metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23:Dic7, C23.2D14, (C2xC4):Dic7, (C2xC28):1C4, C7:2(C23:C4), (C2xD4).3D7, (C2xC14).2D4, (C22xC14):2C4, (D4xC14).6C2, C23.D7:2C2, C22.2(C7:D4), C2.5(C23.D7), C22.3(C2xDic7), C14.15(C22:C4), (C22xC14).6C22, (C2xC14).29(C2xC4), SmallGroup(224,40)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23:Dic7
G = < a,b,c,d,e | a2=b2=c2=d14=1, e2=d7, ab=ba, dad-1=ac=ca, eae-1=abc, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >
Quotients: C1, C2, C4, C22, C2xC4, D4, D7, C22:C4, Dic7, D14, C23:C4, C2xDic7, C7:D4, C23.D7, C23:Dic7
2C2
2C2
2C2
4C2
2C22
2C4
4C22
4C22
28C4
28C4
2C14
2C14
2C14
4C14
2D4
2D4
14C2xC4
14C2xC4
2C28
4Dic7
4Dic7
Smallest permutation representation of C23:Dic7
►On 56 points
Generators in S56
(1 54)(2 22)(3 56)(4 24)(5 44)(6 26)(7 46)(8 28)(9 48)(10 16)(11 50)(12 18)(13 52)(14 20)(15 30)(17 32)(19 34)(21 36)(23 38)(25 40)(27 42)(29 47)(31 49)(33 51)(35 53)(37 55)(39 43)(41 45)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 55)(16 56)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 43)(25 44)(26 45)(27 46)(28 47)
(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)
(1 54 8 47)(2 53 9 46)(3 52 10 45)(4 51 11 44)(5 50 12 43)(6 49 13 56)(7 48 14 55)(15 35 22 42)(16 34 23 41)(17 33 24 40)(18 32 25 39)(19 31 26 38)(20 30 27 37)(21 29 28 36)
G:=sub<Sym(56)| (1,54)(2,22)(3,56)(4,24)(5,44)(6,26)(7,46)(8,28)(9,48)(10,16)(11,50)(12,18)(13,52)(14,20)(15,30)(17,32)(19,34)(21,36)(23,38)(25,40)(27,42)(29,47)(31,49)(33,51)(35,53)(37,55)(39,43)(41,45), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,55)(16,56)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,43)(25,44)(26,45)(27,46)(28,47), (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), (1,54,8,47)(2,53,9,46)(3,52,10,45)(4,51,11,44)(5,50,12,43)(6,49,13,56)(7,48,14,55)(15,35,22,42)(16,34,23,41)(17,33,24,40)(18,32,25,39)(19,31,26,38)(20,30,27,37)(21,29,28,36)>;
G:=Group( (1,54)(2,22)(3,56)(4,24)(5,44)(6,26)(7,46)(8,28)(9,48)(10,16)(11,50)(12,18)(13,52)(14,20)(15,30)(17,32)(19,34)(21,36)(23,38)(25,40)(27,42)(29,47)(31,49)(33,51)(35,53)(37,55)(39,43)(41,45), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,55)(16,56)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,43)(25,44)(26,45)(27,46)(28,47), (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), (1,54,8,47)(2,53,9,46)(3,52,10,45)(4,51,11,44)(5,50,12,43)(6,49,13,56)(7,48,14,55)(15,35,22,42)(16,34,23,41)(17,33,24,40)(18,32,25,39)(19,31,26,38)(20,30,27,37)(21,29,28,36) );
G=PermutationGroup([[(1,54),(2,22),(3,56),(4,24),(5,44),(6,26),(7,46),(8,28),(9,48),(10,16),(11,50),(12,18),(13,52),(14,20),(15,30),(17,32),(19,34),(21,36),(23,38),(25,40),(27,42),(29,47),(31,49),(33,51),(35,53),(37,55),(39,43),(41,45)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,55),(16,56),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,43),(25,44),(26,45),(27,46),(28,47)], [(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)], [(1,54,8,47),(2,53,9,46),(3,52,10,45),(4,51,11,44),(5,50,12,43),(6,49,13,56),(7,48,14,55),(15,35,22,42),(16,34,23,41),(17,33,24,40),(18,32,25,39),(19,31,26,38),(20,30,27,37),(21,29,28,36)]])
C23:Dic7 is a maximal subgroup of
C23.D28 C23.2D28 C23.3D28 C23.4D28 C24:Dic7 (C22xC28):C4 C42:2Dic7 C42:3Dic7 C23:C4:5D7 D7xC23:C4 C24:D14 C22:C4:D14 (D4xC14):10C4 2+ 1+4.2D7 2+ 1+4:2D7
C23:Dic7 is a maximal quotient of
C24.Dic7 C24.D14 (C2xC28):C8 C24:Dic7 (D4xC14):C4 C4:C4:Dic7 (C22xC28):C4 C42:2Dic7 C42.Dic7 C42:3Dic7 C42.3Dic7
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | D7 | Dic7 | Dic7 | D14 | C7:D4 | C23:C4 | C23:Dic7 |
| kernel | C23:Dic7 | C23.D7 | D4xC14 | C2xC28 | C22xC14 | C2xC14 | C2xD4 | C2xC4 | C23 | C23 | C22 | C7 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 12 | 1 | 6 |
Matrix representation of C23:Dic7 ►in GL4(F29) generated by
| 9 | 14 | 0 | 0 |
| 15 | 20 | 0 | 0 |
| 18 | 4 | 4 | 1 |
| 25 | 0 | 14 | 25 |
| 24 | 13 | 21 | 22 |
| 16 | 5 | 0 | 8 |
| 0 | 0 | 20 | 24 |
| 0 | 0 | 16 | 9 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 16 | 5 | 24 | 0 |
| 24 | 2 | 15 | 19 |
| 0 | 0 | 19 | 22 |
| 0 | 0 | 5 | 21 |
| 2 | 11 | 3 | 1 |
| 23 | 11 | 2 | 0 |
| 18 | 4 | 4 | 1 |
| 3 | 3 | 18 | 12 |
G:=sub<GL(4,GF(29))| [9,15,18,25,14,20,4,0,0,0,4,14,0,0,1,25],[24,16,0,0,13,5,0,0,21,0,20,16,22,8,24,9],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[16,24,0,0,5,2,0,0,24,15,19,5,0,19,22,21],[2,23,18,3,11,11,4,3,3,2,4,18,1,0,1,12] >;
C23:Dic7 in GAP, Magma, Sage, TeX
C_2^3\rtimes {\rm Dic}_7 % in TeX
G:=Group("C2^3:Dic7"); // GroupNames label
G:=SmallGroup(224,40);
// by ID
G=gap.SmallGroup(224,40);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,188,579,6917]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^14=1,e^2=d^7,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e^-1=a*b*c,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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