C₄×#16810421 #27299275

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₄×#16810421

Splittings

Sylow

C₅₃ ⋊ C₄×#356

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133131
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×Q₈	C₂²×D₅₃
Fitting	C₄×#101×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂²×Q₈×C₅₃	C₂³
hypercenter	C₄×#101	D₅₃
hypocenter	C₅₃	C₄×#356
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ C₄×#101

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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)(58,60,63,62)(59,64,67,72)(61,70,71,69)(65,66,73,68);(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)(58,59,62,66,63,67,60,68)(61,65,69,72,71,73,70,64);(54,55,56,57)

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84751);