Dic₅₉₂₅ #37557126

order	23700
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₉₂₅
unambiguous semidirect product	C₃⋊Dic₁₉₇₅
unambiguous semidirect product	C₇₉⋊Dic₇₅

Splittings

derived, Hall	C₃×C₂₅×C₇₉ ⋊ C₄
Hall	C₂₅×C₇₉ ⋊ Dic₃
Hall	C₃×C₇₉ ⋊ Dic₂₅
Sylow	C₇₉ ⋊ Dic₇₅
Hall	C₃×C₂₅ ⋊ Dic₇₉
Sylow	C₂₅ ⋊ Dic₂₃₇
Sylow	C₃ ⋊ Dic₁₉₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₉₂₅
commutator, hypocenter	C₃×C₂₅×C₇₉	C₄
Frattini	C₂×C₅	D₁₁₈₅
Fitting, cosocle	C₂×C₃×C₂₅×C₇₉	C₂
socle, prime radical	C₂×C₃×C₅×C₇₉	D₅
LCS	G ▷ C₃×C₂₅×C₇₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 23700

A minimal faithful permutation representation — 2 generators on 111 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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(1,3,2,4)(6,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)

Polycyclic presentation:

2,3,5,79,2,5|a^2=e,c^5=f^4,b^a=b^2,c^a=c^4f,d^a=d^78,f^a=f^4

Conjugacy classes & character table

5928 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₈₅₀
maximal normal subgroup of	Dic₁₁₈₅₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23700, 9);

Dic5925 #37557126