Abstract group 2048.cqj: $C_2^8.D_4$

• Label: 2048.cqj
• Order: \( 2^{11} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: 16
• $\card{\Aut(G)}$: \( 2^{32} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{25} \cdot 3 \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $6$

Group information

Description:	$C_2^8.D_4$
Order:	\(2048\)\(\medspace = 2^{11} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	Group of order \(12884901888\)\(\medspace = 2^{32} \cdot 3 \)
Composition factors:	$C_2$ x 11
Nilpotency class:	$3$
Derived length:	$2$

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

Group statistics

Order	1	2	4
Elements	1	703	1344	2048
Conjugacy classes	1	199	96	296
Divisions	1	199	96	296
Autjugacy classes	1	14	6	21

Dimension	1	2	4
Irr. complex chars.	64	144	88	296
Irr. rational chars.	64	144	88	296

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	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid b^{4}=c^{2}=d^{4}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(1,15)(2,7)(3,5)(4,11)(8,14)(9,18), (4,11)(6,12)(8,14)(10,16), (1,7)(2,15) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^8$ . $D_4$	$C_2^9$ . $C_2^2$	$C_2^8$ . $C_2^3$ (15)	$(C_2^5:D_4)$ . $D_4$ (104)	all 93
Aut. group:	$\Aut(C_4^2.C_2^3)$	$\Aut(C_4^2.C_2^3)$	$\Aut(C_3\times C_4^2:C_2^2)$	$\Aut(C_2^4.(C_2\times D_4))$	all 20

Elements of the group are displayed as permutations of degree 18.

Homology

Abelianization:	$C_{2}^{6} $
Schur multiplier:	$C_{2}^{19}$
Commutator length:	$1$

Subgroups

There are 8648 normal subgroups (18 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 $C_2^4$
Commutator:	a subgroup isomorphic to $C_2^5$
Frattini:	a subgroup isomorphic to $C_2^5$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^8.D_4$

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 4 (order at least 512) are shown, as well as normal subgroups of index up to 8 or of order up to 16.

Order 2048: $C_2^8.D_4$

Order 1024: $C_2^7:D_4$ x 32, $C_2^7:D_4$ x 12, $C_2^7.C_2^3$ x 8, $C_2^7:D_4$ x 6, $C_2^9.C_2$ x 2, $C_2^8.C_2^2$, $C_2^6.C_2^4$, $C_2^8:C_4$

Order 512: $C_2^6:D_4$ x 192, $C_2^6:D_4$ x 192, $C_2^6:D_4$ x 138, $C_2^5.C_2^4$ x 96, $C_2^4\wr C_2$ x 48, $C_2^5:C_2^4$ x 40, $C_2^7.C_2^2$ x 32, $C_2^4.C_2^5$ x 24, $C_2^5.C_2^4$ x 24, $C_2^8.C_2$ x 23, $C_2^7:C_4$ x 22, $C_2^7:C_4$ x 16, $C_2^2\times C_2^5.C_2^2$ x 12, $C_2^4.C_2^5$ x 12, $C_2^4.C_2^5$ x 12, $C_2\times C_2^6.C_2^2$ x 12, $C_2^3.C_2^6$ x 9, $C_2^3.C_2^6$ x 8, $C_2^3:D_4^2$ x 8, $C_2^6:D_4$ x 4, $C_2\times C_2^4.C_2^4$ x 3, $C_2^3.C_2^6$ x 3, $C_4^2:C_2^5$ x 3, $D_4\times C_2^6$ x 2, $C_2^8.C_2$ x 2, $C_2^3.C_2^6$, $C_2^9$

Order 256: $C_2^5:D_4$ x 230, $C_2^5.D_4$ x 192, $C_2^5:D_4$ x 192, $C_2^5:D_4$ x 128, $C_2^6:C_4$ x 112, $C_2^4.C_4^2$ x 96, $C_2^5.D_4$ x 96, $C_2^5.D_4$ x 96, $C_2^6:C_4$ x 96, $C_2^6:C_4$ x 50, $C_2^5:D_4$ x 48, $C_2^5:D_4$ x 32, $D_4\times C_2^5$ x 28, $C_4^2:C_2^4$ x 24, $C_2^6:C_4$ x 24, $C_2^3.C_2^5$ x 24, $C_2^4.C_4^2$ x 24, $C_2^8$ x 16, $C_2^6:C_4$ x 14, $C_2^5:D_4$ x 6, $C_2^5.D_4$ x 6, $C_4^2:C_2^4$ x 3, $D_4:C_2^5$, $C_2^6\times C_4$

Order 16: $C_2^4$ x 699, $C_2\times D_4$ x 64, $C_2^2\times C_4$ x 40, $C_2^2:C_4$ x 32

Order 8: $C_2^3$ x 363, $C_2\times C_4$ x 8

Order 4: $C_2^2$ x 107

Order 2: $C_2$ x 15

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 2048: $C_2^8.D_4$ ($C_1$)

Order 1024: $C_2^7:D_4$ ($C_2$) x 32, $C_2^7:D_4$ ($C_2$) x 12, $C_2^7.C_2^3$ ($C_2$) x 8, $C_2^7:D_4$ ($C_2$) x 6, $C_2^9.C_2$ ($C_2$) x 2, $C_2^8.C_2^2$ ($C_2$), $C_2^6.C_2^4$ ($C_2$), $C_2^8:C_4$ ($C_2$)

Order 512: $C_2^6:D_4$ ($C_2^2$) x 192, $C_2^6:D_4$ ($C_2^2$) x 96, $C_2^5.C_2^4$ ($C_2^2$) x 96, $C_2^4\wr C_2$ ($C_2^2$) x 32, $C_2^7.C_2^2$ ($C_2^2$) x 32, $C_2^6:D_4$ ($C_2^2$) x 28, $C_2^4.C_2^5$ ($C_2^2$) x 24, $C_2^5.C_2^4$ ($C_2^2$) x 24, $C_2^5:C_2^4$ ($C_2^2$) x 16, $C_2^7:C_4$ ($C_2^2$) x 16, $C_2^8.C_2$ ($C_2^2$) x 14, $C_2^7:C_4$ ($C_2^2$) x 14, $C_2^2\times C_2^5.C_2^2$ ($C_2^2$) x 12, $C_2^4.C_2^5$ ($C_2^2$) x 12, $C_2^4.C_2^5$ ($C_2^2$) x 12, $C_2\times C_2^6.C_2^2$ ($C_2^2$) x 12, $C_2^3.C_2^6$ ($C_2^2$) x 6, $C_2^6:D_4$ ($C_2^2$) x 4, $C_2^3.C_2^6$ ($C_2^2$) x 3, $C_4^2:C_2^5$ ($C_2^2$) x 3, $C_2^9$ ($C_2^2$), $D_4\times C_2^6$ ($C_2^2$), $C_2^8.C_2$ ($C_2^2$)

Order 256: $C_2^5.D_4$ ($C_2^3$) x 192, $C_2^5:D_4$ ($C_2^3$) x 192, $C_2^5:D_4$ ($C_2^3$) x 126, $C_2^6:C_4$ ($C_2^3$) x 112, $C_2^5:D_4$ ($C_2^3$) x 112, $C_2^5:D_4$ ($D_4$) x 104, $C_2^4.C_4^2$ ($C_2^3$) x 96, $C_2^5.D_4$ ($C_2^3$) x 96, $C_2^5.D_4$ ($C_2^3$) x 96, $C_2^6:C_4$ ($C_2^3$) x 96, $C_2^5:D_4$ ($C_2^3$) x 48, $C_2^6:C_4$ ($C_2^3$) x 42, $C_2^5:D_4$ ($C_2^3$) x 32, $C_4^2:C_2^4$ ($C_2^3$) x 24, $C_2^6:C_4$ ($C_2^3$) x 24, $C_2^3.C_2^5$ ($C_2^3$) x 24, $C_2^4.C_4^2$ ($C_2^3$) x 24, $C_2^5:D_4$ ($D_4$) x 16, $C_2^8$ ($C_2^3$) x 15, $D_4\times C_2^5$ ($C_2^3$) x 14, $D_4\times C_2^5$ ($D_4$) x 14, $C_2^6:C_4$ ($C_2^3$) x 14, $C_2^6:C_4$ ($D_4$) x 8, $C_2^5:D_4$ ($C_2^3$) x 6, $C_2^5.D_4$ ($C_2^3$) x 6, $C_4^2:C_2^4$ ($C_2^3$) x 3, $C_2^8$ ($D_4$), $D_4:C_2^5$ ($C_2^3$), $C_2^6\times C_4$ ($D_4$)

Order 128: ? (unidentified group of order 16) x 1731

Order 64: ? (unidentified group of order 32) x 1859

Order 32: ? (unidentified group of order 64) x 1475

Order 16: $C_2^4$ ($C_2^4:D_4$) x 192, $C_2^4$ ($C_2\times D_4^2$) x 144, $C_2^4$ ($C_2^4:D_4$) x 136, $C_2^4$ ($C_2^4:D_4$) x 112, $C_2^4$ ($D_4^2:C_2$) x 48, $C_2^4$ ($C_2^4:D_4$) x 36, $C_2^2:C_4$ ($C_2^4:D_4$) x 32, $C_2\times D_4$ ($C_2^4:D_4$) x 32, $C_2\times D_4$ ($C_2^3\wr C_2$) x 32, $C_2^2\times C_4$ ($C_2^4:D_4$) x 24, $C_2^4$ ($C_2^3\wr C_2$) x 16, $C_2^2\times C_4$ ($C_2^3\wr C_2$) x 16, $C_2^4$ ($D_4\times C_2^4$) x 9, $C_2^4$ ($D_4:C_2^4$) x 6

Order 8: $C_2^3$ ($C_2^5:D_4$) x 96, $C_2^3$ ($C_2^5:D_4$) x 72, $C_2^3$ ($C_2^2:D_4^2$) x 64, $C_2^3$ ($C_2^5:D_4$) x 56, $C_2^3$ ($C_2^5:D_4$) x 24, $C_2^3$ ($C_2^5:D_4$) x 16, $C_2^3$ ($C_4^2:C_2^4$) x 12, $C_2^3$ ($C_2^2\times D_4^2$) x 12, $C_2^3$ ($C_2^5:D_4$) x 8, $C_2\times C_4$ ($C_2^5:D_4$) x 8, $C_2^3$ ($C_2^5:D_4$) x 3

Order 4: $C_2^2$ ($C_2^6:D_4$) x 72, $C_2^2$ ($C_2^6:D_4$) x 16, $C_2^2$ ($C_2^3:D_4^2$) x 8, $C_2^2$ ($C_2^6:D_4$) x 4, $C_2^2$ ($C_2^3.C_2^6$) x 3, $C_2^2$ ($C_2^3.C_2^6$) x 3, $C_2^2$ ($C_2^8.C_2$)

Order 2: $C_2$ ($C_2^7:D_4$) x 8, $C_2$ ($C_2^7:D_4$) x 6, $C_2$ ($C_2^7:D_4$)

Order 1: $C_1$ ($C_2^8.D_4$)

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 9 larger groups in the database.

This group is a maximal quotient of 6 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

See the $296 \times 296$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.