D₁₅₂₁₁ #44478368

order	30422
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₂₁₁
unambiguous semidirect product	C₇⋊D₂₁₇₃
unambiguous semidirect product	C₄₁⋊D₃₇₁
unambiguous semidirect product	C₅₃⋊D₂₈₇

Splittings

Fitting, socle, derived, Hall	C₇×C₄₁×C₅₃ ⋊ C₂
Hall	C₄₁×C₅₃ ⋊ D₇
Hall	C₇×C₅₃ ⋊ D₄₁
Hall	C₇×C₄₁ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₂₈₇
Sylow	C₄₁ ⋊ D₃₇₁
Sylow	C₇ ⋊ D₂₁₇₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₄₁×C₅₃	C₂
LCS	G ▷ C₇×C₄₁×C₅₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 15211

A minimal faithful permutation representation — 2 generators on 101 points:

(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);(2,7)(3,6)(4,5)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2,7,41,53|b^a=b^6,c^a=c^40,d^a=d^52

Conjugacy classes & character table

7607 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₂₁₁
maximal normal subgroup of	C₂×D₁₅₂₁₁
maximal quotient of	Dic₁₅₂₁₁, C₂×D₁₅₂₁₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30422, 8);

D15211 #44478368