C₂²×C₈×#88322 #27349658

order	13568
isoclinism	isoclinic class of D₅₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈² · Schur multiplier (H₂) = C₂⁵×C₈

Aliases

direct product

C₂²×C₈×#88322

Splittings

derived, Sylow

C₅₃ ⋊ C₂²×C₈²

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₄×C₈	D₅₃
commutator, hypocenter	C₅₃	C₂²×C₈²
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄×C₈

1342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 4 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)(66,67,68,70,69,71,72,73);(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)(58,59,60,62,61,63,64,65);(56,57);(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, 135134);

C22×C8×#88322 #27349658