C₂×C₄⋊Q₈×C₄₇ #15688786

order	3008
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₄⋊Q₈×C₄₇

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₄₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ G

290 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 4 generators on 61 points:

(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61);(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

1316 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₄₇×#332, C₂×C₄₇×#333, C₄₇×#1546, …
maximal normal subgroup of	C₂×C₄₇×#332, C₂×C₄₇×#333, C₄₇×#1546, …
maximal quotient of	C₂×C₄₇×#384, C₂×C₄₇×#389, C₂×C₄₇×#391, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3008, 212);

C2×C4⋊Q8×C47 #15688786