Q₈×#13133132 #27400641

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

Q₈×#13133132

Splittings

Sylow

C₅₃ ⋊ Q₈×#99

Subgroups

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₂×D₄×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂²×D₄×C₅₃	C₂³
cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×D₄×Q₈	D₅₃
hypocenter	C₅₃	Q₈×#99
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ C₂×D₄×Q₈

1354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 3392

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

Conjugacy classes & character table

1355 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 186117);

Q8×#13133132 #27400641