Abstract group 4096.bqv: $C_2^6.D_4^2$

• Label: 4096.bqv
• Order: \( 2^{12} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: 4
• $\card{\Aut(G)}$: \( 2^{22} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \)
• Perm deg.: $24$
• Trans deg.: not computed
• Rank: $4$

Group information

Description:	$C_2^6.D_4^2$
Order:	\(4096\)\(\medspace = 2^{12} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^{12}.C_2^6.C_2^4$, of order \(4194304\)\(\medspace = 2^{22} \)
Composition factors:	$C_2$ x 12
Nilpotency class:	$4$
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	703	2880	512	4096
Conjugacy classes	1	98	63	4	166
Divisions	1	98	62	3	164
Autjugacy classes	1	30	25	2	58

Dimension	1	2	4	8
Irr. complex chars.	16	44	60	46	166
Irr. rational chars.	16	40	62	46	164

Minimal presentations

Permutation degree:	$24$
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	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid b^{4}=c^{4}=e^{4}=f^{2}=g^{2}=h^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,4,10,5)(2,7,12,16)(3,9,11,13)(6,8,15,14)(17,19)(18,21)(20,22)(23,24) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^9$ . $D_4$	$C_2^6$ . $D_4^2$ (3)	$(C_2^6:D_4)$ . $D_4$ (8)	$(C_2^6:D_4)$ . $D_4$ (2)	all 115
Aut. group:	$\Aut(C_4^2:C_4)$

Elements of the group are displayed as permutations of degree 24.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{11}$
Commutator length:	$2$

Subgroups

There are 443 normal subgroups (199 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 $D_4\times C_2^5$
Frattini:	a subgroup isomorphic to $D_4\times C_2^5$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.D_4^2$

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 512) are shown, as well as all normal subgroups of any index.

Order 4096: $C_2^6.D_4^2$

Order 2048: $C_2^5.C_2^4.C_2^2$ x 3, $C_2.C_2^6.C_2^4$ x 3, $C_2^6.C_2^4.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^9.C_2^2$, $C_2^6.(C_4\times D_4)$, $C_2^3.C_2^6.C_2^2$, $C_2^5.D_4^2$, $C_2^5.D_4^2$

Order 1024: $C_2^8.C_2^2$ x 29, $C_2.C_2^6.C_2^3$ x 20, $C_2^7.C_2^3$ x 17, $C_2^5.C_2^4.C_2$ x 11, $C_2^4.C_2^5.C_2$ x 8, $C_2^4.C_2^4.C_2^2$ x 8, $C_2^8:C_4$ x 4, $C_2^7:D_4$ x 4, $C_2^6.C_2^4$ x 2, $C_2^2.C_2^6.C_2^2$ x 2, $C_2^7:D_4$ x 2, $C_2^7:D_4$ x 2, $C_2^9.C_2$ x 2, $C_2^6.C_2^3.C_2$, $C_2^5.C_2^3.C_2^2$, $C_2^3.C_2^5.C_2^2$, $C_2^7:D_4$

Order 512: $C_2^8.C_2$ x 60, $C_2^6:D_4$ x 54, $C_2^4\wr C_2$ x 40, $C_2^6:D_4$ x 33, $C_2^7.C_2^2$ x 32, $C_2^5.C_2^4$ x 32, $C_2^7:C_4$ x 24, $C_2^5.C_2^4$ x 24, $C_2^7.C_2^2$ x 24, $C_2^7:C_4$ x 22, $C_2^5.C_2^4$ x 18, $C_2^3.C_2^6$ x 16, $C_2^7:C_2^2$ x 16, $C_2^7:C_4$ x 16, $C_2^7.C_2^2$ x 16, $C_2^5.C_2^4$ x 16, $C_2\times C_2^4.C_2^4$ x 12, $C_2^5.C_2^4$ x 12, $C_2^3.C_2^6$ x 8, $C_2^3.C_2^6$ x 8, $C_2^6.D_4$ x 8, $C_2^6:D_4$ x 8, $C_2^6.D_4$ x 8, $D_4^2:D_4$ x 8, $C_2^7.C_2^2$ x 8, $C_2^7.C_2^2$ x 8, $C_2^7.C_2^2$ x 8, $C_2^7.C_2^2$ x 8, $C_2^7.C_2^2$ x 8, $C_2^6.C_2^3$ x 8, $C_2^5.C_2^4$ x 8, $C_2\times C_2^4.C_2^3.C_2$ x 6, $C_2\times C_2^4.C_2^3.C_2$ x 6, $C_2^4.C_2^5$ x 5, $C_2^7:C_4$ x 5, $C_2\times C_2^2.C_2^5.C_2$ x 4, $C_2^5.C_2^4$ x 4, $C_2^8.C_2$ x 4, $D_4^2:C_2^3$ x 4, $C_2^4.C_2^5$ x 4, $C_2^4.C_2^5$ x 4, $C_2^4.C_2^5$ x 4, $C_2^5.C_2^3.C_2$ x 4, $C_2^5.C_2^3.C_2$ x 4, $C_2^5.C_2^3.C_2$ x 4, $C_2^5.C_2^3.C_2$ x 4, $C_2^6.D_4$ x 4, $C_2^5.C_2^3.C_2$ x 4, $C_2^6:D_4$ x 4, $C_2^4.C_2^4.C_2$ x 4, $C_2^3.D_4^2$ x 4, $C_2^4.C_2^4.C_2$ x 4, $C_2^6:D_4$ x 4, $C_2^4.C_2^4.C_2$ x 4, $(C_2^2\times D_4).C_2^4$ x 4, $C_2^5.C_2^3.C_2$ x 4, $C_2^5.C_2^3.C_2$ x 4, $C_2^5.C_2^3.C_2$ x 4, $C_2^5.C_2^3.C_2$ x 4, $C_2.C_2^6.C_2^2$ x 4, $(C_2^2\times D_4).C_2^4$ x 4, $C_2^3.C_2^4.C_2^2$ x 4, $(C_2^2\times D_4).C_2^4$ x 4, $C_2\times (C_2^2\times D_4).D_4$ x 4, $C_2\times C_2^4.C_2^3.C_2$ x 4, $C_2^7.C_2^2$ x 4, $C_2^7.C_2^2$ x 4, $C_2^6.C_2^3$ x 4, $C_2^6.C_2^3$ x 4, $C_2^7.C_2^2$ x 4, $C_2^6:D_4$ x 4, $C_2^3.C_2^6$ x 3, $C_2^5:C_2^4$ x 3, $C_2^4.C_2^5$ x 3, $C_2^5.C_2^3.C_2$ x 3, $C_2^5.C_2^3.C_2$ x 3, $C_2^5.C_2^4$ x 3, $C_2^4.C_2^4.C_2$ x 3, $C_2^6.C_2^3$ x 3, $C_2^6:D_4$ x 3, $C_2^6.C_2^3$ x 2, $C_2^3.C_2^6$ x 2, $C_2^4.C_2^5$ x 2, $C_2\times C_2^3.C_2^4.C_2$ x 2, $C_2^2\times C_2^5.C_2^2$ x 2, $C_2^4.C_2^5$ x 2, $C_2^7.C_2^2$ x 2, $C_2^5.C_2^3.C_2$ x 2, $C_2^5.C_2^3.C_2$ x 2, $C_2^7.C_2^2$ x 2, $C_2^4.C_2^5$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^5.C_2^3.C_2$ x 2, $C_2^5.C_2^3.C_2$ x 2, $C_4.C_2^6.C_2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.C_2^3$ x 2, $C_2\times C_2^6.C_2^2$ x 2, $C_2^6.C_2^3$ x 2, $C_2^2.C_2^5.C_2^2$ x 2, $C_2^6.C_2^3$, $C_2^4.C_2^4.C_2$, $C_2^2.C_2^6.C_2$, $C_2^7.C_2^2$, $C_2^3.C_2^6$, $C_2^7.C_2^2$, $C_2^4.C_2^5$, $C_2^3.C_2^5.C_2$, $C_2^4.C_2^5$, $C_2^3.C_2^5.C_2$, $C_2^4.C_2^5$, $C_2^6:D_4$, $C_2^4.C_2^5$, $C_2^6.C_2^3$, $C_2^7.C_2^2$, $(C_2\times C_4).D_4^2$, $C_2^6.C_2^3$, $C_4.C_2^5.C_2^2$, $C_2^6.C_2^3$, $C_2^3.C_2^5.C_2$, $C_2^6.C_2^3$, $C_2\times C_2^6.C_2^2$, $C_2^6.D_4$, $C_2^5.C_2^3.C_2$, $C_2^5.C_2^3.C_2$, $C_2^6.C_2^3$, $C_2^9$, $C_4^2:C_2^5$, $C_4^2:C_2^5$

Order 256: $C_2^5:D_4$ x 20, $C_2^6:C_4$ x 14, $C_4^2:C_2^4$ x 6, $C_2^3.C_2^5$ x 6, $C_2^6:C_4$ x 4, $C_2^8$ x 3, $C_2^6:C_4$ x 3, $C_2^5.D_4$ x 2, $C_2^6.C_4$ x 2, $D_4:C_2^5$, $D_4\times C_2^5$, $C_4^2:C_2^4$

Order 128: $C_2^4:D_4$ x 16, $C_2^7$ x 12, $D_4\times C_2^4$ x 12, $C_2^5:C_4$ x 10, $C_2^5:C_4$ x 8, $C_4^2:C_2^3$ x 6, $D_4:C_2^4$ x 2, $C_2^5\times C_4$

Order 64: $C_2^6$ x 37, $D_4\times C_2^3$ x 16, $C_2^3:D_4$ x 8, $C_2^4\times C_4$ x 6, $C_2^4:C_4$ x 4

Order 32: $C_2^5$ x 43, $C_2^2\times D_4$ x 4, $C_2^3\times C_4$ x 4, $C_2^3:C_4$ x 4

Order 16: $C_2^4$ x 41, $C_2^2\times C_4$ x 2

Order 8: $C_2^3$ x 23

Order 4: $C_2^2$ x 11

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 4096: $C_2^6.D_4^2$ ($C_1$)

Order 2048: $C_2^5.C_2^4.C_2^2$ ($C_2$) x 3, $C_2.C_2^6.C_2^4$ ($C_2$) x 3, $C_2^6.C_2^4.C_2$ ($C_2$) x 2, $C_2^5.C_2^5.C_2$ ($C_2$) x 2, $C_2^9.C_2^2$ ($C_2$), $C_2^6.(C_4\times D_4)$ ($C_2$), $C_2^3.C_2^6.C_2^2$ ($C_2$), $C_2^5.D_4^2$ ($C_2$), $C_2^5.D_4^2$ ($C_2$)

Order 1024: $C_2.C_2^6.C_2^3$ ($C_2^2$) x 11, $C_2^7.C_2^3$ ($C_2^2$) x 7, $C_2^8:C_4$ ($C_2^2$) x 4, $C_2^7:D_4$ ($C_2^2$) x 4, $C_2^6.C_2^4$ ($C_2^2$) x 2, $C_2^5.C_2^4.C_2$ ($C_2^2$) x 2, $C_2^7:D_4$ ($C_2^2$) x 2, $C_2^8.C_2^2$ ($C_2^2$), $C_2^6.C_2^3.C_2$ ($C_2^2$), $C_2^4.C_2^5.C_2$ ($C_2^2$)

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

Order 256: $C_2^5:D_4$ ($C_2\times D_4$) x 18, $C_2^6:C_4$ ($C_2\times D_4$) x 14, $C_4^2:C_2^4$ ($C_2\times D_4$) x 6, $C_2^3.C_2^5$ ($C_2\times D_4$) x 6, $C_2^6:C_4$ ($C_2\times D_4$) x 4, $C_2^8$ ($C_2\times D_4$) x 3, $C_2^6:C_4$ ($C_2\times D_4$) x 3, $C_2^5:D_4$ ($D_4:C_2$) x 2, $C_2^5.D_4$ ($C_2\times D_4$) x 2, $C_2^6.C_4$ ($C_2\times D_4$) x 2, $D_4:C_2^5$ ($C_2\times D_4$), $D_4\times C_2^5$ ($C_2^4$), $C_4^2:C_2^4$ ($C_2\times D_4$)

Order 128: $D_4\times C_2^4$ ($C_2^2\times D_4$) x 9, $C_2^7$ ($C_2^2\wr C_2$) x 8, $C_2^4:D_4$ ($C_4:D_4$) x 8, $C_2^5:C_4$ ($C_2^2\wr C_2$) x 8, $C_2^5:C_4$ ($C_2^2\wr C_2$) x 8, $C_4^2:C_2^3$ ($C_2^2\wr C_2$) x 6, $C_2^4:D_4$ ($C_4:D_4$) x 4, $C_2^4:D_4$ ($C_2^2\wr C_2$) x 4, $C_2^7$ ($C_4:D_4$) x 2, $D_4:C_2^4$ ($C_2^2\wr C_2$) x 2, $D_4\times C_2^4$ ($D_4:C_2^2$) x 2, $C_2^5:C_4$ ($C_4:D_4$) x 2, $C_2^7$ ($D_4:C_2^2$), $C_2^7$ ($C_2^2\times D_4$), $D_4\times C_2^4$ ($D_4:C_2^2$), $C_2^5\times C_4$ ($D_4:C_2^2$)

Order 64: $C_2^6$ ($C_2\wr C_2^2$) x 20, $C_2^3:D_4$ ($C_2\wr C_2^2$) x 8, $D_4\times C_2^3$ ($D_4^2$) x 6, $D_4\times C_2^3$ ($C_2^3:D_4$) x 5, $C_2^6$ ($C_2^3:D_4$) x 4, $C_2^6$ ($C_2^3:D_4$) x 4, $C_2^4:C_4$ ($C_2^3:D_4$) x 4, $C_2^6$ ($D_4^2$) x 3, $C_2^6$ ($C_2^3:D_4$) x 3, $C_2^4\times C_4$ ($D_4^2$) x 3, $D_4\times C_2^3$ ($D_4:D_4$) x 2, $D_4\times C_2^3$ ($C_2^3:D_4$) x 2, $C_2^6$ ($D_4:D_4$), $C_2^6$ ($C_4^2:C_2^2$), $C_2^6$ ($C_2^3:D_4$), $D_4\times C_2^3$ ($C_4^2:C_2^2$), $C_2^4\times C_4$ ($D_4:D_4$), $C_2^4\times C_4$ ($C_4^2:C_2^2$), $C_2^4\times C_4$ ($C_2^3:D_4$)

Order 32: $C_2^5$ ($C_2^4:D_4$) x 16, $C_2^5$ ($C_2^4:D_4$) x 10, $C_2^5$ ($C_2^4:D_4$) x 6, $C_2^2\times D_4$ ($C_2^4:D_4$) x 4, $C_2^3:C_4$ ($C_2^4:D_4$) x 4, $C_2^5$ ($C_2^4:D_4$) x 3, $C_2^5$ ($C_2^4:D_4$) x 3, $C_2^5$ ($C_2^3\wr C_2$), $C_2^5$ ($C_2^4:D_4$), $C_2^5$ ($C_2.D_4^2$), $C_2^5$ ($C_2.D_4^2$), $C_2^5$ ($C_2^4:D_4$), $C_2^3\times C_4$ ($C_2.D_4^2$), $C_2^3\times C_4$ ($C_2^4:D_4$), $C_2^3\times C_4$ ($C_2.D_4^2$), $C_2^3\times C_4$ ($C_2^4:D_4$)

Order 16: $C_2^4$ ($C_2^5:D_4$) x 8, $C_2^4$ ($C_2^5:D_4$) x 8, $C_2^4$ ($C_2^5:D_4$) x 6, $C_2^4$ ($C_2^5:D_4$) x 5, $C_2^4$ ($C_2^5:D_4$) x 4, $C_2^4$ ($C_2^5:D_4$) x 4, $C_2^4$ ($C_2^5:D_4$) x 2, $C_2^4$ ($C_2^5:D_4$) x 2, $C_2^2\times C_4$ ($C_2^5:D_4$) x 2, $C_2^4$ ($C_2^5:D_4$), $C_2^4$ ($C_2^5:D_4$)

Order 8: $C_2^3$ ($C_2^6:D_4$) x 4, $C_2^3$ ($C_2^5.C_2^4$) x 4, $C_2^3$ ($C_2^5.C_2^4$) x 4, $C_2^3$ ($C_2^5.C_2^4$) x 2, $C_2^3$ ($C_2^5.C_2^4$) x 2, $C_2^3$ ($C_2^6:D_4$) x 2, $C_2^3$ ($C_2^5.C_2^4$) x 2, $C_2^3$ ($C_2^5.C_2^4$) x 2, $C_2^3$ ($C_2^5.C_2^4$)

Order 4: $C_2^2$ ($C_2^6.C_2^4$) x 5, $C_2^2$ ($C_2^7:D_4$) x 4, $C_2^2$ ($C_2^6.C_2^4$) x 2

Order 2: $C_2$ ($C_2^7.C_2^4$), $C_2$ ($C_2^6.D_4.C_2^2$), $C_2$ ($C_2^6.C_2.C_2^4$)

Order 1: $C_1$ ($C_2^6.D_4^2$)

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 1 larger groups in the database.

Character theory

Complex character table

See the $166 \times 166$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $164 \times 164$ rational character table (warning: may be slow to load).