D₄×Q₈×C₄₇ #15688804

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

Aliases

direct product

D₄×Q₈×C₄₇

Subgroups

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₄₇
prime radical	C₂×D₄×C₄₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ G

280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 1504

A minimal faithful permutation representation — 4 generators on 59 points:

(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59);(2,4);(5,6,7,8)(9,12,11,10);(5,9,7,11)(6,10,8,12)

Conjugacy classes & character table

1175 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄₇×#1627, C₄₇×#1637, C₄₇×#1649, …
maximal normal subgroup of	C₄₇×#1627, C₄₇×#1637, C₄₇×#1649, …
maximal quotient of	Q₈×C₂²⋊C₄×C₄₇, D₄×C₄⋊C₄×C₄₇, Q₈×C₄⋊C₄×C₄₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3008, 230);

D4×Q8×C47 #15688804