C₄×#16811374 #27546803

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₄×#16811374

Splittings

Fitting	C₄×C₂²⋊Q₈×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₄×#563

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₄×C₂²⋊Q₈×C₅₃	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂²×C₂²⋊D₅₃	C₂³
hypercenter	C₄×C₂²⋊Q₈	D₅₃
hypocenter	C₅₃	C₄×#563
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊Q₈

2024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 generators on 81 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)(59,63)(62,65)(67,73)(68,76)(72,81)(75,77);(59,63)(60,64)(66,69,70,78)(67,72,73,81)(68,75,76,77)(71,79,74,80);(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)(59,62)(61,64)(63,65)(66,68,70,76)(67,71,73,74)(69,77,78,75)(72,80,81,79);(58,59)(60,62)(61,63)(64,65)(66,67,70,73)(68,74,76,71)(69,72,78,81)(75,80,77,79);(54,55,56,57)

Conjugacy classes & character table

1532 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 332279);

C4×#16811374 #27546803