Copied to
clipboard

## G = C23⋊Dic7order 224 = 25·7

### The semidirect product of C23 and Dic7 acting via Dic7/C7=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C23⋊Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C23.D7 — C23⋊Dic7
 Lower central C7 — C14 — C2×C14 — C23⋊Dic7
 Upper central C1 — C2 — C23 — C2×D4

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 >

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  (C22×C28)⋊C4  C422Dic7  C423Dic7  C23⋊C45D7  D7×C23⋊C4  C24⋊D14  C22⋊C4⋊D14  (D4×C14)⋊10C4  2+ 1+4.2D7  2+ 1+42D7
C23⋊Dic7 is a maximal quotient of
C24.Dic7  C24.D14  (C2×C28)⋊C8  C24⋊Dic7  (D4×C14)⋊C4  C4⋊C4⋊Dic7  (C22×C28)⋊C4  C422Dic7  C42.Dic7  C423Dic7  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 D4×C14 C2×C28 C22×C14 C2×C14 C2×D4 C2×C4 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

Export

׿
×
𝔽