Abstract group 640.19150: $D_{10}.(C_4\times D_4)$

• Label: 640.19150
• Order: \( 2^{7} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $15$
• Trans deg.: $80$
• Rank: $4$

Group information

Description:	$D_{10}.(C_4\times D_4)$
Order:	\(640\)\(\medspace = 2^{7} \cdot 5 \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$C_5:(C_2^6.C_2^5.C_2^3)$, of order \(81920\)\(\medspace = 2^{14} \cdot 5 \)
Composition factors:	$C_2$ x 7, $C_5$
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	10	20
Elements	1	95	416	4	60	64	640
Conjugacy classes	1	23	32	1	11	8	76
Divisions	1	23	20	1	11	4	60
Autjugacy classes	1	9	8	1	4	1	24

Dimension	1	2	4	8
Irr. complex chars.	32	24	16	4	76
Irr. rational chars.	16	24	12	8	60

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$80$
Rank:	$4$
Inequivalent generating quadruples:	$624960$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{2}=d^{4}=e^{20}=[a,b]=[a,c]=[b,c]=[b,d]=[b,e]=[c,d]=[c,e]=1, d^{a}=bd, e^{a}=be, e^{d}=e^{3} \rangle$
Permutation group:	Degree $15$ $\langle(2,3,5,4)(6,7)(8,9)(10,13)(11,12)(14,15), (2,4,5,3)(9,12)(13,15), (6,7) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 5 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 10 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_{10}.(C_4\times D_4))$
Semidirect product:	$(D_{10}.D_4)$ $\,\rtimes\,$ $C_4$	$D_5$ $\,\rtimes\,$ $(C_2^3.D_4)$	$C_5$ $\,\rtimes\,$ $(C_2^4.D_4)$	$(C_2^3:C_4)$ $\,\rtimes\,$ $F_5$	all 29
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{10}$ . $(C_4\times D_4)$	$D_5$ . $(C_2^3:Q_8)$	$D_5$ . $(C_2^3:D_4)$	$D_5$ . $(C_2^3.D_4)$	all 46

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{20}\Z)$.

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 5744 subgroups in 1232 conjugacy classes, 274 normal (28 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_2^2\times F_5$
Commutator:	$G' \simeq$ $C_2\times C_{10}$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^3\times F_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3:C_{20}$	$G/\operatorname{Fit} \simeq$ $C_4$
Radical:	$R \simeq$ $D_{10}.(C_4\times D_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.D_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 640: $D_{10}.(C_4\times D_4)$

Order 320: $C_{10}.(C_4\times D_4)$ x 8, $D_{10}.C_2^4$ x 2, $D_{10}.C_4^2$ x 2, $C_2^4\times F_5$, $C_2^4:F_5$, $C_2^4.D_{10}$

Order 160: $C_2^3\times F_5$ x 20, $D_{10}.D_4$ x 16, $C_{10}.C_4^2$ x 8, $C_2^3:F_5$ x 8, $D_{10}.D_4$ x 8, $C_{20}:C_2^3$ x 2, $C_2^2.D_{20}$ x 2, $C_2^3\times D_{10}$, $C_2^3:C_{20}$, $C_2^3.D_{10}$

Order 128: $C_2^4.D_4$

Order 80: $C_2^2\times F_5$ x 92, $C_2^2\times D_{10}$ x 22, $C_4\times D_{10}$ x 16, $C_{20}:C_4$ x 16, $D_{10}:C_4$ x 8, $D_{10}:C_4$ x 8, $C_2^2:C_{20}$ x 4, $C_{10}.D_4$ x 4, $C_2^2\times C_{20}$ x 2, $C_2^2.D_{10}$ x 2, $C_2^3\times C_{10}$

Order 64: $C_2^3.D_4$ x 8, $C_2^2.C_4^2$ x 2, $C_2^3.D_4$ x 2, $C_2^4:C_4$ x 2, $C_2^4\times C_4$

Order 40: $C_2\times D_{10}$ x 88, $C_2\times F_5$ x 84, $C_4\times D_5$ x 16, $C_2^2\times C_{10}$ x 11, $C_2\times C_{20}$ x 8, $C_{10}:C_4$ x 8

Order 32: $C_2^3\times C_4$ x 22, $C_2^3:C_4$ x 20, $C_2^2.D_4$ x 16, $C_2.C_4^2$ x 8, $C_2^5$

Order 20: $D_{10}$ x 76, $C_2\times C_{10}$ x 23, $F_5$ x 12, $C_{20}$ x 4, $C_5:C_4$ x 4

Order 16: $C_2^2\times C_4$ x 112, $C_2^2:C_4$ x 24, $C_2^4$ x 23, $C_4:C_4$ x 16

Order 10: $D_5$ x 12, $C_{10}$ x 11

Order 8: $C_2\times C_4$ x 116, $C_2^3$ x 99

Order 5: $C_5$

Order 4: $C_2^2$ x 99, $C_4$ x 20

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 640: $D_{10}.(C_4\times D_4)$

Order 320: $C_2^4\times F_5$, $C_2^4:F_5$, $D_{10}.C_2^4$, $C_2^4.D_{10}$, $D_{10}.C_4^2$, $C_{10}.(C_4\times D_4)$

Order 160: $C_2^3\times F_5$ x 6, $C_2^3:F_5$ x 2, $D_{10}.D_4$ x 2, $D_{10}.D_4$ x 2, $C_{10}.C_4^2$, $C_2^3\times D_{10}$, $C_{20}:C_2^3$, $C_2^3:C_{20}$, $C_2^3.D_{10}$, $C_2^2.D_{20}$

Order 128: $C_2^4.D_4$

Order 80: $C_2^2\times F_5$ x 13, $C_2^2\times D_{10}$ x 8, $C_4\times D_{10}$ x 4, $C_2^3\times C_{10}$, $C_2^2\times C_{20}$, $C_2^2.D_{10}$, $D_{10}:C_4$, $C_{20}:C_4$, $C_2^2:C_{20}$, $C_{10}.D_4$, $D_{10}:C_4$

Order 64: $C_2^4:C_4$ x 2, $C_2^3.D_4$, $C_2^2.C_4^2$, $C_2^4\times C_4$, $C_2^3.D_4$

Order 40: $C_2\times D_{10}$ x 18, $C_2\times F_5$ x 11, $C_2^2\times C_{10}$ x 4, $C_4\times D_5$ x 3, $C_2\times C_{20}$ x 2, $C_{10}:C_4$ x 2

Order 32: $C_2^3\times C_4$ x 7, $C_2^3:C_4$ x 7, $C_2^2.D_4$ x 2, $C_2^5$, $C_2.C_4^2$

Order 20: $D_{10}$ x 16, $C_2\times C_{10}$ x 6, $F_5$ x 3, $C_{20}$, $C_5:C_4$

Order 16: $C_2^2\times C_4$ x 19, $C_2^4$ x 9, $C_2^2:C_4$ x 4, $C_4:C_4$

Order 10: $D_5$ x 5, $C_{10}$ x 4

Order 8: $C_2^3$ x 22, $C_2\times C_4$ x 18

Order 5: $C_5$

Order 4: $C_2^2$ x 22, $C_4$ x 5

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 640: $D_{10}.(C_4\times D_4)$ ($C_1$)

Order 320: $C_{10}.(C_4\times D_4)$ ($C_2$) x 8, $D_{10}.C_2^4$ ($C_2$) x 2, $D_{10}.C_4^2$ ($C_2$) x 2, $C_2^4\times F_5$ ($C_2$), $C_2^4:F_5$ ($C_2$), $C_2^4.D_{10}$ ($C_2$)

Order 160: $C_{10}.C_4^2$ ($C_2^2$) x 8, $C_2^3\times F_5$ ($C_2^2$) x 8, $D_{10}.D_4$ ($C_2^2$) x 8, $C_2^3:F_5$ ($C_2^2$) x 4, $D_{10}.D_4$ ($C_2^2$) x 4, $D_{10}.D_4$ ($C_4$) x 4, $C_{20}:C_2^3$ ($C_2^2$) x 2, $C_2^2.D_{20}$ ($C_4$) x 2, $C_2^3\times D_{10}$ ($C_2^2$), $C_2^3:C_{20}$ ($C_4$), $C_2^3.D_{10}$ ($C_4$)

Order 80: $C_2^2\times F_5$ ($C_2^3$) x 8, $C_2^2\times F_5$ ($D_4$) x 8, $D_{10}:C_4$ ($C_2\times C_4$) x 8, $C_2^2\times D_{10}$ ($Q_8$) x 4, $C_2^2\times D_{10}$ ($D_4$) x 4, $C_4\times D_{10}$ ($C_2^3$) x 4, $C_4\times D_{10}$ ($C_2\times C_4$) x 4, $C_2^2:C_{20}$ ($C_2\times C_4$) x 4, $C_{10}.D_4$ ($C_2\times C_4$) x 4, $C_2^2\times D_{10}$ ($C_2^3$) x 3, $C_2^2\times D_{10}$ ($C_2\times C_4$) x 3, $C_2^2\times C_{20}$ ($C_2\times C_4$) x 2, $C_2^2.D_{10}$ ($C_2\times C_4$) x 2, $C_2^3\times C_{10}$ ($C_2\times C_4$)

Order 40: $C_2\times D_{10}$ ($C_4:C_4$) x 12, $C_2\times D_{10}$ ($C_2\times D_4$) x 10, $C_2\times F_5$ ($C_2\times D_4$) x 8, $C_2\times D_{10}$ ($C_2\times Q_8$) x 6, $C_2\times C_{20}$ ($C_2^2\times C_4$) x 4, $C_{10}:C_4$ ($C_2^2\times C_4$) x 4, $C_2^2\times C_{10}$ ($C_4:C_4$) x 4, $C_2\times D_{10}$ ($D_4:C_2$) x 4, $C_2^2\times C_{10}$ ($C_2^2\times C_4$) x 3, $C_2\times D_{10}$ ($C_2^2\times C_4$) x 3, $C_2\times D_{10}$ ($C_2^4$)

Order 32: $C_2^3:C_4$ ($F_5$)

Order 20: $D_{10}$ ($C_2^2:Q_8$) x 8, $C_2\times C_{10}$ ($C_2^2.D_4$) x 6, $D_{10}$ ($C_2^2.D_4$) x 6, $C_2\times C_{10}$ ($C_4\times D_4$) x 4, $D_{10}$ ($C_2^2.D_4$) x 4, $D_{10}$ ($C_2^2\wr C_2$) x 4, $D_{10}$ ($C_4\times D_4$) x 4, $D_{10}$ ($C_2^2\times D_4$) x 3, $D_{10}$ ($D_4:C_2^2$) x 2, $C_2\times C_{10}$ ($C_2^3\times C_4$), $D_{10}$ ($C_2^2\times Q_8$)

Order 16: $C_2^2:C_4$ ($C_2\times F_5$) x 4, $C_2^2\times C_4$ ($C_2\times F_5$) x 2, $C_2^4$ ($C_2\times F_5$)

Order 10: $C_{10}$ ($C_2^3.D_4$) x 4, $D_5$ ($C_2^3.D_4$) x 4, $C_{10}$ ($C_4^2:C_2^2$) x 2, $D_5$ ($C_2^3:Q_8$) x 2, $C_{10}$ ($C_2^3.D_4$), $D_5$ ($C_2^3.D_4$), $D_5$ ($C_2^3:D_4$)

Order 8: $C_2^3$ ($C_{20}:C_4$) x 4, $C_2\times C_4$ ($C_2^2\times F_5$) x 4, $C_2^3$ ($C_2^2\times F_5$) x 3

Order 5: $C_5$ ($C_2^4.D_4$)

Order 4: $C_2^2$ ($D_{10}.D_4$) x 6, $C_2^2$ ($D_4\times F_5$) x 4, $C_2^2$ ($C_2^3\times F_5$)

Order 2: $C_2$ ($C_{10}.(C_4\times D_4)$) x 4, $C_2$ ($D_{10}.C_2^4$) x 2, $C_2$ ($D_{10}.C_2^4$)

Order 1: $C_1$ ($D_{10}.(C_4\times D_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 640: $D_{10}.(C_4\times D_4)$ ($C_1$)

Order 320: $C_2^4\times F_5$ ($C_2$), $C_2^4:F_5$ ($C_2$), $D_{10}.C_2^4$ ($C_2$), $C_2^4.D_{10}$ ($C_2$), $D_{10}.C_4^2$ ($C_2$), $C_{10}.(C_4\times D_4)$ ($C_2$)

Order 160: $C_2^3\times F_5$ ($C_2^2$) x 4, $C_{10}.C_4^2$ ($C_2^2$), $C_2^3\times D_{10}$ ($C_2^2$), $C_{20}:C_2^3$ ($C_2^2$), $C_2^3:F_5$ ($C_2^2$), $D_{10}.D_4$ ($C_2^2$), $C_2^3:C_{20}$ ($C_4$), $C_2^3.D_{10}$ ($C_4$), $C_2^2.D_{20}$ ($C_4$), $D_{10}.D_4$ ($C_2^2$), $D_{10}.D_4$ ($C_4$)

Order 80: $C_2^2\times F_5$ ($C_2^3$) x 3, $C_2^2\times D_{10}$ ($C_2^3$) x 2, $C_2^2\times D_{10}$ ($C_2\times C_4$) x 2, $C_2^3\times C_{10}$ ($C_2\times C_4$), $C_2^2\times D_{10}$ ($Q_8$), $C_2^2\times D_{10}$ ($D_4$), $C_2^2\times F_5$ ($D_4$), $C_2^2\times C_{20}$ ($C_2\times C_4$), $C_2^2.D_{10}$ ($C_2\times C_4$), $C_4\times D_{10}$ ($C_2^3$), $C_4\times D_{10}$ ($C_2\times C_4$), $C_2^2:C_{20}$ ($C_2\times C_4$), $C_{10}.D_4$ ($C_2\times C_4$), $D_{10}:C_4$ ($C_2\times C_4$)

Order 40: $C_2\times D_{10}$ ($C_2\times D_4$) x 3, $C_2^2\times C_{10}$ ($C_2^2\times C_4$) x 2, $C_2\times D_{10}$ ($C_4:C_4$) x 2, $C_2\times D_{10}$ ($C_2\times Q_8$) x 2, $C_2\times D_{10}$ ($C_2^2\times C_4$) x 2, $C_2\times C_{20}$ ($C_2^2\times C_4$), $C_{10}:C_4$ ($C_2^2\times C_4$), $C_2^2\times C_{10}$ ($C_4:C_4$), $C_2\times D_{10}$ ($C_2^4$), $C_2\times D_{10}$ ($D_4:C_2$), $C_2\times F_5$ ($C_2\times D_4$)

Order 32: $C_2^3:C_4$ ($F_5$)

Order 20: $C_2\times C_{10}$ ($C_2^2.D_4$) x 2, $D_{10}$ ($C_2^2\times D_4$) x 2, $D_{10}$ ($C_2^2.D_4$) x 2, $C_2\times C_{10}$ ($C_2^3\times C_4$), $C_2\times C_{10}$ ($C_4\times D_4$), $D_{10}$ ($D_4:C_2^2$), $D_{10}$ ($C_2^2\times Q_8$), $D_{10}$ ($C_2^2.D_4$), $D_{10}$ ($C_2^2:Q_8$), $D_{10}$ ($C_2^2\wr C_2$), $D_{10}$ ($C_4\times D_4$)

Order 16: $C_2^2:C_4$ ($C_2\times F_5$), $C_2^4$ ($C_2\times F_5$), $C_2^2\times C_4$ ($C_2\times F_5$)

Order 10: $C_{10}$ ($C_2^3.D_4$), $C_{10}$ ($C_4^2:C_2^2$), $C_{10}$ ($C_2^3.D_4$), $D_5$ ($C_2^3.D_4$), $D_5$ ($C_2^3.D_4$), $D_5$ ($C_2^3:Q_8$), $D_5$ ($C_2^3:D_4$)

Order 8: $C_2^3$ ($C_2^2\times F_5$) x 2, $C_2^3$ ($C_{20}:C_4$), $C_2\times C_4$ ($C_2^2\times F_5$)

Order 5: $C_5$ ($C_2^4.D_4$)

Order 4: $C_2^2$ ($D_{10}.D_4$) x 2, $C_2^2$ ($C_2^3\times F_5$), $C_2^2$ ($D_4\times F_5$)

Order 2: $C_2$ ($D_{10}.C_2^4$), $C_2$ ($D_{10}.C_2^4$), $C_2$ ($C_{10}.(C_4\times D_4)$)

Order 1: $C_1$ ($D_{10}.(C_4\times D_4)$)

Series

Derived series

$D_{10}.(C_4\times D_4)$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_1$

Chief series

$D_{10}.(C_4\times D_4)$

$\rhd$

$C_2^4:F_5$

$\rhd$

$C_2^3\times D_{10}$

$\rhd$

$C_2^2\times D_{10}$

$\rhd$

$C_2\times D_{10}$

$\rhd$

$D_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$D_{10}.(C_4\times D_4)$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^3:C_4$

Supergroups

This group is a maximal subgroup of 9 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $76 \times 76$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $60 \times 60$ rational character table.