Abstract group 2048.cle: $D_{128}:C_8$

• Label: 2048.cle
• Order: \( 2^{11} \)
• Exponent: \( 2^{7} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{16} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $256$
• Trans deg.: $256$
• Rank: $3$

Group information

Description:	$D_{128}:C_8$
Order:	\(2048\)\(\medspace = 2^{11} \)
Exponent:	\(128\)\(\medspace = 2^{7} \)
Automorphism group:	$C_{64}.C_8.C_2^3.C_2^4$, of order \(65536\)\(\medspace = 2^{16} \)
Composition factors:	$C_2$ x 11
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	131	140	304	576	128	256	512	2048
Conjugacy classes	1	4	9	30	44	64	128	256	536
Divisions	1	4	6	10	8	5	5	5	44
Autjugacy classes	1	3	4	6	7	4	4	4	33

Dimension	1	2	4	8	16	32	64	128
Irr. complex chars.	32	504	0	0	0	0	0	0	536
Irr. rational chars.	8	6	7	6	5	5	3	4	44

Minimal presentations

Permutation degree:	$256$
Transitive degree:	$256$
Rank:	$3$
Inequivalent generating triples:	$43008$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	128
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{8}=c^{128}=[b,c]=1, b^{a}=bc^{112}, c^{a}=c^{63} \rangle$
Permutation group:	Degree $256$ $\langle(1,2,5,11,23,43,70,101,133,165,184,152,120,88,58,34,18,27,47,74,105,137,169,198,221,233,213,188,156,124,92,62,38,51,78,109,141,173,201,223,241,249,237,217,192,160,128,96,66,82,113,145,177,205,227,243,253,256,252,240,220,195,163,131,99,116,148,180,208,230,246,254,255,251,239,219,194,162,130,98,68,84,115,147,179,207,229,245,247,235,215,190,158,126,94,64,40,53,80,111,143,175,203,225,231,211,186,154,122,90,60,36,20,29,49,76,107,139,171,196,164,132,100,69,42,22,10,4) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 64 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 200 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{257})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$Q_{256}$ $\,\rtimes\,$ $C_8$	$D_{128}$ $\,\rtimes\,$ $C_8$	$\SD_{256}$ $\,\rtimes\,$ $C_8$ (2)	$(D_{64}:C_8)$ $\,\rtimes\,$ $C_2$ (2)	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{16}$ . $D_{64}$ (2)	$C_8^2$ . $D_{16}$	$(C_4.D_{64})$ . $C_4$ (2)	$(C_8\times C_{32})$ . $D_4$	all 50

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

Homology

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

Subgroups

There are 1492 subgroups in 201 conjugacy classes, 74 normal (54 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{16}$	$G/Z \simeq$ $D_{64}$
Commutator:	$G' \simeq$ $C_{64}$	$G/G' \simeq$ $C_2^2\times C_8$
Frattini:	$\Phi \simeq$ $C_4\times C_{64}$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $D_{128}:C_8$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $D_{128}:C_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_{16}.(C_8\times D_4)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_{128}:C_8$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 2048: $D_{128}:C_8$

Order 1024: $C_8.D_{64}$ x 2, $D_{64}:C_8$ x 2, $C_8\times C_{128}$, $C_{128}.C_8$, $D_{128}:C_4$

Order 512: $D_{64}:C_4$ x 2, $D_{32}:C_8$ x 2, $C_4.D_{64}$ x 2, $C_4\times C_{128}$ x 2, $C_{64}.C_8$ x 2, $C_8.D_{32}$ x 2, $D_{128}:C_2$, $C_{128}.C_4$, $C_8\times C_{64}$

Order 256: $C_2\times C_{128}$ x 3, $D_{64}:C_2$ x 2, $D_{32}:C_4$ x 2, $\SD_{256}$ x 2, $C_{64}.C_4$ x 2, $C_4.D_{32}$ x 2, $D_{16}:C_8$ x 2, $C_4\times C_{64}$ x 2, $C_{32}.C_8$ x 2, $C_8.D_{16}$ x 2, $Q_{256}$, $D_{128}$, $C_8\times C_{32}$

Order 128: $C_{128}$ x 5, $C_2\times C_{64}$ x 3, $D_{32}:C_2$ x 2, $D_{16}:C_4$ x 2, $C_{16}.D_4$ x 2, $C_8\wr C_2$ x 2, $Q_{128}$ x 2, $\SD_{128}$ x 2, $D_{64}$ x 2, $C_{32}.C_4$ x 2, $C_4.D_{16}$ x 2, $C_4\times C_{32}$ x 2, $C_{16}.C_8$ x 2, $C_8\times C_{16}$

Order 64: $C_{64}$ x 5, $C_2\times C_{32}$ x 3, $Q_{64}$ x 2, $\SD_{64}$ x 2, $D_{32}$ x 2, $C_{16}.C_4$ x 2, $C_8.C_8$ x 2, $C_4.D_8$ x 2, $\OD_{32}:C_2$ x 2, $C_4\times C_{16}$ x 2, $D_{16}:C_2$ x 2, $\OD_{32}:C_2$ x 2, $D_8:C_4$ x 2, $C_8^2$

Order 32: $C_2\times C_{16}$ x 6, $C_{32}$ x 5, $\OD_{32}$ x 4, $D_8:C_2$ x 2, $\OD_{16}:C_2$ x 2, $C_4\times C_8$ x 2, $Q_{32}$ x 2, $\SD_{32}$ x 2, $D_{16}$ x 2, $C_8.C_4$ x 2, $C_4\wr C_2$ x 2

Order 16: $C_{16}$ x 8, $C_2\times C_8$ x 6, $\OD_{16}$ x 4, $Q_{16}$ x 2, $\SD_{16}$ x 2, $D_8$ x 2, $D_4:C_2$ x 2, $C_4^2$

Order 8: $C_8$ x 10, $D_4$ x 4, $C_2\times C_4$ x 4, $Q_8$ x 2

Order 4: $C_4$ x 6, $C_2^2$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2048: $D_{128}:C_8$

Order 1024: $C_8\times C_{128}$, $C_{128}.C_8$, $D_{128}:C_4$, $C_8.D_{64}$, $D_{64}:C_8$

Order 512: $C_4\times C_{128}$ x 2, $D_{128}:C_2$, $D_{64}:C_4$, $D_{32}:C_8$, $C_{128}.C_4$, $C_4.D_{64}$, $C_{64}.C_8$, $C_8.D_{32}$, $C_8\times C_{64}$

Order 256: $C_2\times C_{128}$ x 3, $C_4\times C_{64}$ x 2, $D_{64}:C_2$, $D_{32}:C_4$, $Q_{256}$, $\SD_{256}$, $D_{128}$, $C_{64}.C_4$, $C_4.D_{32}$, $D_{16}:C_8$, $C_{32}.C_8$, $C_8.D_{16}$, $C_8\times C_{32}$

Order 128: $C_{128}$ x 4, $C_2\times C_{64}$ x 3, $C_4\times C_{32}$ x 2, $D_{32}:C_2$, $D_{16}:C_4$, $C_{16}.D_4$, $C_8\wr C_2$, $C_8\times C_{16}$, $Q_{128}$, $\SD_{128}$, $D_{64}$, $C_{32}.C_4$, $C_4.D_{16}$, $C_{16}.C_8$

Order 64: $C_{64}$ x 4, $C_2\times C_{32}$ x 3, $C_4\times C_{16}$ x 2, $Q_{64}$, $\SD_{64}$, $D_{32}$, $C_{16}.C_4$, $C_8.C_8$, $C_4.D_8$, $\OD_{32}:C_2$, $C_8^2$, $D_{16}:C_2$, $\OD_{32}:C_2$, $D_8:C_4$

Order 32: $C_2\times C_{16}$ x 5, $C_{32}$ x 4, $C_4\times C_8$ x 2, $\OD_{32}$ x 2, $D_8:C_2$, $\OD_{16}:C_2$, $Q_{32}$, $\SD_{32}$, $D_{16}$, $C_8.C_4$, $C_4\wr C_2$

Order 16: $C_{16}$ x 7, $C_2\times C_8$ x 4, $\OD_{16}$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$, $C_4^2$, $D_4:C_2$

Order 8: $C_8$ x 6, $C_2\times C_4$ 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 2048: $D_{128}:C_8$ ($C_1$)

Order 1024: $C_8.D_{64}$ ($C_2$) x 2, $D_{64}:C_8$ ($C_2$) x 2, $C_8\times C_{128}$ ($C_2$), $C_{128}.C_8$ ($C_2$), $D_{128}:C_4$ ($C_2$)

Order 512: $D_{64}:C_4$ ($C_2^2$) x 2, $C_4.D_{64}$ ($C_4$) x 2, $C_4\times C_{128}$ ($C_2^2$) x 2, $C_{64}.C_8$ ($C_2^2$) x 2, $D_{128}:C_2$ ($C_4$), $C_{128}.C_4$ ($C_4$), $C_8\times C_{64}$ ($C_2^2$)

Order 256: $D_{64}:C_2$ ($C_2\times C_4$) x 2, $\SD_{256}$ ($C_8$) x 2, $C_2\times C_{128}$ ($C_2\times C_4$) x 2, $C_{64}.C_4$ ($C_2\times C_4$) x 2, $Q_{256}$ ($C_8$), $D_{128}$ ($C_8$), $C_4\times C_{64}$ ($C_2^3$), $C_4\times C_{64}$ ($D_4$), $C_8\times C_{32}$ ($D_4$)

Order 128: $Q_{128}$ ($C_2\times C_8$) x 2, $D_{64}$ ($C_2\times C_8$) x 2, $C_{128}$ ($C_2\times C_8$) x 2, $C_8\times C_{16}$ ($D_8$), $C_2\times C_{64}$ ($D_4:C_2$), $C_2\times C_{64}$ ($C_2^2\times C_4$), $C_4\times C_{32}$ ($D_8$), $C_4\times C_{32}$ ($C_2\times D_4$)

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

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

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

Order 8: $C_2\times C_4$ ($C_4\times D_{32}$), $C_8$ ($D_{64}:C_2$), $C_8$ ($C_2\times D_{64}$), $C_8$ ($D_{32}:C_4$), $C_8$ ($C_8\times D_{16}$)

Order 4: $C_2^2$ ($D_{64}:C_4$), $C_4$ ($C_8.(C_4\times D_8)$), $C_4$ ($C_8\times D_{32}$)

Order 2: $C_2$ ($C_{16}.(C_8\times D_4)$)

Order 1: $C_1$ ($D_{128}:C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2048: $D_{128}:C_8$ ($C_1$)

Order 1024: $C_8\times C_{128}$ ($C_2$), $C_{128}.C_8$ ($C_2$), $D_{128}:C_4$ ($C_2$), $C_8.D_{64}$ ($C_2$), $D_{64}:C_8$ ($C_2$)

Order 512: $C_4\times C_{128}$ ($C_2^2$) x 2, $D_{128}:C_2$ ($C_4$), $D_{64}:C_4$ ($C_2^2$), $C_{128}.C_4$ ($C_4$), $C_4.D_{64}$ ($C_4$), $C_{64}.C_8$ ($C_2^2$), $C_8\times C_{64}$ ($C_2^2$)

Order 256: $C_2\times C_{128}$ ($C_2\times C_4$) x 2, $D_{64}:C_2$ ($C_2\times C_4$), $Q_{256}$ ($C_8$), $\SD_{256}$ ($C_8$), $D_{128}$ ($C_8$), $C_{64}.C_4$ ($C_2\times C_4$), $C_4\times C_{64}$ ($C_2^3$), $C_4\times C_{64}$ ($D_4$), $C_8\times C_{32}$ ($D_4$)

Order 128: $C_{128}$ ($C_2\times C_8$) x 2, $C_8\times C_{16}$ ($D_8$), $Q_{128}$ ($C_2\times C_8$), $D_{64}$ ($C_2\times C_8$), $C_2\times C_{64}$ ($D_4:C_2$), $C_2\times C_{64}$ ($C_2^2\times C_4$), $C_4\times C_{32}$ ($D_8$), $C_4\times C_{32}$ ($C_2\times D_4$)

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

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

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

Order 8: $C_2\times C_4$ ($C_4\times D_{32}$), $C_8$ ($D_{64}:C_2$), $C_8$ ($C_2\times D_{64}$), $C_8$ ($D_{32}:C_4$), $C_8$ ($C_8\times D_{16}$)

Order 4: $C_2^2$ ($D_{64}:C_4$), $C_4$ ($C_8.(C_4\times D_8)$), $C_4$ ($C_8\times D_{32}$)

Order 2: $C_2$ ($C_{16}.(C_8\times D_4)$)

Order 1: $C_1$ ($D_{128}:C_8$)

Series

Derived series

$D_{128}:C_8$

$\rhd$

$C_{64}$

$\rhd$

$C_1$

Chief series

$D_{128}:C_8$

$\rhd$

$C_8\times C_{128}$

$\rhd$

$C_8\times C_{64}$

$\rhd$

$C_4\times C_{64}$

$\rhd$

$C_2\times C_{64}$

$\rhd$

$C_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$D_{128}:C_8$

$\rhd$

$C_{64}$

$\rhd$

$C_{32}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{16}$

$\lhd$

$C_2\times C_{16}$

$\lhd$

$C_4\times C_{16}$

$\lhd$

$C_8\times C_{16}$

$\lhd$

$C_8\times C_{32}$

$\lhd$

$C_8\times C_{64}$

$\lhd$

$D_{128}:C_8$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

The $536 \times 536$ character table is not available for this group.

Rational character table

See the $44 \times 44$ rational character table.