C₂×#11924213 #72228763

order	2352
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#11924213

Splittings

Hall	C₂×#87648 ⋊ C₃
Hall	#3861 ⋊ C₂×Q₈
Sylow	C₇² ⋊ C₂×C₃×Q₈

Subgroups

	subgroup	G/H
center	C₂²	C₂×#64424
commutator, cosocle	C₂×C₇²	C₂³×C₃
Frattini	C₂	C₂²×#64424
Fitting	C₂×C₄×C₇²	C₂×C₃
socle	C₂²×C₇²	C₂²×C₃
prime radical	C₂²×#3861	C₂²
hypercenter	C₂×C₄	#64424
hypocenter	C₇²	C₂×C₃×Q₈
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 784

A minimal faithful permutation representation — 3 generators on 24 points:

(3,5,6,10)(4,7,8,9)(12,14,16)(13,15,17)(18,19,24)(21,22,23);(1,2)(11,12,14,17,16,15,13);(3,4,6,8)(5,9,10,7)(12,13)(14,15)(16,17)(19,20)(21,22)(23,24)

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×Q₈×#64424, C₂×#72228598, C₂×#72228708, …
maximal normal subgroup of	C₂×Q₈×#64424, C₂×#72228598, C₂×#72228708, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,5,6,10)(4,7,8,9)(12,14,16)(13,15,17)(18,19,24)(21,22,23), (1,2)(11,12,14,17,16,15,13), (3,4,6,8)(5,9,10,7)(12,13)(14,15)(16,17)(19,20)(21,22)(23,24));

C2×#11924213 #72228763