C₄×#16811055 #27537376

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₄×#16811055

Splittings

Fitting	C₂×C₄×C₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₄×#508

Subgroups

	subgroup	G/H
center	C₄²	D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#16811055	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	C₄×#508
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 89 points:

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

Conjugacy classes & character table

3416 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322852);

C4×#16811055 #27537376