Abstract group 1024.dco: $C_8.D_{64}$

• Label: 1024.dco
• Order: \( 2^{10} \)
• Exponent: \( 2^{7} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{14} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $256$
• Trans deg.: $256$
• Rank: $2$

Group information

Description:	$C_8.D_{64}$
Order:	\(1024\)\(\medspace = 2^{10} \)
Exponent:	\(128\)\(\medspace = 2^{7} \)
Automorphism group:	$C_{32}.C_{32}.C_2^4$, of order \(16384\)\(\medspace = 2^{14} \)
Composition factors:	$C_2$ x 10
Nilpotency class:	$7$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8	16	32	64	128
Elements	1	67	76	144	288	64	128	256	1024
Conjugacy classes	1	3	8	12	20	32	64	128	268
Divisions	1	3	5	5	4	3	3	2	26
Autjugacy classes	1	3	4	5	4	3	3	1	24

Dimension	1	2	4	8	16	32	64	128
Irr. complex chars.	16	252	0	0	0	0	0	0	268
Irr. rational chars.	4	4	7	2	3	3	1	2	26

Minimal presentations

Permutation degree:	$256$
Transitive degree:	$256$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	128
Arbitrary	2	4	128

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{128}=c^{4}=[b,c]=1, b^{a}=b^{47}c, c^{a}=b^{96}c \rangle$
Permutation group:	Degree $256$ $\langle(1,2)(3,9)(4,13)(5,7)(6,16)(8,20)(10,25)(11,23)(12,29)(14,17)(15,34)(18,40) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 207 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 81 & 0 \\ 0 & 165 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{257})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{64}.C_8)$ $\,\rtimes\,$ $C_2$	$(C_4\times C_{128})$ $\,\rtimes\,$ $C_2$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{64}$ . $C_8$	$C_8$ . $D_{64}$	$Q_{128}$ . $C_8$	$C_8$ . $\SD_{128}$	all 33

Elements of the group are displayed as matrices in $\GL_{2}(\F_{257})$.

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 617 subgroups in 88 conjugacy classes, 37 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_8$	$G/Z \simeq$ $D_{64}$
Commutator:	$G' \simeq$ $C_{64}$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_4\times C_{64}$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_8.D_{64}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_8.D_{64}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $D_{32}:C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8.D_{64}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 1024: $C_8.D_{64}$

Order 512: $D_{64}:C_4$, $C_4\times C_{128}$, $C_{64}.C_8$

Order 256: $D_{64}:C_2$, $D_{32}:C_4$, $C_2\times C_{128}$, $C_{64}.C_4$, $C_4.D_{32}$, $C_4\times C_{64}$, $C_{32}.C_8$

Order 128: $C_2\times C_{64}$ x 2, $C_{128}$ x 2, $D_{32}:C_2$, $D_{16}:C_4$, $Q_{128}$, $\SD_{128}$, $D_{64}$, $C_{32}.C_4$, $C_4.D_{16}$, $C_4\times C_{32}$, $C_{16}.C_8$

Order 64: $C_{64}$ x 3, $C_2\times C_{32}$ x 2, $Q_{64}$, $\SD_{64}$, $D_{32}$, $C_{16}.C_4$, $C_8.C_8$, $C_4.D_8$, $C_4\times C_{16}$, $D_{16}:C_2$, $D_8:C_4$

Order 32: $C_{32}$ x 3, $C_2\times C_{16}$ x 2, $D_8:C_2$, $\OD_{16}:C_2$, $C_4\times C_8$, $Q_{32}$, $\SD_{32}$, $D_{16}$, $\OD_{32}$, $C_8.C_4$, $C_4\wr C_2$

Order 16: $C_{16}$ x 4, $C_2\times C_8$ x 3, $\OD_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$, $C_4^2$, $D_4:C_2$

Order 8: $C_8$ x 5, $C_2\times C_4$ x 3, $D_4$ x 2, $Q_8$

Order 4: $C_4$ x 5, $C_2^2$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1024: $C_8.D_{64}$

Order 512: $D_{64}:C_4$, $C_4\times C_{128}$, $C_{64}.C_8$

Order 256: $D_{64}:C_2$, $D_{32}:C_4$, $C_2\times C_{128}$, $C_{64}.C_4$, $C_4.D_{32}$, $C_4\times C_{64}$, $C_{32}.C_8$

Order 128: $C_2\times C_{64}$ x 2, $D_{32}:C_2$, $D_{16}:C_4$, $Q_{128}$, $\SD_{128}$, $D_{64}$, $C_{32}.C_4$, $C_4.D_{16}$, $C_4\times C_{32}$, $C_{16}.C_8$, $C_{128}$

Order 64: $C_{64}$ x 3, $C_2\times C_{32}$ x 2, $Q_{64}$, $\SD_{64}$, $D_{32}$, $C_{16}.C_4$, $C_8.C_8$, $C_4.D_8$, $C_4\times C_{16}$, $D_{16}:C_2$, $D_8:C_4$

Order 32: $C_{32}$ x 3, $C_2\times C_{16}$ x 2, $D_8:C_2$, $\OD_{16}:C_2$, $C_4\times C_8$, $Q_{32}$, $\SD_{32}$, $D_{16}$, $\OD_{32}$, $C_8.C_4$, $C_4\wr C_2$

Order 16: $C_{16}$ x 4, $C_2\times C_8$ x 3, $\OD_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$, $C_4^2$, $D_4:C_2$

Order 8: $C_8$ x 5, $C_2\times C_4$ x 3, $D_4$ x 2, $Q_8$

Order 4: $C_4$ x 4, $C_2^2$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1024: $C_8.D_{64}$ ($C_1$)

Order 512: $D_{64}:C_4$ ($C_2$), $C_4\times C_{128}$ ($C_2$), $C_{64}.C_8$ ($C_2$)

Order 256: $D_{64}:C_2$ ($C_4$), $C_{64}.C_4$ ($C_4$), $C_4\times C_{64}$ ($C_2^2$)

Order 128: $Q_{128}$ ($C_8$), $D_{64}$ ($C_8$), $C_2\times C_{64}$ ($D_4$), $C_2\times C_{64}$ ($C_2\times C_4$), $C_4\times C_{32}$ ($D_4$)

Order 64: $C_2\times C_{32}$ ($\SD_{16}$), $C_2\times C_{32}$ ($C_2^2:C_4$), $C_4\times C_{16}$ ($D_8$), $C_{64}$ ($\OD_{16}$), $C_{64}$ ($C_2\times C_8$)

Order 32: $C_4\times C_8$ ($D_{16}$), $C_2\times C_{16}$ ($D_4:C_4$), $C_2\times C_{16}$ ($\SD_{32}$), $C_{32}$ ($C_2^2:C_8$), $C_{32}$ ($C_4\wr C_2$)

Order 16: $C_2\times C_8$ ($D_{32}$), $C_2\times C_8$ ($D_8:C_4$), $C_4^2$ ($\SD_{64}$), $C_{16}$ ($D_4:C_8$), $C_{16}$ ($C_4.D_8$)

Order 8: $C_2\times C_4$ ($D_{16}:C_4$), $C_8$ ($D_8:C_8$), $C_8$ ($\SD_{128}$), $C_8$ ($D_{64}$), $C_8$ ($C_4.D_{16}$)

Order 4: $C_2^2$ ($C_4.D_{32}$), $C_4$ ($D_{32}:C_4$), $C_4$ ($D_{16}:C_8$)

Order 2: $C_2$ ($D_{32}:C_8$)

Order 1: $C_1$ ($C_8.D_{64}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1024: $C_8.D_{64}$ ($C_1$)

Order 512: $D_{64}:C_4$ ($C_2$), $C_4\times C_{128}$ ($C_2$), $C_{64}.C_8$ ($C_2$)

Order 256: $D_{64}:C_2$ ($C_4$), $C_{64}.C_4$ ($C_4$), $C_4\times C_{64}$ ($C_2^2$)

Order 128: $Q_{128}$ ($C_8$), $D_{64}$ ($C_8$), $C_2\times C_{64}$ ($D_4$), $C_2\times C_{64}$ ($C_2\times C_4$), $C_4\times C_{32}$ ($D_4$)

Order 64: $C_2\times C_{32}$ ($\SD_{16}$), $C_2\times C_{32}$ ($C_2^2:C_4$), $C_4\times C_{16}$ ($D_8$), $C_{64}$ ($\OD_{16}$), $C_{64}$ ($C_2\times C_8$)

Order 32: $C_4\times C_8$ ($D_{16}$), $C_2\times C_{16}$ ($D_4:C_4$), $C_2\times C_{16}$ ($\SD_{32}$), $C_{32}$ ($C_2^2:C_8$), $C_{32}$ ($C_4\wr C_2$)

Order 16: $C_2\times C_8$ ($D_{32}$), $C_2\times C_8$ ($D_8:C_4$), $C_4^2$ ($\SD_{64}$), $C_{16}$ ($D_4:C_8$), $C_{16}$ ($C_4.D_8$)

Order 8: $C_2\times C_4$ ($D_{16}:C_4$), $C_8$ ($D_8:C_8$), $C_8$ ($\SD_{128}$), $C_8$ ($D_{64}$), $C_8$ ($C_4.D_{16}$)

Order 4: $C_2^2$ ($C_4.D_{32}$), $C_4$ ($D_{32}:C_4$), $C_4$ ($D_{16}:C_8$)

Order 2: $C_2$ ($D_{32}:C_8$)

Order 1: $C_1$ ($C_8.D_{64}$)

Series

Derived series

$C_8.D_{64}$

$\rhd$

$C_{64}$

$\rhd$

$C_1$

Chief series

$C_8.D_{64}$

$\rhd$

$C_4\times C_{128}$

$\rhd$

$C_4\times C_{64}$

$\rhd$

$C_2\times C_{64}$

$\rhd$

$C_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_8.D_{64}$

$\rhd$

$C_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_8$

$\lhd$

$C_2\times C_8$

$\lhd$

$C_4\times C_8$

$\lhd$

$C_4\times C_{16}$

$\lhd$

$C_4\times C_{32}$

$\lhd$

$C_4\times C_{64}$

$\lhd$

$C_8.D_{64}$

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

See the $268 \times 268$ 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 $26 \times 26$ rational character table.