Abstract group 720.441: $D_{10}.S_3^2$

• Label: 720.441
• Order: \( 2^{4} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $60$
• Rank: $2$

Group information

Description:	$D_{10}.S_3^2$
Order:	\(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$F_5\times D_6^2$, of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $C_5$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	11	8	276	4	88	4	192	32	24	32	48	720
Conjugacy classes	1	3	3	6	1	9	1	8	5	2	5	4	48
Divisions	1	3	3	4	1	9	1	3	3	1	3	1	33
Autjugacy classes	1	3	3	6	1	9	1	4	3	1	3	1	36

Dimension	1	2	4	8	16
Irr. complex chars.	8	18	16	6	0	48
Irr. rational chars.	4	10	11	5	3	33

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$60$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	16
Arbitrary	6	8	8

Constructions

Presentation:	$\langle a, b, c \mid a^{4}=b^{60}=c^{3}=[a,c]=1, b^{a}=b^{47}, c^{b}=c^{2} \rangle$
Permutation group:	Degree $15$ $\langle(2,3,4,5)(7,8)(9,10)(11,12), (9,11,12,10)(14,15), (2,4)(3,5), (9,12)(10,11), (13,14,15), (6,7,8), (1,2,5,3,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 15 \\ 30 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 26 & 0 \\ 0 & 26 \end{array}\right), \left(\begin{array}{rr} 41 & 19 \\ 10 & 13 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 31 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 34 & 40 \\ 10 & 29 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/45\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3$ $\,\rtimes\,$ $(D_{10}.D_6)$	$C_{15}$ $\,\rtimes\,$ $(C_6.D_4)$	$C_5$ $\,\rtimes\,$ $(C_6.D_{12})$	$(C_3:C_{60})$ $\,\rtimes\,$ $C_4$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{10}$ . $S_3^2$	$C_6$ . $(S_3\times F_5)$	$(C_6.D_{10})$ . $S_3$	$(C_{30}:C_4)$ . $S_3$	all 23

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

Homology

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

Subgroups

There are 880 subgroups in 120 conjugacy classes, 40 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_5.S_3^2$
Commutator:	$G' \simeq$ $C_3\times C_{30}$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_5.S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_{30}$	$G/\operatorname{Fit} \simeq$ $C_2\times C_4$
Radical:	$R \simeq$ $D_{10}.S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4:C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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 720: $D_{10}.S_3^2$

Order 360: $C_{30}.D_6$, $C_2\times C_3^2:F_5$, $C_{30}:C_{12}$

Order 240: $D_{10}.D_6$, $C_{60}:C_4$

Order 180: $C_3^2\times D_{10}$, $C_3^2:F_5$, $C_{15}:C_{12}$, $C_{15}:C_{12}$, $C_3:C_{60}$

Order 144: $C_6.D_{12}$

Order 120: $C_{30}:C_4$ x 4, $C_6.D_{10}$, $C_6\times F_5$, $D_5\times C_{12}$

Order 90: $C_3^2\times D_5$ x 2, $C_3\times C_{30}$

Order 80: $C_{20}:C_4$

Order 72: $C_6:C_{12}$ x 2, $C_6.D_6$

Order 60: $C_{15}:C_4$ x 5, $C_3\times D_{10}$ x 3, $C_3\times F_5$, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $C_3:C_{20}$

Order 48: $C_{12}:C_4$, $C_6.D_4$

Order 45: $C_3\times C_{15}$

Order 40: $C_2\times F_5$ x 2, $C_4\times D_5$

Order 36: $C_3:C_{12}$ x 3, $C_3^2:C_4$, $C_6^2$

Order 30: $C_3\times D_5$ x 6, $C_{30}$ x 3

Order 24: $C_6:C_4$ x 5, $C_2\times C_{12}$ x 2

Order 20: $F_5$ x 2, $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times C_6$ x 3

Order 16: $C_4:C_4$

Order 15: $C_{15}$ x 3

Order 12: $C_3:C_4$ x 7, $C_2\times C_6$ x 3, $C_{12}$ x 3

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 3

Order 6: $C_6$ x 9

Order 5: $C_5$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 720: $D_{10}.S_3^2$

Order 360: $C_{30}.D_6$, $C_2\times C_3^2:F_5$, $C_{30}:C_{12}$

Order 240: $D_{10}.D_6$, $C_{60}:C_4$

Order 180: $C_3^2\times D_{10}$, $C_3^2:F_5$, $C_{15}:C_{12}$, $C_{15}:C_{12}$, $C_3:C_{60}$

Order 144: $C_6.D_{12}$

Order 120: $C_{30}:C_4$ x 4, $C_6.D_{10}$, $C_6\times F_5$, $D_5\times C_{12}$

Order 90: $C_3^2\times D_5$ x 2, $C_3\times C_{30}$

Order 80: $C_{20}:C_4$

Order 72: $C_6:C_{12}$ x 2, $C_6.D_6$

Order 60: $C_{15}:C_4$ x 4, $C_3\times D_{10}$ x 3, $C_3\times F_5$, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $C_3:C_{20}$

Order 48: $C_{12}:C_4$, $C_6.D_4$

Order 45: $C_3\times C_{15}$

Order 40: $C_2\times F_5$ x 2, $C_4\times D_5$

Order 36: $C_3:C_{12}$ x 3, $C_3^2:C_4$, $C_6^2$

Order 30: $C_3\times D_5$ x 6, $C_{30}$ x 3

Order 24: $C_6:C_4$ x 5, $C_2\times C_{12}$ x 2

Order 20: $F_5$ x 2, $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times C_6$ x 3

Order 16: $C_4:C_4$

Order 15: $C_{15}$ x 3

Order 12: $C_3:C_4$ x 6, $C_2\times C_6$ x 3, $C_{12}$ x 3

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 3

Order 6: $C_6$ x 9

Order 5: $C_5$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 720: $D_{10}.S_3^2$ ($C_1$)

Order 360: $C_{30}.D_6$ ($C_2$), $C_2\times C_3^2:F_5$ ($C_2$), $C_{30}:C_{12}$ ($C_2$)

Order 180: $C_3^2\times D_{10}$ ($C_2^2$), $C_{15}:C_{12}$ ($C_4$), $C_3:C_{60}$ ($C_4$)

Order 120: $C_6.D_{10}$ ($S_3$), $C_{30}:C_4$ ($S_3$)

Order 90: $C_3^2\times D_5$ ($Q_8$), $C_3^2\times D_5$ ($D_4$), $C_3\times C_{30}$ ($C_2\times C_4$)

Order 60: $C_3\times D_{10}$ ($D_6$) x 2, $C_{15}:C_4$ ($C_3:C_4$), $C_3:C_{20}$ ($C_3:C_4$)

Order 45: $C_3\times C_{15}$ ($C_4:C_4$)

Order 36: $C_3:C_{12}$ ($F_5$)

Order 30: $C_3\times D_5$ ($C_3:Q_8$) x 2, $C_{30}$ ($C_6:C_4$), $C_{30}$ ($C_4\times S_3$), $C_3\times D_5$ ($C_3:D_4$), $C_3\times D_5$ ($D_{12}$)

Order 20: $D_{10}$ ($S_3^2$)

Order 18: $C_3\times C_6$ ($C_2\times F_5$)

Order 15: $C_{15}$ ($C_{12}:C_4$), $C_{15}$ ($C_6.D_4$)

Order 12: $C_3:C_4$ ($C_{15}:C_4$)

Order 10: $C_{10}$ ($C_6.D_6$), $D_5$ ($C_3^2:Q_8$), $D_5$ ($C_3:D_{12}$)

Order 9: $C_3^2$ ($C_{20}:C_4$)

Order 6: $C_6$ ($C_{30}:C_4$), $C_6$ ($S_3\times F_5$)

Order 5: $C_5$ ($C_6.D_{12}$)

Order 3: $C_3$ ($D_{10}.D_6$), $C_3$ ($C_{60}:C_4$)

Order 2: $C_2$ ($D_5.S_3^2$)

Order 1: $C_1$ ($D_{10}.S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 720: $D_{10}.S_3^2$ ($C_1$)

Order 360: $C_{30}.D_6$ ($C_2$), $C_2\times C_3^2:F_5$ ($C_2$), $C_{30}:C_{12}$ ($C_2$)

Order 180: $C_3^2\times D_{10}$ ($C_2^2$), $C_{15}:C_{12}$ ($C_4$), $C_3:C_{60}$ ($C_4$)

Order 120: $C_6.D_{10}$ ($S_3$), $C_{30}:C_4$ ($S_3$)

Order 90: $C_3^2\times D_5$ ($Q_8$), $C_3^2\times D_5$ ($D_4$), $C_3\times C_{30}$ ($C_2\times C_4$)

Order 60: $C_3\times D_{10}$ ($D_6$) x 2, $C_{15}:C_4$ ($C_3:C_4$), $C_3:C_{20}$ ($C_3:C_4$)

Order 45: $C_3\times C_{15}$ ($C_4:C_4$)

Order 36: $C_3:C_{12}$ ($F_5$)

Order 30: $C_3\times D_5$ ($C_3:Q_8$) x 2, $C_{30}$ ($C_6:C_4$), $C_{30}$ ($C_4\times S_3$), $C_3\times D_5$ ($C_3:D_4$), $C_3\times D_5$ ($D_{12}$)

Order 20: $D_{10}$ ($S_3^2$)

Order 18: $C_3\times C_6$ ($C_2\times F_5$)

Order 15: $C_{15}$ ($C_{12}:C_4$), $C_{15}$ ($C_6.D_4$)

Order 12: $C_3:C_4$ ($C_{15}:C_4$)

Order 10: $C_{10}$ ($C_6.D_6$), $D_5$ ($C_3^2:Q_8$), $D_5$ ($C_3:D_{12}$)

Order 9: $C_3^2$ ($C_{20}:C_4$)

Order 6: $C_6$ ($C_{30}:C_4$), $C_6$ ($S_3\times F_5$)

Order 5: $C_5$ ($C_6.D_{12}$)

Order 3: $C_3$ ($D_{10}.D_6$), $C_3$ ($C_{60}:C_4$)

Order 2: $C_2$ ($D_5.S_3^2$)

Order 1: $C_1$ ($D_{10}.S_3^2$)

Series

Derived series

$D_{10}.S_3^2$

$\rhd$

$C_3\times C_{30}$

$\rhd$

$C_1$

Chief series

$D_{10}.S_3^2$

$\rhd$

$C_{30}:C_{12}$

$\rhd$

$C_3^2\times D_{10}$

$\rhd$

$C_3^2\times D_5$

$\rhd$

$C_3\times C_{15}$

$\rhd$

$C_{15}$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_{10}.S_3^2$

$\rhd$

$C_3\times C_{30}$

$\rhd$

$C_3\times C_{15}$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 17 larger groups in the database.

This group is a maximal quotient of 8 larger groups in the database.

Character theory

Complex character table

See the $48 \times 48$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $33 \times 33$ rational character table.