C₂×#135×C₃₇ #14557237

order	2368
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#135×C₃₇

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃₇
socle	C₂²×C₃₇	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 4 generators on 55 points:

(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55);(3,4)(5,12)(6,10)(7,15)(8,11)(9,13)(14,17)(16,18);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)

Conjugacy classes & character table

1036 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#1876, C₃₇×#1974, C₃₇×#1975, …
maximal normal subgroup of	C₃₇×#1876, C₃₇×#1974, C₃₇×#1975, …
maximal quotient of	C₂×C₃₇×#416, C₂×C₃₇×#427, C₂×C₄×D₈×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 253);

C2×#135×C37 #14557237