Abstract group 16384.lp: $C_2^4.C_2^2\wr C_4$

• Label: 16384.lp
• Order: \( 2^{14} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: 4
• $\card{\Aut(G)}$: \( 2^{24} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $4$

Group information

Description:	$C_2^4.C_2^2\wr C_4$
Order:	\(16384\)\(\medspace = 2^{14} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	Group of order \(16777216\)\(\medspace = 2^{24} \)
Composition factors:	$C_2$ x 14
Nilpotency class:	$6$
Derived length:	$3$

This group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).

Group statistics

Order	1	2	4	8
Elements	1	1535	8704	6144	16384
Conjugacy classes	1	131	136	24	292
Divisions	1	131	124	8	264
Autjugacy classes	1	47	25	2	75

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid b^{4}=d^{4}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $64$ $\langle(1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^8$ . $(D_4:D_4)$	$(C_2^9.D_4)$ . $C_4$	$C_2^8$ . $(C_2\wr C_4)$ (9)	$(C_2^7.C_2^4)$ . $D_4$ (24)	all 133
Aut. group:	$\Aut(C_4^3.C_2)$

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{11}$
Commutator length:	$1$

Subgroups

There are 809 normal subgroups (191 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_2^2$
Commutator:	a subgroup isomorphic to 512.7532419
Frattini:	not computed
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.C_2^2\wr C_4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 8 (order at least 2048) are shown, as well as all normal subgroups of any index.

Order 16384: $C_2^4.C_2^2\wr C_4$

Order 8192: $C_2^8.C_2^4.C_2$ x 10, $C_2^7.C_2^5.C_2$ x 4, $C_2^7.C_2^6$

Order 4096: $C_2^7.C_2^4.C_2$ x 30, $C_2^8.C_2^4$ x 21, $C_2^7.C_2^5$ x 16, $C_2^8.C_2^3.C_2$ x 12, $C_2^6.C_2^5.C_2$ x 10, $C_2^9.D_4$ x 2

Order 2048: $C_2^7.C_2^4$ x 376, $C_2^6.C_2^4.C_2$ x 92, $C_2^8.C_2^3$ x 79, $C_2^5.D_4^2$ x 72, $C_2^6.C_2^5$ x 44, $C_2^7.C_2^3.C_2$ x 32, $C_2^5.C_2^5.C_2$ x 13, $C_2^9.C_2^2$ x 6, $C_2^5.D_4^2$ x 6, $C_2\times C_2^8.C_2^2$ x 5, $C_2\times C_2^7.C_2^3$ x 4, $C_2\times C_2^6.C_2^4$ x 3, $C_2\times C_2^5.C_2^4.C_2$ x 2, $C_2^8.D_4$ x 2, $C_2^9.C_4$ x 2, $C_2^3\times C_2^5.C_2^3$

Order 1024: $C_2^7.C_2^3$ x 52, $C_2^8.C_2^2$ x 33, $C_2^3\times C_2^5.C_2^2$ x 5, $C_2\times C_2^7.C_2^2$ x 4, $C_2\times C_2^8.C_2$ x 2, $C_2^9.C_2$, $C_2^3\times C_2^6.C_2$, $C_2^{10}$

Order 512: $C_2^8.C_2$ x 10, $C_2\times C_2^6.C_2^2$ x 8, $C_2^7.C_2^2$ x 8, $C_2^7.C_2^2$ x 8, $C_2^4.C_2^5$ x 8, $C_2\times C_2^6.C_2^2$ x 8, $C_2^6:D_4$ x 8, $C_2^9$ x 7, $C_2^7:C_4$ x 6, $C_2\times C_2^6.C_2^2$ x 4, $C_2\times C_2^6.C_2^2$ x 4, $D_4\times C_2^6$ x 2, $C_2^8.C_2$ x 2

Order 256: $C_2^8$ x 21, $C_2^5:Q_8$ x 16, $C_2^5:D_4$ x 16, $C_2^6:C_4$ x 16, $C_2^6:C_4$ x 12, $D_4\times C_2^5$ x 8, $C_2^5.Q_8$ x 4, $C_2^5:Q_8$ x 4, $C_2^5.D_4$ x 4, $C_2^5:D_4$ x 4, $C_2^6\times C_4$ x 2

Order 128: $C_2^7$ x 39, $C_2^5:C_4$ x 24, $C_2^5:C_4$ x 8, $C_2^4.Q_8$ x 8, $C_2^4:Q_8$ x 8, $C_2^4.D_4$ x 8, $C_2^4:D_4$ x 8, $C_2^5\times C_4$ x 4

Order 64: $C_2^6$ x 67, $C_2^4:C_4$ x 16

Order 32: $C_2^5$ x 99

Order 16: $C_2^4$ x 67

Order 8: $C_2^3$ x 27

Order 4: $C_2^2$ x 7

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 16384: $C_2^4.C_2^2\wr C_4$ ($C_1$)

Order 8192: $C_2^8.C_2^4.C_2$ ($C_2$) x 10, $C_2^7.C_2^5.C_2$ ($C_2$) x 4, $C_2^7.C_2^6$ ($C_2$)

Order 4096: $C_2^7.C_2^4.C_2$ ($C_2^2$) x 20, $C_2^8.C_2^3.C_2$ ($C_2^2$) x 8, $C_2^8.C_2^4$ ($C_2^2$) x 7, $C_2^8.C_2^4$ ($C_4$) x 7, $C_2^9.D_4$ ($C_4$)

Order 2048: $C_2^8.C_2^3$ ($C_2\times C_4$) x 25, $C_2^7.C_2^4$ ($D_4$) x 16, $C_2^7.C_2^3.C_2$ ($C_2^3$) x 8, $C_2^8.C_2^3$ ($C_2^3$) x 7, $C_2^8.C_2^3$ ($D_4$) x 7, $C_2\times C_2^8.C_2^2$ ($C_2\times C_4$) x 3, $C_2^3\times C_2^5.C_2^3$ ($D_4$)

Order 1024: $C_2^7.C_2^3$ ($C_2^2:C_4$) x 28, $C_2^7.C_2^3$ ($C_2\times D_4$) x 24, $C_2^8.C_2^2$ ($C_2^2\times C_4$) x 13, $C_2^8.C_2^2$ ($C_2^2:C_4$) x 10, $C_2^8.C_2^2$ ($C_2\times D_4$) x 9, $C_2\times C_2^7.C_2^2$ ($C_2^2:C_4$) x 4, $C_2^3\times C_2^5.C_2^2$ ($C_2\times D_4$) x 3, $C_2^3\times C_2^5.C_2^2$ ($C_2^2:C_4$) x 2, $C_2^9.C_2$ ($C_2^2:C_4$), $C_2^8.C_2^2$ ($C_2^4$), $C_2^3\times C_2^6.C_2$ ($C_2^2:C_4$), $C_2\times C_2^8.C_2$ ($C_2^2:C_4$), $C_2\times C_2^8.C_2$ ($C_2^2\times C_4$), $C_2^{10}$ ($C_2^2:C_4$)

Order 512: $C_2\times C_2^6.C_2^2$ ($C_2^3:C_4$) x 8, $C_2^8.C_2$ ($C_2^3:C_4$) x 7, $C_2^7.C_2^2$ ($C_2^3:C_4$) x 4, $C_2^7.C_2^2$ ($C_2^3:C_4$) x 4, $C_2^7.C_2^2$ ($C_2^3:C_4$) x 4, $C_2^7.C_2^2$ ($C_2^3:C_4$) x 4, $C_2\times C_2^6.C_2^2$ ($C_2^3:C_4$) x 4, $C_2\times C_2^6.C_2^2$ ($C_2^3:C_4$) x 4, $C_2^4.C_2^5$ ($C_2^2\wr C_2$) x 4, $C_2\times C_2^6.C_2^2$ ($C_2^2\wr C_2$) x 4, $C_2^9$ ($C_2^3:C_4$) x 4, $C_2^7:C_4$ ($C_2^3:C_4$) x 3, $C_2^9$ ($C_2^3:C_4$) x 3, $C_2^6:D_4$ ($C_2^3:C_4$) x 3, $C_2^8.C_2$ ($C_2^2\wr C_2$) x 2, $C_2^4.C_2^5$ ($C_2^2\times D_4$) x 2, $C_2^4.C_2^5$ ($C_2^3:C_4$) x 2, $C_2^7:C_4$ ($C_2^2\wr C_2$) x 2, $C_2\times C_2^6.C_2^2$ ($C_2^2\times D_4$) x 2, $C_2\times C_2^6.C_2^2$ ($C_2^3:C_4$) x 2, $C_2^6:D_4$ ($C_2^3:C_4$) x 2, $C_2^6:D_4$ ($C_2^2\wr C_2$) x 2, $C_2^8.C_2$ ($C_2^3\times C_4$), $C_2^7:C_4$ ($C_2^2\times D_4$), $D_4\times C_2^6$ ($C_2^3:C_4$), $D_4\times C_2^6$ ($C_2^2\wr C_2$), $C_2^6:D_4$ ($C_2^2\times D_4$), $C_2^8.C_2$ ($C_2^3:C_4$), $C_2^8.C_2$ ($C_2^2\wr C_2$)

Order 256: $C_2^8$ ($C_2\wr C_4$) x 9, $C_2^8$ ($C_2^4:C_4$) x 8, $C_2^5:Q_8$ ($C_2^4:C_4$) x 8, $C_2^5:Q_8$ ($C_2\wr C_4$) x 8, $C_2^5:D_4$ ($C_2^4:C_4$) x 8, $C_2^5:D_4$ ($C_2\wr C_4$) x 8, $C_2^6:C_4$ ($C_2^4:C_4$) x 8, $C_2^6:C_4$ ($C_2\wr C_4$) x 4, $C_2^5.Q_8$ ($C_2\wr C_4$) x 4, $C_2^5:Q_8$ ($C_2\wr C_4$) x 4, $C_2^5.D_4$ ($C_2\wr C_4$) x 4, $C_2^5:D_4$ ($C_2\wr C_4$) x 4, $C_2^6:C_4$ ($C_2^4:C_4$) x 4, $C_2^8$ ($C_2^4:C_4$) x 2, $D_4\times C_2^5$ ($C_2^4:C_4$) x 2, $D_4\times C_2^5$ ($C_2\wr C_4$) x 2, $C_2^6:C_4$ ($C_2^4:C_4$) x 2, $C_2^6:C_4$ ($C_4^2:C_2^2$) x 2, $C_2^6:C_4$ ($D_4:D_4$) x 2, $C_2^6:C_4$ ($C_2^3:D_4$) x 2, $C_2^6:C_4$ ($C_2^4:C_4$) x 2, $C_2^8$ ($C_2^4:C_4$), $C_2^8$ ($D_4:D_4$), $D_4\times C_2^5$ ($C_2^4:C_4$), $D_4\times C_2^5$ ($C_2^3:D_4$), $D_4\times C_2^5$ ($C_4^2:C_2^2$), $D_4\times C_2^5$ ($D_4:D_4$), $C_2^6\times C_4$ ($C_2\wr C_4$), $C_2^6\times C_4$ ($C_4^2:C_2^2$), $C_2^6:C_4$ ($C_2^4:C_4$), $C_2^6:C_4$ ($C_2^3:D_4$)

Order 128: $C_2^7$ ($C_2^5:C_4$) x 18, $C_2^5:C_4$ ($C_2^5:C_4$) x 16, $C_2^7$ ($C_2^5:C_4$) x 8, $C_2^4.Q_8$ ($C_2^5:C_4$) x 8, $C_2^4:Q_8$ ($C_2^5:C_4$) x 8, $C_2^4.D_4$ ($C_2^5:C_4$) x 8, $C_2^4:D_4$ ($C_2^5:C_4$) x 8, $C_2^7$ ($C_2^4.D_4$) x 5, $C_2^7$ ($(C_2^2\times C_8):C_2^2$) x 4, $C_2^5:C_4$ ($C_2^5:C_4$) x 4, $C_2^5:C_4$ ($C_2^5:C_4$) x 4, $C_2^5:C_4$ ($C_2^5:C_4$) x 4, $C_2^7$ ($C_2^5:C_4$) x 2, $C_2^5\times C_4$ ($C_2^5:C_4$) x 2, $C_2^5:C_4$ ($C_2^4.D_4$) x 2, $C_2^7$ ($C_4^2:C_2^3$), $C_2^7$ ($C_2^5:C_4$), $C_2^5\times C_4$ ($C_2^4.D_4$), $C_2^5\times C_4$ ($C_4^2:C_2^3$), $C_2^5:C_4$ ($C_4^2:C_2^3$), $C_2^5:C_4$ ($C_4^2:C_2^3$)

Order 64: $C_2^6$ ($C_2^6:C_4$) x 16, $C_2^6$ ($C_2^4.(C_2\times D_4)$) x 16, $C_2^4:C_4$ ($C_2^6:C_4$) x 8, $C_2^4:C_4$ ($C_2^6:C_4$) x 8, $C_2^6$ ($(C_2^2\times D_4).D_4$) x 4, $C_2^6$ ($(C_2^2\times Q_8).D_4$) x 4, $C_2^6$ ($C_2^5.D_4$) x 4, $C_2^6$ ($C_2^6:C_4$) x 4, $C_2^6$ ($C_2^6:C_4$) x 4, $C_2^6$ ($C_2^4.(C_2\times D_4)$) x 4, $C_2^6$ ($C_2^5.D_4$) x 4, $C_2^6$ ($C_2^6:C_4$) x 3, $C_2^6$ ($(C_2^2\times C_8):C_2^3$) x 2, $C_2^6$ ($C_2^5.D_4$) x 2

Order 32: $C_2^5$ ($C_2^7:C_4$) x 12, $C_2^5$ ($C_2^5.C_2^3.C_2$) x 8, $C_2^5$ ($C_2^4.C_2^4.C_2$) x 8, $C_2^5$ ($C_2^3.C_2^5.C_2$) x 8, $C_2^5$ ($C_2^4.C_2^4.C_2$) x 8, $C_2^5$ ($C_2^3.C_2^5.C_2$) x 8, $C_2^5$ ($C_2^4.C_2^4.C_2$) x 8, $C_2^5$ ($C_2^3.C_2^5.C_2$) x 8, $C_2^5$ ($C_2^5.(C_2\times D_4)$) x 8, $C_2^5$ ($C_2^7:C_4$) x 6, $C_2^5$ ($C_2\times C_2^4.C_2^3.C_2$) x 4, $C_2^5$ ($C_2^5.(C_2\times D_4)$) x 4, $C_2^5$ ($C_2\times C_2^3.C_2^4.C_2$) x 2, $C_2^5$ ($C_2\times C_2^3.C_2^4.C_2$) x 2, $C_2^5$ ($C_2\times C_2^4.C_2^3.C_2$) x 2, $C_2^5$ ($C_2^5.C_2^4$), $C_2^5$ ($C_2\times C_2^2.C_2^5.C_2$), $C_2^5$ ($C_2\times C_2^4.C_2^3.C_2$)

Order 16: $C_2^4$ ($C_2^5.C_2^4.C_2$) x 42, $C_2^4$ ($C_2\times C_2^4.C_2^4.C_2$) x 16, $C_2^4$ ($C_2^2\wr C_4$) x 4, $C_2^4$ ($C_2\times C_2^5.C_2^3.C_2$) x 3, $C_2^4$ ($C_2^7.D_4$) x 2

Order 8: $C_2^3$ ($C_2^6.C_2^4.C_2$) x 16, $C_2^3$ ($C_2\times C_2^5.C_2^4.C_2$) x 6, $C_2^3$ ($C_2^5.C_2^5.C_2$) x 4, $C_2^3$ ($C_2^9.C_4$)

Order 4: $C_2^2$ ($C_2^7.C_2^4.C_2$) x 4, $C_2^2$ ($C_2^6.C_2^5.C_2$) x 2, $C_2^2$ ($C_2\times C_2^6.C_2^4.C_2$)

Order 2: $C_2$ ($C_2^8.C_2^4.C_2$) x 2, $C_2$ ($C_2^7.C_2^5.C_2$)

Order 1: $C_1$ ($C_2^4.C_2^2\wr C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $292 \times 292$ character table is not available for this group.

Rational character table

The $264 \times 264$ rational character table is not available for this group.