Abstract group 320.264: $D_{10}.Q_{16}$

• Label: 320.264
• Order: \( 2^{6} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $21$
• Trans deg.: $80$
• Rank: $2$

Group information

Description:	$D_{10}.Q_{16}$
Order:	\(320\)\(\medspace = 2^{6} \cdot 5 \)
Exponent:	\(40\)\(\medspace = 2^{3} \cdot 5 \)
Automorphism group:	$C_5:(C_2^3.C_2^5)$, of order \(1280\)\(\medspace = 2^{8} \cdot 5 \)
Composition factors:	$C_2$ x 6, $C_5$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	4	5	8	10	20
Elements	1	23	152	4	80	12	48	320
Conjugacy classes	1	5	9	1	4	3	6	29
Divisions	1	5	6	1	1	3	2	19
Autjugacy classes	1	5	7	1	2	3	2	21

Dimension	1	2	4	8	16
Irr. complex chars.	8	10	9	2	0	29
Irr. rational chars.	4	4	7	3	1	19

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$80$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	6	12	12

Constructions

Presentation:	$\langle a, b, c \mid a^{4}=b^{4}=c^{20}=1, b^{a}=b^{3}c^{5}, c^{a}=b^{2}c^{13}, c^{b}=c^{19} \rangle$
Permutation group:	Degree $21$ $\langle(2,6,7,14)(3,9)(8,10,15,16)(11,12)(18,19,20,21), (1,2,5,8)(3,10,12,6)(4,7,13,15) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 33 & 3 \\ 30 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 20 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 38 \\ 20 & 11 \end{array}\right), \left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 30 & 1 \end{array}\right), \left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/40\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2\times Q_8)$ $\,\rtimes\,$ $F_5$	$(Q_8\times C_{10})$ $\,\rtimes\,$ $C_4$	$(C_{20}:C_4)$ $\,\rtimes\,$ $C_4$	$C_5$ $\,\rtimes\,$ $(C_2^2.Q_{16})$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{10}$ . $Q_{16}$	$D_{10}$ . $\SD_{16}$	$C_2$ . $(Q_8:F_5)$	$(D_{10}:Q_8)$ . $C_2$	all 18

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 410 subgroups in 80 conjugacy classes, 24 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_{10}:C_4$
Commutator:	$G' \simeq$ $C_2\times C_{20}$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2\times F_5$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{10}$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $D_{10}.Q_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2^2:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2.Q_{16}$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 320: $D_{10}.Q_{16}$

Order 160: $C_{10}.C_4^2$, $D_{10}:C_8$, $D_{10}:Q_8$

Order 80: $C_2^2\times F_5$, $Q_8\times C_{10}$, $C_4\times D_{10}$, $C_{10}:C_8$, $D_{10}:C_4$, $C_{20}:C_4$, $C_{10}.D_4$

Order 64: $C_2^2.Q_{16}$

Order 40: $C_2\times F_5$ x 4, $C_2\times C_{20}$ x 2, $C_{10}:C_4$ x 2, $C_4\times D_5$, $C_5:C_8$, $C_2\times D_{10}$, $C_5\times Q_8$

Order 32: $C_2^2:C_8$, $C_2^2:Q_8$, $C_2.C_4^2$

Order 20: $D_{10}$ x 4, $F_5$ x 2, $C_{20}$ x 2, $C_5:C_4$ x 2, $C_2\times C_{10}$

Order 16: $C_4:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_2\times C_8$, $C_2^2:C_4$, $C_2\times Q_8$

Order 10: $C_{10}$ x 3, $D_5$ x 2

Order 8: $C_2\times C_4$ x 9, $C_2^3$, $Q_8$, $C_8$

Order 5: $C_5$

Order 4: $C_4$ x 6, $C_2^2$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 320: $D_{10}.Q_{16}$

Order 160: $C_{10}.C_4^2$, $D_{10}:C_8$, $D_{10}:Q_8$

Order 80: $C_2^2\times F_5$, $Q_8\times C_{10}$, $C_4\times D_{10}$, $C_{10}:C_8$, $D_{10}:C_4$, $C_{20}:C_4$, $C_{10}.D_4$

Order 64: $C_2^2.Q_{16}$

Order 40: $C_2\times F_5$ x 3, $C_2\times C_{20}$ x 2, $C_{10}:C_4$ x 2, $C_4\times D_5$, $C_5:C_8$, $C_2\times D_{10}$, $C_5\times Q_8$

Order 32: $C_2^2:C_8$, $C_2^2:Q_8$, $C_2.C_4^2$

Order 20: $D_{10}$ x 4, $C_{20}$ x 2, $C_5:C_4$ x 2, $C_2\times C_{10}$, $F_5$

Order 16: $C_4:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_2\times C_8$, $C_2^2:C_4$, $C_2\times Q_8$

Order 10: $C_{10}$ x 3, $D_5$ x 2

Order 8: $C_2\times C_4$ x 8, $C_2^3$, $Q_8$, $C_8$

Order 5: $C_5$

Order 4: $C_2^2$ x 5, $C_4$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 320: $D_{10}.Q_{16}$ ($C_1$)

Order 160: $C_{10}.C_4^2$ ($C_2$), $D_{10}:C_8$ ($C_2$), $D_{10}:Q_8$ ($C_2$)

Order 80: $Q_8\times C_{10}$ ($C_4$), $C_4\times D_{10}$ ($C_2^2$), $C_{20}:C_4$ ($C_4$)

Order 40: $C_2\times C_{20}$ ($C_2\times C_4$), $C_{10}:C_4$ ($D_4$), $C_2\times D_{10}$ ($D_4$)

Order 20: $C_2\times C_{10}$ ($C_2^2:C_4$), $D_{10}$ ($Q_{16}$), $D_{10}$ ($\SD_{16}$)

Order 16: $C_2\times Q_8$ ($F_5$)

Order 10: $C_{10}$ ($C_2^3:C_4$), $C_{10}$ ($C_4\wr C_2$), $C_{10}$ ($Q_8:C_4$)

Order 8: $C_2\times C_4$ ($C_2\times F_5$)

Order 5: $C_5$ ($C_2^2.Q_{16}$)

Order 4: $C_2^2$ ($D_{10}:C_4$)

Order 2: $C_2$ ($C_2^3:F_5$), $C_2$ ($D_{20}:C_4$), $C_2$ ($Q_8:F_5$)

Order 1: $C_1$ ($D_{10}.Q_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 320: $D_{10}.Q_{16}$ ($C_1$)

Order 160: $C_{10}.C_4^2$ ($C_2$), $D_{10}:C_8$ ($C_2$), $D_{10}:Q_8$ ($C_2$)

Order 80: $Q_8\times C_{10}$ ($C_4$), $C_4\times D_{10}$ ($C_2^2$), $C_{20}:C_4$ ($C_4$)

Order 40: $C_2\times C_{20}$ ($C_2\times C_4$), $C_{10}:C_4$ ($D_4$), $C_2\times D_{10}$ ($D_4$)

Order 20: $C_2\times C_{10}$ ($C_2^2:C_4$), $D_{10}$ ($Q_{16}$), $D_{10}$ ($\SD_{16}$)

Order 16: $C_2\times Q_8$ ($F_5$)

Order 10: $C_{10}$ ($C_2^3:C_4$), $C_{10}$ ($C_4\wr C_2$), $C_{10}$ ($Q_8:C_4$)

Order 8: $C_2\times C_4$ ($C_2\times F_5$)

Order 5: $C_5$ ($C_2^2.Q_{16}$)

Order 4: $C_2^2$ ($D_{10}:C_4$)

Order 2: $C_2$ ($C_2^3:F_5$), $C_2$ ($D_{20}:C_4$), $C_2$ ($Q_8:F_5$)

Order 1: $C_1$ ($D_{10}.Q_{16}$)

Series

Derived series

$D_{10}.Q_{16}$

$\rhd$

$C_2\times C_{20}$

$\rhd$

$C_1$

Chief series

$D_{10}.Q_{16}$

$\rhd$

$C_{10}.C_4^2$

$\rhd$

$C_4\times D_{10}$

$\rhd$

$C_2\times D_{10}$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$D_{10}.Q_{16}$

$\rhd$

$C_2\times C_{20}$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2\times Q_8$

Supergroups

This group is a maximal subgroup of 13 larger groups in the database.

This group is a maximal quotient of 13 larger groups in the database.

Character theory

Complex character table

See the $29 \times 29$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $19 \times 19$ rational character table.