D₇₅⋊D₆₇ #34117174

order	20100
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	D₇₅⋊D₆₇
unambiguous semidirect product	D₂₀₁⋊D₂₅
unambiguous semidirect product	D₁₆₇₅⋊D₃

Splittings

Fitting, derived, Hall	C₃×C₂₅×C₆₇ ⋊ C₂²
Hall	C₂₅×C₆₇ ⋊ C₂×D₃
Hall	C₃×C₆₇ ⋊ C₂×D₂₅
Hall	C₃×C₂₅ ⋊ C₂×D₆₇
Sylow	C₆₇ ⋊ D₃×D₂₅
Sylow	C₂₅ ⋊ D₃×D₆₇
Sylow	C₃ ⋊ D₂₅×D₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₂₅×C₆₇	C₂²
Frattini	C₅	D₁₅⋊D₆₇
socle	C₃×C₅×C₆₇	C₂×D₅
LCS	G ▷ C₃×C₂₅×C₆₇
UCS	C₁

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 5025

A minimal faithful permutation representation — 2 generators on 95 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(68,88)(69,83)(70,81)(71,92)(72,87)(73,75)(74,82)(76,86)(77,90)(78,91)(79,84)(80,89)(94,95);(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)(69,71)(70,74)(72,77)(73,80)(75,83)(76,82)(78,79)(81,87)(84,88)(85,89)(86,91)(90,92)(93,94,95)

Polycyclic presentation:

2^2,3,5,67,5|d^5=f^4,d^a=d^4f,e^a=e^66,f^a=f^4,c^b=c^2,d^b=d^4f,f^b=f^4

Conjugacy classes & character table

1329 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20100, 30);

D75⋊D67 #34117174