C₂²×#16810435 #27537042

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#16810435

Splittings

Fitting	C₂²×#108×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂²×#361

Subgroups

	subgroup	G/H
center	C₂³	#13133137
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂³×C₅₃	#102
prime radical	C₂³×D₄₂₄	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	C₂²×#361
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

1390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 73 points:

(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)(59,63)(60,69)(61,72)(64,65)(66,73)(67,68)(70,71);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(58,59,61,65,62,67,72,66)(60,68,70,64,71,63,69,73);(56,57);(54,55)

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322518);

C22×#16810435 #27537042