D₄×C₂²⋊C₄×D₅₃ #27543259

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

D₄×C₂²⋊C₄×D₅₃

Splittings

Fitting	D₄×C₂²⋊C₄×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×D₄×C₂²⋊C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Fitting	D₄×C₂²⋊C₄×C₅₃	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂³×D₄×D₅₃	C₂
hypercenter	D₄×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	C₂×D₄×C₂²⋊C₄
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ D₄×C₂²⋊C₄

8774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 5 generators on 65 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);(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)(63,65);(54,55,56,57)(62,63)(64,65);(59,61);(58,59)(60,61)

Conjugacy classes & character table

1400 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 328735);

D4×C22⋊C4×D53 #27543259