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G = C8⋊D7order 112 = 24·7

3rd semidirect product of C8 and D7 acting via D7/C7=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C83D7, C564C2, D14.C4, Dic7.C4, C71M4(2), C4.13D14, C28.13C22, C7⋊C84C2, C2.3(C4×D7), C14.2(C2×C4), (C4×D7).2C2, SmallGroup(112,4)

Series: Derived Chief Lower central Upper central

C1C14 — C8⋊D7
C1C7C14C28C4×D7 — C8⋊D7
C7C14 — C8⋊D7
C1C4C8

Generators and relations for C8⋊D7
 G = < a,b,c | a8=b7=c2=1, ab=ba, cac=a5, cbc=b-1 >

14C2
7C22
7C4
2D7
7C2×C4
7C8
7M4(2)

Smallest permutation representation of C8⋊D7
On 56 points
Generators in S56
(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 49 17 33 44 13 31)(2 50 18 34 45 14 32)(3 51 19 35 46 15 25)(4 52 20 36 47 16 26)(5 53 21 37 48 9 27)(6 54 22 38 41 10 28)(7 55 23 39 42 11 29)(8 56 24 40 43 12 30)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 53)(10 50)(11 55)(12 52)(13 49)(14 54)(15 51)(16 56)(17 44)(18 41)(19 46)(20 43)(21 48)(22 45)(23 42)(24 47)(34 38)(36 40)

G:=sub<Sym(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), (1,49,17,33,44,13,31)(2,50,18,34,45,14,32)(3,51,19,35,46,15,25)(4,52,20,36,47,16,26)(5,53,21,37,48,9,27)(6,54,22,38,41,10,28)(7,55,23,39,42,11,29)(8,56,24,40,43,12,30), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,53)(10,50)(11,55)(12,52)(13,49)(14,54)(15,51)(16,56)(17,44)(18,41)(19,46)(20,43)(21,48)(22,45)(23,42)(24,47)(34,38)(36,40)>;

G:=Group( (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), (1,49,17,33,44,13,31)(2,50,18,34,45,14,32)(3,51,19,35,46,15,25)(4,52,20,36,47,16,26)(5,53,21,37,48,9,27)(6,54,22,38,41,10,28)(7,55,23,39,42,11,29)(8,56,24,40,43,12,30), (1,31)(2,28)(3,25)(4,30)(5,27)(6,32)(7,29)(8,26)(9,53)(10,50)(11,55)(12,52)(13,49)(14,54)(15,51)(16,56)(17,44)(18,41)(19,46)(20,43)(21,48)(22,45)(23,42)(24,47)(34,38)(36,40) );

G=PermutationGroup([(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)], [(1,49,17,33,44,13,31),(2,50,18,34,45,14,32),(3,51,19,35,46,15,25),(4,52,20,36,47,16,26),(5,53,21,37,48,9,27),(6,54,22,38,41,10,28),(7,55,23,39,42,11,29),(8,56,24,40,43,12,30)], [(1,31),(2,28),(3,25),(4,30),(5,27),(6,32),(7,29),(8,26),(9,53),(10,50),(11,55),(12,52),(13,49),(14,54),(15,51),(16,56),(17,44),(18,41),(19,46),(20,43),(21,48),(22,45),(23,42),(24,47),(34,38),(36,40)])

C8⋊D7 is a maximal subgroup of
D28.2C4  D7×M4(2)  D28.C4  D8⋊D7  D56⋊C2  SD16⋊D7  Q16⋊D7  C8⋊F7  C28.32D6  D42.C4  C56⋊S3
C8⋊D7 is a maximal quotient of
Dic7⋊C8  C56⋊C4  D14⋊C8  C28.32D6  D42.C4  C56⋊S3

34 conjugacy classes

class 1 2A2B4A4B4C7A7B7C8A8B8C8D14A14B14C28A···28F56A···56L
order122444777888814141428···2856···56
size111411142222214142222···22···2

34 irreducible representations

dim11111122222
type++++++
imageC1C2C2C2C4C4D7M4(2)D14C4×D7C8⋊D7
kernelC8⋊D7C7⋊C8C56C4×D7Dic7D14C8C7C4C2C1
# reps111122323612

Matrix representation of C8⋊D7 in GL2(𝔽13) generated by

412
89
,
66
42
,
24
911
G:=sub<GL(2,GF(13))| [4,8,12,9],[6,4,6,2],[2,9,4,11] >;

C8⋊D7 in GAP, Magma, Sage, TeX

C_8\rtimes D_7
% in TeX

G:=Group("C8:D7");
// GroupNames label

G:=SmallGroup(112,4);
// by ID

G=gap.SmallGroup(112,4);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,101,26,42,2404]);
// Polycyclic

G:=Group<a,b,c|a^8=b^7=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C8⋊D7 in TeX

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