C₂²×C₄×#11719908 #27536715

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₆ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂²×C₄×#11719908

Splittings

derived, Sylow

C₅₃ ⋊ C₂²×C₄×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₄×C₈	D₅₃
commutator, hypocenter	C₅₃	C₂²×C₄×C₁₆
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄×C₈

1022 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 4 generators on 77 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)(62,63,64,67,65,68,70,73,66,69,71,74,72,75,76,77);(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)(58,59,60,61);(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, 322191);

C22×C4×#11719908 #27536715