C₂×C₄⋊Q₈×C₄₁ #14579999

order	2624
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₄⋊Q₈×C₄₁

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₁	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₄₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ G

290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 2624

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

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

Conjugacy classes & character table

1148 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₄₁×#332, C₂×C₄₁×#333, C₄₁×#1546, …
maximal normal subgroup of	C₂×C₄₁×#332, C₂×C₄₁×#333, C₄₁×#1546, …
maximal quotient of	C₂×C₄₁×#384, C₂×C₄₁×#389, C₂×C₄₁×#391, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 212);

C2×C4⋊Q8×C41 #14579999