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Properties

Label 128.147
Order \( 2^{7} \)
Exponent \( 2^{5} \)
Nilpotent yes
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{3} \)
$\card{Z(G)}$ \( 2^{2} \)
$\card{\mathrm{Aut}(G)}$ \( 2^{10} \)
$\card{\mathrm{Out}(G)}$ \( 2^{5} \)
Perm deg. $36$
Trans deg. $64$
Rank $2$

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Group information

Description:$D_{16}:C_4$
Order: \(128\)\(\medspace = 2^{7} \)
Exponent: \(32\)\(\medspace = 2^{5} \)
Automorphism group:$C_2^2\times D_{16}:C_8$, of order \(1024\)\(\medspace = 2^{10} \) (generators)
Outer automorphisms:$C_2^2\times C_8$, of order \(32\)\(\medspace = 2^{5} \)
Composition factors:$C_2$ x 7
Nilpotency class:$5$
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
Elements 1 35 36 8 16 32 128
Conjugacy classes   1 5 4 4 8 16 38
Divisions 1 5 3 2 2 1 14
Autjugacy classes 1 4 3 2 2 1 13

Dimension 1 2 4 8 16
Irr. complex chars.   8 30 0 0 0 38
Irr. rational chars. 4 4 2 2 2 14

Minimal Presentations

Permutation degree:$36$
Transitive degree:$64$
Rank: $2$
Inequivalent generating pairs: $6$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible none none none
Arbitrary 3 4 18

Constructions

Presentation: $\langle a, b, c \mid a^{2}=b^{2}=c^{32}=[a,b]=[b,c]=1, c^{a}=bc^{31} \rangle$ Copy content Toggle raw display
Permutation group:Degree $36$ $\langle(1,2,4,9)(3,7,14,20)(5,18,16,32)(6,19,17,11)(8,23,24,12)(10,27,26,13)(15,28,29,22) \!\cdots\! \rangle$ Copy content Toggle raw display
Matrix group:$\left\langle \left(\begin{array}{rr} 31 & 0 \\ 0 & 31 \end{array}\right), \left(\begin{array}{rr} 23 & 7 \\ 2 & 9 \end{array}\right), \left(\begin{array}{rr} 23 & 15 \\ 10 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 24 \\ 16 & 17 \end{array}\right), \left(\begin{array}{rr} 25 & 28 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 29 & 27 \\ 2 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
$\left\langle \left(\begin{array}{rr} 34 & 9 \\ 54 & 28 \end{array}\right), \left(\begin{array}{rr} 32 & 0 \\ 0 & 32 \end{array}\right), \left(\begin{array}{rr} 66 & 28 \\ 5 & 27 \end{array}\right), \left(\begin{array}{rr} 0 & 25 \\ 5 & 0 \end{array}\right), \left(\begin{array}{rr} 61 & 0 \\ 0 & 61 \end{array}\right), \left(\begin{array}{rr} 16 & 36 \\ 30 & 85 \end{array}\right), \left(\begin{array}{rr} 77 & 57 \\ 51 & 47 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/93\Z)$
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $D_{16}$ $\,\rtimes\,$ $C_4$ (2) $(C_{16}:C_4)$ $\,\rtimes\,$ $C_2$ $(C_2\times C_{32})$ $\,\rtimes\,$ $C_2$ more information
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_{16}$ . $D_4$ $C_2$ . $D_{32}$ $C_8$ . $\SD_{16}$ $C_4$ . $\SD_{32}$ all 14

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

Homology

Abelianization: $C_{2} \times C_{4} $
Schur multiplier: $C_{2}^{2}$
Commutator length: $1$

Subgroups

There are 172 subgroups in 41 conjugacy classes, 20 normal (18 characteristic).

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

Special subgroups

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

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $D_{16}:C_4$ $\rhd$ $C_{16}$ $\rhd$ $C_1$
Chief series $D_{16}:C_4$ $\rhd$ $C_2\times C_{32}$ $\rhd$ $C_2\times C_{16}$ $\rhd$ $C_{16}$ $\rhd$ $C_8$ $\rhd$ $C_4$ $\rhd$ $C_2$ $\rhd$ $C_1$
Lower central series $D_{16}:C_4$ $\rhd$ $C_{16}$ $\rhd$ $C_8$ $\rhd$ $C_4$ $\rhd$ $C_2$ $\rhd$ $C_1$
Upper central series $C_1$ $\lhd$ $C_2^2$ $\lhd$ $C_2\times C_4$ $\lhd$ $C_2\times C_8$ $\lhd$ $C_2\times C_{16}$ $\lhd$ $D_{16}:C_4$

Supergroups

This group is a maximal subgroup of 36 larger groups in the database.

This group is a maximal quotient of 18 larger groups in the database.

Character theory

Complex character table

See the $38 \times 38$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

1A 2A 2B 2C 2D 2E 4A 4B 4C 8A 8B 16A 16B 32A
Size 1 1 1 1 16 16 2 2 32 4 4 8 8 32
2 P 1A 1A 1A 1A 1A 1A 2A 2A 2C 4B 4B 8B 8B 16B
128.147.1a 1 1 1 1 1 1 1 1 1 1 1 1 1 1
128.147.1b 1 1 1 1 −1 −1 1 1 −1 1 1 1 1 1
128.147.1c 1 1 1 1 −1 −1 1 1 1 1 1 1 1 −1
128.147.1d 1 1 1 1 1 1 1 1 −1 1 1 1 1 −1
128.147.1e 2 2 −2 −2 −2 2 −2 2 0 −2 2 −2 2 0
128.147.1f 2 2 −2 −2 2 −2 −2 2 0 −2 2 −2 2 0
128.147.2a 2 2 −2 −2 0 0 −2 2 0 −2 2 2 −2 0
128.147.2b 2 2 2 2 0 0 2 2 0 2 2 −2 −2 0
128.147.2c 4 4 4 4 0 0 4 4 0 −4 −4 0 0 0
128.147.2d 4 4 −4 −4 0 0 −4 4 0 4 −4 0 0 0
128.147.2e 8 8 −8 −8 0 0 8 −8 0 0 0 0 0 0
128.147.2f 8 8 8 8 0 0 −8 −8 0 0 0 0 0 0
128.147.2g 16 −16 16 −16 0 0 0 0 0 0 0 0 0 0
128.147.2h 16 −16 −16 16 0 0 0 0 0 0 0 0 0 0