C₂×C₈×#11719908 #27293064

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₆ · Schur multiplier (H₂) = C₂²×C₈

Aliases

direct product

C₂×C₈×#11719908

Splittings

derived, Sylow

C₅₃ ⋊ C₂×C₈×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₈²	D₅₃
commutator, hypocenter	C₅₃	C₂×C₈×C₁₆
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₄×C₈×C₅₃	C₂³
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂×C₈²

398 conjugacy classes of subgroups; only the 339 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 339 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) = 79
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 79 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)(64,65,66,69,67,70,72,75,68,71,73,76,74,77,78,79);(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)(56,57,58,60,59,61,62,63);(54,55)

Conjugacy classes & character table

3584 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78540);