D₅₃×#2222 #27538651

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₅₃×#2222

Splittings

Fitting	C₅₃×#2222 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2222

Subgroups

	subgroup	G/H
center	C₂	#466×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2222	C₂
socle	C₂×C₅₃	C₂×#466
prime radical	#41×D₅₃	D₄
hypercenter	#2222	D₅₃
hypocenter	C₅₃	C₂×#2222
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #2222

638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 117 points:

(1,4)(2,10)(3,17)(5,23)(6,29)(7,24)(8,25)(9,31)(11,35)(12,39)(13,22)(14,36)(15,40)(16,45)(18,47)(19,51)(20,48)(21,49)(26,53)(27,56)(28,55)(30,57)(32,60)(33,52)(34,59)(37,61)(38,54)(41,63)(42,46)(43,62)(44,64)(50,58)(65,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,110,112,114,116,117,115,113,111,109,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67);(1,3,16,44,7,20,28,30,8,21,27,50,6,19,5,18)(2,9,26,42,13,33,38,43,14,34,37,41,12,32,11,15)(4,17,45,64,24,48,55,57,25,49,56,58,29,51,23,47)(10,31,53,46,22,52,54,62,36,59,61,63,39,60,35,40);(1,2,8,14)(3,15,21,43)(4,22,25,39)(5,26,28,37)(6,13,7,12)(9,30,34,18)(10,29,36,24)(11,27,38,16)(17,46,49,63)(19,42,20,41)(23,54,55,35)(31,58,59,64)(32,50,33,44)(40,51,62,48)(45,53,56,61)(47,52,57,60);(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)(112,113)(114,115)(116,117)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324127);

D53×#2222 #27538651