metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.21D14, (C2×C28)⋊6C4, (C2×C4)⋊4Dic7, C28.37(C2×C4), C4⋊Dic7⋊17C2, (C22×C4).7D7, C7⋊4(C42⋊C2), (C4×Dic7)⋊15C2, C2.4(C4○D28), (C2×C4).102D14, C4.15(C2×Dic7), C23.D7.5C2, C14.16(C4○D4), (C2×C14).44C23, C14.24(C22×C4), (C22×C28).10C2, (C2×C28).93C22, C22.5(C2×Dic7), C2.5(C22×Dic7), C22.22(C22×D7), (C22×C14).36C22, (C2×Dic7).35C22, (C2×C14).35(C2×C4), SmallGroup(224,121)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.21D14
G = < a,b,c,d,e | a2=b2=c2=1, d14=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=d13 >
Subgroups: 206 in 76 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2×C14, C42⋊C2, C2×Dic7, C2×C28, C2×C28, C22×C14, C4×Dic7, C4⋊Dic7, C23.D7, C22×C28, C23.21D14
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, Dic7, D14, C42⋊C2, C2×Dic7, C22×D7, C4○D28, C22×Dic7, C23.21D14
►On 112 points
(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 100)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 110)(22 111)(23 112)(24 85)(25 86)(26 87)(27 88)(28 89)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 57)(56 58)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(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 71 104 55)(2 84 105 40)(3 69 106 53)(4 82 107 38)(5 67 108 51)(6 80 109 36)(7 65 110 49)(8 78 111 34)(9 63 112 47)(10 76 85 32)(11 61 86 45)(12 74 87 30)(13 59 88 43)(14 72 89 56)(15 57 90 41)(16 70 91 54)(17 83 92 39)(18 68 93 52)(19 81 94 37)(20 66 95 50)(21 79 96 35)(22 64 97 48)(23 77 98 33)(24 62 99 46)(25 75 100 31)(26 60 101 44)(27 73 102 29)(28 58 103 42)
G:=sub<Sym(112)| (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,85)(25,86)(26,87)(27,88)(28,89)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,57)(56,58), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,71,104,55)(2,84,105,40)(3,69,106,53)(4,82,107,38)(5,67,108,51)(6,80,109,36)(7,65,110,49)(8,78,111,34)(9,63,112,47)(10,76,85,32)(11,61,86,45)(12,74,87,30)(13,59,88,43)(14,72,89,56)(15,57,90,41)(16,70,91,54)(17,83,92,39)(18,68,93,52)(19,81,94,37)(20,66,95,50)(21,79,96,35)(22,64,97,48)(23,77,98,33)(24,62,99,46)(25,75,100,31)(26,60,101,44)(27,73,102,29)(28,58,103,42)>;
G:=Group( (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,85)(25,86)(26,87)(27,88)(28,89)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,57)(56,58), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,71,104,55)(2,84,105,40)(3,69,106,53)(4,82,107,38)(5,67,108,51)(6,80,109,36)(7,65,110,49)(8,78,111,34)(9,63,112,47)(10,76,85,32)(11,61,86,45)(12,74,87,30)(13,59,88,43)(14,72,89,56)(15,57,90,41)(16,70,91,54)(17,83,92,39)(18,68,93,52)(19,81,94,37)(20,66,95,50)(21,79,96,35)(22,64,97,48)(23,77,98,33)(24,62,99,46)(25,75,100,31)(26,60,101,44)(27,73,102,29)(28,58,103,42) );
G=PermutationGroup([[(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84)], [(1,90),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,100),(12,101),(13,102),(14,103),(15,104),(16,105),(17,106),(18,107),(19,108),(20,109),(21,110),(22,111),(23,112),(24,85),(25,86),(26,87),(27,88),(28,89),(29,59),(30,60),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(55,57),(56,58)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)], [(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)], [(1,71,104,55),(2,84,105,40),(3,69,106,53),(4,82,107,38),(5,67,108,51),(6,80,109,36),(7,65,110,49),(8,78,111,34),(9,63,112,47),(10,76,85,32),(11,61,86,45),(12,74,87,30),(13,59,88,43),(14,72,89,56),(15,57,90,41),(16,70,91,54),(17,83,92,39),(18,68,93,52),(19,81,94,37),(20,66,95,50),(21,79,96,35),(22,64,97,48),(23,77,98,33),(24,62,99,46),(25,75,100,31),(26,60,101,44),(27,73,102,29),(28,58,103,42)]])
C23.21D14 is a maximal subgroup of
C28.8C42 C42⋊Dic7 C28.3C42 (C2×C56)⋊C4 C23.9D28 M4(2)⋊4Dic7 C56⋊C4⋊C2 C23.10D28 C23.13D28 C4.Dic7⋊C4 C28.45(C4⋊C4) (C2×D28)⋊13C4 (C2×D4).D14 C14.(C4○D8) C28.12C42 Dic7⋊C8⋊C2 C23.22D28 C23.23D28 C28.439(C2×D4) C23.46D28 C23.47D28 C28.7C42 C23.48D28 C23.20D28 (D4×C14)⋊6C4 (Q8×C14)⋊6C4 C28.(C2×D4) (D4×C14)⋊9C4 (D4×C14)⋊10C4 C42.274D14 C4×C4○D28 C24.31D14 C14.72+ 1+4 C14.82+ 1+4 C14.52- 1+4 C42.87D14 C42.88D14 C42.90D14 D7×C42⋊C2 C42⋊7D14 C42.102D14 C42.105D14 C42.106D14 C42.229D14 C42.117D14 C42.119D14 C14.712- 1+4 C14.432+ 1+4 C14.452+ 1+4 C14.472+ 1+4 C14.152- 1+4 C14.212- 1+4 C14.232- 1+4 C14.242- 1+4 C14.802- 1+4 C14.1222+ 1+4 C24.72D14 C24.38D14 C24.41D14 C14.422- 1+4 C14.442- 1+4 C14.1052- 1+4 C4○D4×Dic7 C14.1062- 1+4 (C2×C28)⋊15D4 C14.1462+ 1+4
C23.21D14 is a maximal quotient of
C42.6Dic7 C42.7Dic7 C42⋊4Dic7 C4×C4⋊Dic7 C42⋊9Dic7 C42⋊5Dic7 C24.8D14 C4⋊C4⋊5Dic7 C42.187D14 C4×C23.D7 C24.63D14 C23.27D28
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D7 | C4○D4 | Dic7 | D14 | D14 | C4○D28 |
| kernel | C23.21D14 | C4×Dic7 | C4⋊Dic7 | C23.D7 | C22×C28 | C2×C28 | C22×C4 | C14 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 3 | 4 | 12 | 6 | 3 | 24 |
Matrix representation of C23.21D14 ►in GL3(𝔽29) generated by
| 28 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 28 |
| 28 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 28 | 0 |
| 0 | 0 | 28 |
| 1 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 19 |
| 12 | 0 | 0 |
| 0 | 0 | 19 |
| 0 | 3 | 0 |
G:=sub<GL(3,GF(29))| [28,0,0,0,1,0,0,0,28],[28,0,0,0,1,0,0,0,1],[1,0,0,0,28,0,0,0,28],[1,0,0,0,3,0,0,0,19],[12,0,0,0,0,3,0,19,0] >;
C23.21D14 in GAP, Magma, Sage, TeX
C_2^3._{21}D_{14} % in TeX
G:=Group("C2^3.21D14"); // GroupNames label
G:=SmallGroup(224,121);
// by ID
G=gap.SmallGroup(224,121);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,103,362,6917]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^14=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^13>;
// generators/relations