C₄×#16811066 #27537425

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

Aliases

direct product

C₄×#16811066

Splittings

Fitting	C₄×M₃₂×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₄×#510

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#13133239	C₂²
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	C₄×#510
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

674 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)(59,65)(64,79)(66,81)(67,77)(68,85)(72,84)(80,83)(82,88);(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)(58,60,63,71)(59,64,65,79)(61,73,76,89)(62,78,74,70)(66,82,81,88)(67,83,77,80)(68,84,85,72)(69,86,87,75);(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)(58,59)(60,68)(61,72)(62,77)(63,65)(64,69)(66,75)(67,74)(70,88)(71,85)(73,83)(76,84)(78,82)(79,87)(80,89)(81,86);(54,55,56,57)

Conjugacy classes & character table

2144 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322901);

C4×#16811066 #27537425