C₂×C₄×#13133163 #27537365

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₄×#13133163

Splittings

Sylow

C₅₃ ⋊ C₂×C₄×Q₃₂

Subgroups

	subgroup	G/H
center	C₂²×C₄	#11719922
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₄×Q₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₅₃
hypocenter	C₅₃	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 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)(60,62,65,73)(61,66,69,83)(63,75,78,91)(64,80,76,72)(67,85,86,90)(68,87,79,82)(70,88,89,74)(71,84,81,77);(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)(60,61,65,69)(62,70,73,89)(63,74,78,88)(64,79,76,68)(66,81,83,71)(67,84,86,77)(72,90,80,85)(75,87,91,82);(56,57,58,59);(54,55)

Conjugacy classes & character table

1544 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322841);

C2×C4×#13133163 #27537365