direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C23×D7, C7⋊C24, C14⋊C23, (C22×C14)⋊3C2, (C2×C14)⋊4C22, SmallGroup(112,42)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C23×D7 |
Generators and relations for C23×D7
G = < a,b,c,d,e | a2=b2=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 440 in 134 conjugacy classes, 83 normal (5 characteristic)
C1, C2, C2, C22, C22, C7, C23, C23, D7, C14, C24, D14, C2×C14, C22×D7, C22×C14, C23×D7
Quotients: C1, C2, C22, C23, D7, C24, D14, C22×D7, C23×D7
►On 56 points
Generators in S56
(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 35)(7 34)(8 38)(9 37)(10 36)(11 42)(12 41)(13 40)(14 39)(15 45)(16 44)(17 43)(18 49)(19 48)(20 47)(21 46)(22 52)(23 51)(24 50)(25 56)(26 55)(27 54)(28 53)
G:=sub<Sym(56)| (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56), (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,35)(7,34)(8,38)(9,37)(10,36)(11,42)(12,41)(13,40)(14,39)(15,45)(16,44)(17,43)(18,49)(19,48)(20,47)(21,46)(22,52)(23,51)(24,50)(25,56)(26,55)(27,54)(28,53)>;
G:=Group( (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56), (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,35)(7,34)(8,38)(9,37)(10,36)(11,42)(12,41)(13,40)(14,39)(15,45)(16,44)(17,43)(18,49)(19,48)(20,47)(21,46)(22,52)(23,51)(24,50)(25,56)(26,55)(27,54)(28,53) );
G=PermutationGroup([[(1,34),(2,35),(3,29),(4,30),(5,31),(6,32),(7,33),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56)], [(1,27),(2,28),(3,22),(4,23),(5,24),(6,25),(7,26),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49)], [(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56)], [(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,35),(7,34),(8,38),(9,37),(10,36),(11,42),(12,41),(13,40),(14,39),(15,45),(16,44),(17,43),(18,49),(19,48),(20,47),(21,46),(22,52),(23,51),(24,50),(25,56),(26,55),(27,54),(28,53)]])
C23×D7 is a maximal subgroup of
C22⋊D28 C23⋊D14
C23×D7 is a maximal quotient of D4⋊6D14 Q8.10D14 D4⋊8D14 D4.10D14
40 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 7A | 7B | 7C | 14A | ··· | 14U |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 7 | 7 | 7 | 14 | ··· | 14 |
| size | 1 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D7 | D14 |
| kernel | C23×D7 | C22×D7 | C22×C14 | C23 | C22 |
| # reps | 1 | 14 | 1 | 3 | 21 |
Matrix representation of C23×D7 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 18 |
| 28 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,0,28,0,0,1,18],[28,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
C23×D7 in GAP, Magma, Sage, TeX
C_2^3\times D_7
% in TeX
G:=Group("C2^3xD7"); // GroupNames label
G:=SmallGroup(112,42);
// by ID
G=gap.SmallGroup(112,42);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,2404]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations