C₂×#20274327 #27371768

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#20274327

Splittings

Fitting	C₂×C₈²×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1711

Subgroups

	subgroup	G/H
center	C₂³	#13133214
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle	C₂³×C₅₃	#127
prime radical	C₂×#13133214	C₂²
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	C₂×#1711
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

802 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 235 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) = 87
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 87 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)(57,62)(61,69)(63,68)(64,70)(72,74,76,82,75,81,83,87)(73,78,79,80,77,85,84,86);(56,58,65,71,60,67,59,66)(57,61,68,70,62,69,63,64)(73,77)(78,85)(79,84)(80,86);(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)(56,57)(58,64)(59,68)(60,62)(61,66)(63,65)(67,70)(69,71)(72,73)(74,80)(75,77)(76,84)(78,82)(79,83)(81,86)(85,87);(54,55)

Conjugacy classes & character table

3404 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 157244);