C₄×#11715754 #72310700

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₄×#11715754

Splittings

Fitting	C₄×#3549 ⋊ D₃
derived	#5464 ⋊ C₂²×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₄	#11715754
commutator, hypocenter	#5464	C₂²×C₄
Frattini, socle	C₂³	C₂²×#1020
Fitting	C₄×#3549	D₃
prime radical	C₂×#11715754	C₂
cosocle	C₂×#5464	C₂³
LCS	G ▷ #5464
UCS	C₁ ◁ C₄

5504 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 5504 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 28 points:

(5,9,8,12)(6,17,10,19)(7,28,14,20,16,24,22,27)(11,23,18,25,15,26,21,13);(1,2,3,4)(5,7,13)(6,11,20)(8,15,27)(9,18,23)(10,16,25)(12,22,28)(14,24,17)(19,21,26);(5,6)(7,11)(8,10)(9,17)(12,19)(13,20)(14,18)(15,16)(21,22)(23,24)(25,27)(26,28)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#72218857, #72341172, C₂×C₄×#11715754
maximal normal subgroup of	C₄×#72218857, #72341172, C₂×C₄×#11715754
maximal quotient of	C₂×C₄×#11715754

GAP

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

gap> G := Group((5,9,8,12)(6,17,10,19)(7,28,14,20,16,24,22,27)(11,23,18,25,15,26,21,13), (1,2,3,4)(5,7,13)(6,11,20)(8,15,27)(9,18,23)(10,16,25)(12,22,28)(14,24,17)(19,21,26), (5,6)(7,11)(8,10)(9,17)(12,19)(13,20)(14,18)(15,16)(21,22)(23,24)(25,27)(26,28));