Abstract group 8192.zk: $C_2^{10}.D_4$

• Label: 8192.zk
• Order: \( 2^{13} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: 8
• $\card{\Aut(G)}$: \( 2^{31} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{21} \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $6$

Group information

Description:	$C_2^{10}.D_4$
Order:	\(8192\)\(\medspace = 2^{13} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	Group of order \(2147483648\)\(\medspace = 2^{31} \)
Composition factors:	$C_2$ x 13
Nilpotency class:	$4$
Derived length:	$3$

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

Group statistics

Order	1	2	4	8
Elements	1	1535	5632	1024	8192
Conjugacy classes	1	227	216	8	452
Divisions	1	227	216	8	452
Autjugacy classes	1	49	25	1	76

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
Rank:	$6$
Inequivalent generating 6-tuples:	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:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid a^{2}=b^{4}=c^{4}=d^{2}=e^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $64$ $\langle(3,6)(5,8)(11,14)(13,16)(19,22)(21,24)(27,30)(29,32)(35,38)(37,40)(43,46) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$C_2^{10}$ . $D_4$	$C_2$ . $(C_2^9.D_4)$ (2)	$C_2^4$ . $(D_4^2:D_4)$ (16)	$(C_2^5.C_2^5)$ . $D_4$ (239)	all 32
Aut. group:	$\Aut(C_2^3.\OD_{16})$	$\Aut(C_4^2:C_8)$	$\Aut(C_{12}.(C_2^3\times C_4))$	$\Aut((C_2\times C_4).S_3^3)$

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

Homology

Abelianization:	$C_{2}^{6} $
Schur multiplier:	$C_{2}^{20}$
Commutator length:	$1$

Subgroups

There are 16654 normal subgroups (47 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 $C_2^3$
Commutator:	a subgroup isomorphic to $D_4\times C_2^4$
Frattini:	a subgroup isomorphic to $D_4\times C_2^4$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^{10}.D_4$

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 2 (order at least 4096) are shown, as well as normal subgroups of index up to 8 or of order up to 16.

Order 8192: $C_2^{10}.D_4$

Order 4096: $C_2^6.C_2.C_2^5$ x 46, $C_2.C_2^6.C_2^5$ x 13, $C_2^6.C_2^6$ x 3, $C_2^6.C_2^5.C_2$

Order 2048: $C_2.C_2^6.C_2^4$ x 244, $C_2^6.C_2.C_2^4$ x 224, $C_2^6.C_2^5$ x 66, $C_2^5.D_4^2$ x 64, $C_2^6.C_2^4.C_2$ x 16, $C_2^5.D_4^2$ x 16, $C_2^5.C_2^5.C_2$ x 9, $C_2^5.C_2^6$ x 6, $C_2^9.C_4$ x 4, $C_2^2\times C_2^5.C_2^4$, $C_2\times C_2^5.C_2^5$

Order 1024: $C_2.C_2^6.C_2^3$ x 720, $C_2^6.C_2^4$ x 310, $C_2^5.C_2^5$ x 214, $C_2^7:D_4$ x 128, $C_2\times C_2^5.C_2^4$ x 83, $C_2^5.C_2^4.C_2$ x 72, $C_2^8:C_4$ x 32, $C_2^4.C_2^5.C_2$ x 22, $C_2^7:D_4$ x 16, $C_2^2\times C_2^5.C_2^3$ x 9, $C_2^7.C_2^3$ x 4, $C_2^4.C_2^6$ x 4, $C_2\times C_2^7.C_2^2$ x 4, $C_2^3\times C_2^4.C_2^3$ x 3, $C_2^2\times C_2^4.C_2^4$ x 3, $C_2\times C_2^6.C_2^3$ x 3, $C_2^2\times C_2^6.C_2^2$ x 2, $C_2\times C_2^4.C_2^5$ x 2, $C_2^8:C_4$ x 2, $C_2^2\times C_2^7.C_2$, $C_2^{10}$

Order 16: $C_2^4$ x 211, $C_2^2\times C_4$ x 8

Order 8: $C_2^3$ x 91

Order 4: $C_2^2$ x 27

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

All normal subgroups of index up to 8 or order up to 16 are shown.

Order 8192: $C_2^{10}.D_4$ ($C_1$)

Order 4096: $C_2^6.C_2.C_2^5$ ($C_2$) x 46, $C_2.C_2^6.C_2^5$ ($C_2$) x 13, $C_2^6.C_2^6$ ($C_2$) x 3, $C_2^6.C_2^5.C_2$ ($C_2$)

Order 2048: $C_2.C_2^6.C_2^4$ ($C_2^2$) x 244, $C_2^6.C_2.C_2^4$ ($C_2^2$) x 224, $C_2^6.C_2^5$ ($C_2^2$) x 66, $C_2^5.D_4^2$ ($C_2^2$) x 64, $C_2^6.C_2^4.C_2$ ($C_2^2$) x 16, $C_2^5.D_4^2$ ($C_2^2$) x 16, $C_2^5.C_2^5.C_2$ ($C_2^2$) x 9, $C_2^5.C_2^6$ ($C_2^2$) x 6, $C_2^9.C_4$ ($C_2^2$) x 4, $C_2^2\times C_2^5.C_2^4$ ($C_2^2$), $C_2\times C_2^5.C_2^5$ ($C_2^2$)

Order 1024: $C_2.C_2^6.C_2^3$ ($C_2^3$) x 720, $C_2^6.C_2^4$ ($C_2^3$) x 306, $C_2^5.C_2^5$ ($D_4$) x 150, $C_2^7:D_4$ ($C_2^3$) x 128, $C_2^5.C_2^4.C_2$ ($C_2^3$) x 72, $C_2^5.C_2^5$ ($C_2^3$) x 64, $C_2\times C_2^5.C_2^4$ ($D_4$) x 61, $C_2^8:C_4$ ($C_2^3$) x 32, $C_2^4.C_2^5.C_2$ ($C_2^3$) x 22, $C_2\times C_2^5.C_2^4$ ($C_2^3$) x 22, $C_2^7:D_4$ ($C_2^3$) x 16, $C_2^2\times C_2^5.C_2^3$ ($D_4$) x 5, $C_2^7.C_2^3$ ($D_4$) x 4, $C_2^6.C_2^4$ ($D_4$) x 4, $C_2^4.C_2^6$ ($C_2^3$) x 4, $C_2^2\times C_2^5.C_2^3$ ($C_2^3$) x 4, $C_2\times C_2^7.C_2^2$ ($D_4$) x 4, $C_2^3\times C_2^4.C_2^3$ ($C_2^3$) x 3, $C_2^2\times C_2^4.C_2^4$ ($D_4$) x 3, $C_2\times C_2^6.C_2^3$ ($D_4$) x 3, $C_2^2\times C_2^6.C_2^2$ ($D_4$) x 2, $C_2\times C_2^4.C_2^5$ ($D_4$) x 2, $C_2^8:C_4$ ($C_2^3$) x 2, $C_2^2\times C_2^7.C_2$ ($D_4$), $C_2^{10}$ ($D_4$)

Order 512: ? (unidentified group of order 16) x 2451

Order 256: ? (unidentified group of order 32) x 4403

Order 128: ? (unidentified group of order 64) x 4395

Order 64: ? (unidentified group of order 128) x 2075

Order 32: ? (unidentified group of order 256) x 635

Order 16: $C_2^4$ ($C_2^6:D_4$) x 108, $C_2^4$ ($C_2^4.C_2^5$) x 36, $C_2^4$ ($D_4^2:D_4$) x 16, $C_2^4$ ($C_2^6:D_4$) x 16, $C_2^4$ ($C_2^4\wr C_2$) x 12, $C_2^4$ ($C_2^8.C_2$) x 11, $C_2^4$ ($C_2^3.C_2^6$) x 9, $C_2^2\times C_4$ ($C_2^8.C_2$) x 4, $C_2^2\times C_4$ ($C_2^4\wr C_2$) x 4, $C_2^4$ ($C_2^6:D_4$) x 3

Order 8: $C_2^3$ ($C_2^6.C_2^4$) x 40, $C_2^3$ ($C_2^7:D_4$) x 24, $C_2^3$ ($C_2^7:D_4$) x 12, $C_2^3$ ($C_2^7.C_2^3$) x 8, $C_2^3$ ($C_2\times C_2^6.C_2^3$) x 5, $C_2^3$ ($C_2^2\times C_2^5.C_2^3$), $C_2^3$ ($C_2^9.C_2$)

Order 4: $C_2^2$ ($C_2^7.C_2^4$) x 13, $C_2^2$ ($C_2^8.D_4$) x 8, $C_2^2$ ($C_2^8.D_4$) x 4, $C_2^2$ ($C_2\times C_2^7.C_2^3$), $C_2^2$ ($C_2^8.D_4$)

Order 2: $C_2$ ($C_2^6.C_2.C_2^5$) x 4, $C_2$ ($C_2^9.D_4$) x 2, $C_2$ ($C_2^8.C_2^4$)

Order 1: $C_1$ ($C_2^{10}.D_4$)

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 5 larger groups in the database.

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

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

The $452 \times 452$ rational character table is not available for this group.