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Properties

Label 30.2
Order \( 2 \cdot 3 \cdot 5 \)
Exponent \( 2 \cdot 3 \cdot 5 \)
Nilpotent no
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2 \cdot 3 \)
$\card{Z(G)}$ \( 3 \)
$\card{\mathrm{Aut}(G)}$ \( 2^{3} \cdot 5 \)
$\card{\mathrm{Out}(G)}$ \( 2^{2} \)
Perm deg. $8$
Trans deg. $15$
Rank $2$

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Group information

Description:$C_3\times D_5$
Order: \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \)
Exponent: \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \)
Automorphism group:$C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) (generators)
Outer automorphisms:$C_2^2$, of order \(4\)\(\medspace = 2^{2} \)
Composition factors:$C_2$, $C_3$, $C_5$
Derived length:$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.

Group statistics

Order 1 2 3 5 6 15
Elements 1 5 2 4 10 8 30
Conjugacy classes   1 1 2 2 2 4 12
Divisions 1 1 1 1 1 1 6
Autjugacy classes 1 1 1 1 1 1 6

Dimension 1 2 4 8
Irr. complex chars.   6 6 0 0 12
Irr. rational chars. 2 2 1 1 6

Minimal Presentations

Permutation degree:$8$
Transitive degree:$15$
Rank: $2$
Inequivalent generating pairs: $12$

Minimal degrees of faithful linear representations

Over $\mathbb{C}$ Over $\mathbb{R}$ Over $\mathbb{Q}$
Irreducible 2 4 8
Arbitrary 2 4 6

Constructions

Groups of Lie type:$\COMinus(2,4)$
Presentation: $\langle a, b \mid a^{2}=b^{15}=1, b^{a}=b^{4} \rangle$ Copy content Toggle raw display
Permutation group: $\langle(2,3)(4,5), (6,7,8), (1,2,4,5,3)\rangle$ Copy content Toggle raw display
Matrix group:$\left\langle \left(\begin{array}{rrrrrr} 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 1 & 0 & 0 \\ -1 & -1 & -1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
$\left\langle \left(\begin{array}{rr} 14 & 3 \\ 11 & 14 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 18 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{19})$
Transitive group: 15T3 30T4 more information
Direct product: $C_3$ $\, \times\, $ $D_5$
Semidirect product: $C_5$ $\,\rtimes\,$ $C_6$ $C_{15}$ $\,\rtimes\,$ $C_2$ more information
Trans. wreath product: not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization: $C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier: $C_1$
Commutator length: $1$

Subgroups

There are 16 subgroups in 8 conjugacy classes, 6 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center: $Z \simeq$ $C_3$ $G/Z \simeq$ $D_5$
Commutator: $G' \simeq$ $C_5$ $G/G' \simeq$ $C_6$
Frattini: $\Phi \simeq$ $C_1$ $G/\Phi \simeq$ $C_3\times D_5$
Fitting: $\operatorname{Fit} \simeq$ $C_{15}$ $G/\operatorname{Fit} \simeq$ $C_2$
Radical: $R \simeq$ $C_3\times D_5$ $G/R \simeq$ $C_1$
Socle: $\operatorname{soc} \simeq$ $C_{15}$ $G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup: $P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup: $P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup: $P_{ 5 } \simeq$ $C_5$

Subgroup diagram and profile

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Subgroup information

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Series

Derived series $C_3\times D_5$ $\rhd$ $C_5$ $\rhd$ $C_1$
Chief series $C_3\times D_5$ $\rhd$ $C_{15}$ $\rhd$ $C_5$ $\rhd$ $C_1$
Lower central series $C_3\times D_5$ $\rhd$ $C_5$
Upper central series $C_1$ $\lhd$ $C_3$

Supergroups

This group is a maximal subgroup of 64 larger groups in the database.

This group is a maximal quotient of 70 larger groups in the database.

Character theory

Complex character table

1A 2A 3A1 3A-1 5A1 5A2 6A1 6A-1 15A1 15A-1 15A2 15A-2
Size 1 5 1 1 2 2 5 5 2 2 2 2
2 P 1A 1A 3A-1 3A1 5A2 5A1 3A1 3A-1 15A2 15A-2 15A1 15A-1
3 P 1A 2A 1A 1A 5A2 5A1 2A 2A 5A1 5A1 5A2 5A2
5 P 1A 2A 3A-1 3A1 1A 1A 6A-1 6A1 3A1 3A-1 3A-1 3A1
Type
30.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1
30.2.1b R 1 −1 1 1 1 1 −1 −1 1 1 1 1
30.2.1c1 C 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
30.2.1c2 C 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
30.2.1d1 C 1 −1 ζ3−1 ζ3 1 1 −ζ3 −ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
30.2.1d2 C 1 −1 ζ3 ζ3−1 1 1 −ζ3−1 −ζ3 ζ3−1 ζ3 ζ3 ζ3−1
30.2.2a1 R 2 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
30.2.2a2 R 2 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
30.2.2b1 C 2 0 2ζ15−5 2ζ155 ζ15−6+ζ156 ζ15−3+ζ153 0 0 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ154+ζ155 ζ15+ζ154 1−ζ15−ζ152+ζ153−ζ154−ζ157
30.2.2b2 C 2 0 2ζ155 2ζ15−5 ζ15−6+ζ156 ζ15−3+ζ153 0 0 1−ζ15−ζ154+ζ155 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ152+ζ153−ζ154−ζ157 ζ15+ζ154
30.2.2b3 C 2 0 2ζ15−5 2ζ155 ζ15−3+ζ153 ζ15−6+ζ156 0 0 1−ζ15−ζ152+ζ153−ζ154−ζ157 ζ15+ζ154 1−ζ15−ζ154+ζ155 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157
30.2.2b4 C 2 0 2ζ155 2ζ15−5 ζ15−3+ζ153 ζ15−6+ζ156 0 0 ζ15+ζ154 1−ζ15−ζ152+ζ153−ζ154−ζ157 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ154+ζ155

Rational character table

1A 2A 3A 5A 6A 15A
Size 1 5 2 4 10 8
2 P 1A 1A 3A 5A 3A 15A
3 P 1A 2A 1A 5A 2A 5A
5 P 1A 2A 3A 1A 6A 3A
30.2.1a 1 1 1 1 1 1
30.2.1b 1 −1 1 1 −1 1
30.2.1c 2 2 −1 2 −1 −1
30.2.1d 2 −2 −1 2 1 −1
30.2.2a 4 0 4 −1 0 −1
30.2.2b 8 0 −4 −2 0 1