(3,19)(4,20)(5,21)(6,15)(7,16)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,98)(42,85)(43,67)(44,68)(45,69)(46,70)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,60)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,68)(52,69)(53,70)(54,57)(55,58)(56,59)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,99)(82,100)(83,101)(84,102)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98), (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), (1,81,22,106)(2,73,23,112)(3,79,24,104)(4,71,25,110)(5,77,26,102)(6,83,27,108)(7,75,28,100)(8,99,17,74)(9,105,18,80)(10,111,19,72)(11,103,20,78)(12,109,21,84)(13,101,15,76)(14,107,16,82)(29,51,93,44)(30,60,94,67)(31,49,95,56)(32,58,96,65)(33,47,97,54)(34,70,98,63)(35,45,85,52)(36,68,86,61)(37,43,87,50)(38,66,88,59)(39,55,89,48)(40,64,90,57)(41,53,91,46)(42,62,92,69) );
G=PermutationGroup([[(1,38),(2,32),(3,40),(4,34),(5,42),(6,36),(7,30),(8,31),(9,39),(10,33),(11,41),(12,35),(13,29),(14,37),(15,93),(16,87),(17,95),(18,89),(19,97),(20,91),(21,85),(22,88),(23,96),(24,90),(25,98),(26,92),(27,86),(28,94),(43,75),(44,108),(45,77),(46,110),(47,79),(48,112),(49,81),(50,100),(51,83),(52,102),(53,71),(54,104),(55,73),(56,106),(57,72),(58,105),(59,74),(60,107),(61,76),(62,109),(63,78),(64,111),(65,80),(66,99),(67,82),(68,101),(69,84),(70,103)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,15),(7,16),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,86),(30,87),(31,88),(32,89),(33,90),(34,91),(35,92),(36,93),(37,94),(38,95),(39,96),(40,97),(41,98),(42,85),(43,67),(44,68),(45,69),(46,70),(47,57),(48,58),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,27),(16,28),(17,22),(18,23),(19,24),(20,25),(21,26),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,60),(44,61),(45,62),(46,63),(47,64),(48,65),(49,66),(50,67),(51,68),(52,69),(53,70),(54,57),(55,58),(56,59),(71,103),(72,104),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112),(81,99),(82,100),(83,101),(84,102),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98)], [(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)], [(1,81,22,106),(2,73,23,112),(3,79,24,104),(4,71,25,110),(5,77,26,102),(6,83,27,108),(7,75,28,100),(8,99,17,74),(9,105,18,80),(10,111,19,72),(11,103,20,78),(12,109,21,84),(13,101,15,76),(14,107,16,82),(29,51,93,44),(30,60,94,67),(31,49,95,56),(32,58,96,65),(33,47,97,54),(34,70,98,63),(35,45,85,52),(36,68,86,61),(37,43,87,50),(38,66,88,59),(39,55,89,48),(40,64,90,57),(41,53,91,46),(42,62,92,69)]])
C23.18D14 is a maximal subgroup of
(C2×C28).D4 C23.4D28 C23.5D28 2+ 1+4.2D7 C42.102D14 C42.105D14 C42⋊16D14 C42.118D14 C24.56D14 C24.32D14 C24⋊2D14 C24.35D14 C24.36D14 C4⋊C4.178D14 C14.342+ 1+4 C14.352+ 1+4 C14.712- 1+4 C14.402+ 1+4 C14.732- 1+4 C14.422+ 1+4 C14.432+ 1+4 C14.442+ 1+4 C14.492+ 1+4 C14.792- 1+4 C14.802- 1+4 C14.602+ 1+4 D7×C22.D4 C14.832- 1+4 C14.672+ 1+4 C42.137D14 C42.140D14 C42⋊21D14 C42.166D14 C42.168D14 C42⋊28D14 C24⋊7D14 C24.42D14 C14.1042- 1+4 C14.1052- 1+4 (C2×C28)⋊15D4
C23.18D14 is a maximal quotient of
C23.42D28 C24.3D14 C24.46D14 C24.7D14 C24.9D14 C24.10D14 C22.23(Q8×D7) (C2×C4).44D28 (C2×C28).55D4 (C2×C14).D8 C4⋊D4.D7 (C2×D4).D14 C22⋊Q8.D7 (C2×C14).Q16 C14.(C4○D8) C24.18D14 C24.20D14
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | D4⋊2D7 |
| kernel | C23.18D14 | Dic7⋊C4 | C23.D7 | C22×Dic7 | D4×C14 | C2×C14 | C2×D4 | C14 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 3 | 1 | 1 | 2 | 3 | 4 | 3 | 6 | 12 | 6 |
Matrix representation of C23.18D14 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 27 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 24 | 0 | 0 | 0 |
| 27 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 28 |
| 26 | 2 | 0 | 0 |
| 24 | 3 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 12 | 17 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,27,28],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[24,27,0,0,0,6,0,0,0,0,1,1,0,0,0,28],[26,24,0,0,2,3,0,0,0,0,12,12,0,0,0,17] >;
C23.18D14 in GAP, Magma, Sage, TeX
C_2^3._{18}D_{14} % in TeX
G:=Group("C2^3.18D14"); // GroupNames label
G:=SmallGroup(224,130);
// by ID
G=gap.SmallGroup(224,130);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,218,188,6917]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^14=1,e^2=c*b=b*c,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=b*d^-1>;
// generators/relations