C₄×#16810356 #27274967

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₄×#16810356

Splittings

Fitting	C₄×#95×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₄×#327

Subgroups

	subgroup	G/H
center	C₂²×C₄	#11719921
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₄×#95×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂×#13133191	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₄×#95	D₅₃
hypocenter	C₅₃	C₄×#327
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×#95

816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 73 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)(58,60)(59,62)(61,65)(63,64)(67,70)(68,73)(71,72);(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)(58,59)(60,64,65,62)(61,63)(66,67)(68,70)(69,72)(71,73);(54,55,56,57)

Conjugacy classes & character table

2156 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 60443);

C4×#16810356 #27274967