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Properties

Label 64.36
Order \( 2^{6} \)
Exponent \( 2^{3} \)
Nilpotent yes
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{3} \)
$\card{Z(G)}$ \( 2 \)
$\card{\mathrm{Aut}(G)}$ \( 2^{8} \)
$\card{\mathrm{Out}(G)}$ \( 2^{3} \)
Perm deg. $16$
Trans deg. $16$
Rank $2$

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Group information

Description:$C_4^2.C_4$
Order: \(64\)\(\medspace = 2^{6} \)
Exponent: \(8\)\(\medspace = 2^{3} \)
Automorphism group:$D_4^2:C_2^2$, of order \(256\)\(\medspace = 2^{8} \) (generators)
Outer automorphisms:$C_2^3$, of order \(8\)\(\medspace = 2^{3} \)
Composition factors:$C_2$ x 6
Nilpotency class:$4$
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
Elements 1 11 20 32 64
Conjugacy classes   1 3 5 4 13
Divisions 1 3 4 2 10
Autjugacy classes 1 3 3 1 8

Dimension 1 2 4 8
Irr. complex chars.   8 2 3 0 13
Irr. rational chars. 4 4 1 1 10

Minimal Presentations

Permutation degree:$16$
Transitive degree:$16$
Rank: $2$
Inequivalent generating pairs: $6$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 4 8 8
Arbitrary 4 8 8

Constructions

Presentation: $\langle a, b, c \mid b^{4}=c^{4}=[b,c]=1, a^{4}=c^{2}, b^{a}=b^{3}c, c^{a}=b^{2}c^{3} \rangle$ Copy content Toggle raw display
Permutation group:Degree $16$ $\langle(1,12,4,10,2,11,3,9)(5,15,7,14,6,16,8,13), (1,3)(2,4)(5,8)(6,7)(9,16)(10,15) \!\cdots\! \rangle$ Copy content Toggle raw display
Matrix group:$\left\langle \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 1 & 3 & 3 & 3 \\ 1 & 4 & 2 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 4 & 0 & 0 \\ 3 & 0 & 4 & 3 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 3 & 2 & 4 & 2 \\ 1 & 2 & 3 & 1 \\ 1 & 2 & 1 & 1 \\ 4 & 1 & 4 & 4 \end{array}\right), \left(\begin{array}{rrrr} 4 & 4 & 1 & 1 \\ 0 & 3 & 0 & 0 \\ 3 & 2 & 1 & 4 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 2 & 3 & 2 & 1 \\ 0 & 4 & 0 & 4 \\ 3 & 2 & 4 & 4 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
Transitive group: 16T131 16T151 32T140 32T165 more information
Direct product: not isomorphic to a non-trivial direct product
Semidirect product: $(C_4.D_4)$ $\,\rtimes\,$ $C_2$ (2) more information
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product
Non-split product: $C_4^2$ . $C_4$ $(C_2\times D_4)$ . $C_4$ $(C_2\times C_4)$ . $D_4$ (2) $(C_2\times Q_8)$ . $C_2^2$ all 8

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: $C_{2}$
Commutator length: $1$

Subgroups

There are 73 subgroups in 32 conjugacy classes, 13 normal (9 characteristic).

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

Special subgroups

Center: $Z \simeq$ $C_2$ $G/Z \simeq$ $C_2^3:C_4$
Commutator: $G' \simeq$ $C_2\times C_4$ $G/G' \simeq$ $C_2\times C_4$
Frattini: $\Phi \simeq$ $C_2\times Q_8$ $G/\Phi \simeq$ $C_2^2$
Fitting: $\operatorname{Fit} \simeq$ $C_4^2.C_4$ $G/\operatorname{Fit} \simeq$ $C_1$
Radical: $R \simeq$ $C_4^2.C_4$ $G/R \simeq$ $C_1$
Socle: $\operatorname{soc} \simeq$ $C_2$ $G/\operatorname{soc} \simeq$ $C_2^3:C_4$
2-Sylow subgroup: $P_{ 2 } \simeq$ $C_4^2.C_4$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_4^2.C_4$ $\rhd$ $C_2\times C_4$ $\rhd$ $C_1$
Chief series $C_4^2.C_4$ $\rhd$ $C_4^2:C_2$ $\rhd$ $C_2\times Q_8$ $\rhd$ $C_2\times C_4$ $\rhd$ $C_2^2$ $\rhd$ $C_2$ $\rhd$ $C_1$
Lower central series $C_4^2.C_4$ $\rhd$ $C_2\times C_4$ $\rhd$ $C_2^2$ $\rhd$ $C_2$ $\rhd$ $C_1$
Upper central series $C_1$ $\lhd$ $C_2$ $\lhd$ $C_2^2$ $\lhd$ $C_2\times Q_8$ $\lhd$ $C_4^2.C_4$

Supergroups

This group is a maximal subgroup of 55 larger groups in the database.

This group is a maximal quotient of 52 larger groups in the database.

Character theory

Complex character table

1A 2A 2B 2C 4A 4B 4C 4D1 4D-1 8A1 8A-1 8B1 8B-1
Size 1 1 2 8 4 4 4 4 4 8 8 8 8
2 P 1A 1A 1A 1A 2A 2A 2A 2B 2B 4A 4B 4B 4A
Type
64.36.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
64.36.1b R 1 1 1 −1 1 1 1 −1 −1 −1 1 1 −1
64.36.1c R 1 1 1 −1 1 1 1 −1 −1 1 −1 −1 1
64.36.1d R 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
64.36.1e1 C 1 1 1 −1 −1 −1 1 1 1 −i −i i i
64.36.1e2 C 1 1 1 −1 −1 −1 1 1 1 i i −i −i
64.36.1f1 C 1 1 1 1 −1 −1 1 −1 −1 −i i −i i
64.36.1f2 C 1 1 1 1 −1 −1 1 −1 −1 i −i i −i
64.36.2a R 2 2 2 0 −2 2 −2 0 0 0 0 0 0
64.36.2b R 2 2 2 0 2 −2 −2 0 0 0 0 0 0
64.36.4a R 4 4 −4 0 0 0 0 0 0 0 0 0 0
64.36.4b1 C 4 −4 0 0 0 0 0 −2i 2i 0 0 0 0
64.36.4b2 C 4 −4 0 0 0 0 0 2i −2i 0 0 0 0

Rational character table

1A 2A 2B 2C 4A 4B 4C 4D 8A 8B
Size 1 1 2 8 4 4 4 8 16 16
2 P 1A 1A 1A 1A 2A 2A 2A 2B 4A 4B
64.36.1a 1 1 1 1 1 1 1 1 1 1
64.36.1b 1 1 1 −1 1 1 1 −1 −1 1
64.36.1c 1 1 1 −1 1 1 1 −1 1 −1
64.36.1d 1 1 1 1 1 1 1 1 −1 −1
64.36.1e 2 2 2 −2 −2 −2 2 2 0 0
64.36.1f 2 2 2 2 −2 −2 2 −2 0 0
64.36.2a 2 2 2 0 −2 2 −2 0 0 0
64.36.2b 2 2 2 0 2 −2 −2 0 0 0
64.36.4a 4 4 −4 0 0 0 0 0 0 0
64.36.4b 8 −8 0 0 0 0 0 0 0 0