C₂×C₄×#13133123 #27304262

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₃₂ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×C₄×#13133123

Splittings

derived, Sylow

C₅₃ ⋊ C₂×C₄×C₃₂

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄×C₁₆	D₅₃
commutator, hypocenter	C₅₃	C₂×C₄×C₃₂
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₁₆

270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 91 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)(60,61,62,66,63,67,70,76,64,68,71,77,73,79,82,86,65,69,72,78,74,80,83,87,75,81,84,88,85,89,90,91);(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)(56,57,58,59);(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, 89738);

C2×C4×#13133123 #27304262