C₄⋊C₄×C₂²⋊Dic₅₃ #27305509

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

C₄⋊C₄×C₂²⋊Dic₅₃

Splittings

Sylow

C₅₃ ⋊ C₂²⋊C₄×C₄⋊C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle, cosocle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁵×C₅₃	C₂³
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂²⋊C₄×C₄⋊C₄
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄⋊C₄

2574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 69 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)(67,69);(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)(58,59,60,61)(66,67)(68,69);(62,63,64,65);(54,55,56,57)(63,65)

Conjugacy classes & character table

2180 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90985);

C4⋊C4×C22⋊Dic53 #27305509