C₂×#20272937 #27305181

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂×#20272937

Splittings

Sylow

C₅₃ ⋊ C₂×C₂²⋊#95

Subgroups

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₂³×C₂²⋊C₄×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁶×C₅₃	C₂²
cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂³×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	C₂×C₂²⋊#95
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

7630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 71 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)(69,71);(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)(60,61,62,63)(65,67);(56,57,58,59)(64,65)(66,67)(68,69)(70,71);(54,55)

Conjugacy classes & character table

2192 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90657);

C2×#20272937 #27305181