C₂²×#16837311 #28754865

order	15104
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#16837311

Splittings

Fitting	C₂³×C₂²⋊C₄×C₅₉ ⋊ C₂
Sylow	C₅₉ ⋊ C₂²×#392

Subgroups

	subgroup	G/H
center	C₂⁵	C₂²×D₅₉
commutator, cosocle	C₂³×C₅₉	C₂⁵
Frattini	C₂³	C₂⁴×D₅₉
Fitting	C₂³×C₂²⋊C₄×C₅₉	C₂
socle	C₂⁵×C₅₉	C₂³
hypercenter	C₂³×C₂²⋊C₄	D₅₉
hypocenter	C₅₉	C₂²×#392
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

13646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 7552

A minimal faithful permutation representation — 5 generators on 75 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)(69,71)(73,75);(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)(65,67)(72,73)(74,75);(64,65)(66,67)(68,69)(70,71);(62,63);(60,61)

Conjugacy classes & character table

2408 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15104, 335744);

C22×#16837311 #28754865