Abstract group 1000.183: $D_5^3$

• Label: 1000.183
• Order: \( 2^{3} \cdot 5^{3} \)
• Exponent: \( 2 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $20$
• Rank: $3$

Group information

Description:	$D_5^3$
Order:	\(1000\)\(\medspace = 2^{3} \cdot 5^{3} \)
Exponent:	\(10\)\(\medspace = 2 \cdot 5 \)
Automorphism group:	$F_5\wr S_3$, of order \(48000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{3} \)
Composition factors:	$C_2$ x 3, $C_5$ x 3
Derived length:	$2$

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

Group statistics

Order	1	2	5	10
Elements	1	215	124	660	1000
Conjugacy classes	1	7	26	30	64
Divisions	1	7	13	15	36
Autjugacy classes	1	3	3	3	10

Dimension	1	2	4	8	16
Irr. complex chars.	8	24	24	8	0	64
Irr. rational chars.	8	0	12	12	4	36

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$20$
Rank:	$3$
Inequivalent generating triples:	$6048$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	16
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{10}=c^{10}=d^{5}=[a,c]=[a,d]=[b,d]=1, b^{a}=b^{9}, c^{b}=c^{9}, d^{c}=d^{4} \rangle$
Permutation group:	Degree $15$ $\langle(1,2)(3,5)(4,6)(7,9)(8,10), (12,13)(14,15), (1,2)(3,6)(4,5)(7,10)(8,9), (1,3,7,8,4)(2,5,9,10,6), (11,12,14,15,13), (1,4,8,7,3)(2,5,9,10,6)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 3 & 2 & 2 & 2 \\ 1 & 1 & 3 & 4 \\ 0 & 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 2 & 3 & 3 & 3 \\ 1 & 2 & 3 & 4 \\ 3 & 2 & 4 & 3 \end{array}\right), \left(\begin{array}{rrrr} 2 & 3 & 0 & 4 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 3 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 4 & 1 & 1 & 2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 3 & 1 & 1 & 3 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 4 & 4 \\ 3 & 2 & 2 & 2 \\ 4 & 3 & 2 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 3 & 2 & 1 & 1 \\ 0 & 4 & 0 & 0 \\ 3 & 2 & 3 & 2 \\ 1 & 0 & 2 & 3 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
Transitive group:	20T182	40T966			more information
Direct product:	$D_5$ ${}^3$
Semidirect product:	$C_5^3$ $\,\rtimes\,$ $C_2^3$	$(C_5\times D_5)$ $\,\rtimes\,$ $D_{10}$	$(C_5:D_5)$ $\,\rtimes\,$ $D_{10}$	$C_5$ $\,\rtimes\,$ $(D_5\times D_{10})$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 3256 subgroups in 204 conjugacy classes, 38 normal (5 characteristic).

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

Special subgroups

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

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

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 1000: $D_5^3$

Order 500: $C_5:D_5^2$ x 3, $C_5\times D_5^2$ x 3, $C_5:D_5^2$

Order 250: $C_5^2:C_{10}$ x 3, $D_5\times C_5^2$ x 3, $C_5^3:C_2$

Order 200: $D_5\times D_{10}$ x 3

Order 125: $C_5^3$

Order 100: $D_5^2$ x 18, $C_5\times D_{10}$ x 6, $C_5:D_{10}$ x 3

Order 50: $C_5\times D_5$ x 24, $C_5:D_5$ x 16, $C_5\times C_{10}$ x 3

Order 40: $C_2\times D_{10}$ x 3

Order 25: $C_5^2$ x 13

Order 20: $D_{10}$ x 24, $C_2\times C_{10}$ x 3

Order 10: $D_5$ x 28, $C_{10}$ x 15

Order 8: $C_2^3$

Order 5: $C_5$ x 13

Order 4: $C_2^2$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1000: $D_5^3$

Order 500: $C_5:D_5^2$, $C_5:D_5^2$, $C_5\times D_5^2$

Order 250: $C_5^3:C_2$, $C_5^2:C_{10}$, $D_5\times C_5^2$

Order 200: $D_5\times D_{10}$

Order 125: $C_5^3$

Order 100: $D_5^2$ x 4, $C_5:D_{10}$, $C_5\times D_{10}$

Order 50: $C_5:D_5$ x 4, $C_5\times D_5$ x 4, $C_5\times C_{10}$

Order 40: $C_2\times D_{10}$

Order 25: $C_5^2$ x 3

Order 20: $D_{10}$ x 5, $C_2\times C_{10}$

Order 10: $D_5$ x 6, $C_{10}$ x 3

Order 8: $C_2^3$

Order 5: $C_5$ x 3

Order 4: $C_2^2$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1000: $D_5^3$ ($C_1$)

Order 500: $C_5:D_5^2$ ($C_2$) x 3, $C_5\times D_5^2$ ($C_2$) x 3, $C_5:D_5^2$ ($C_2$)

Order 250: $C_5^2:C_{10}$ ($C_2^2$) x 3, $D_5\times C_5^2$ ($C_2^2$) x 3, $C_5^3:C_2$ ($C_2^2$)

Order 125: $C_5^3$ ($C_2^3$)

Order 100: $D_5^2$ ($D_5$) x 3

Order 50: $C_5\times D_5$ ($D_{10}$) x 6, $C_5:D_5$ ($D_{10}$) x 3

Order 25: $C_5^2$ ($C_2\times D_{10}$) x 3

Order 10: $D_5$ ($D_5^2$) x 3

Order 5: $C_5$ ($D_5\times D_{10}$) x 3

Order 1: $C_1$ ($D_5^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1000: $D_5^3$ ($C_1$)

Order 500: $C_5:D_5^2$ ($C_2$), $C_5:D_5^2$ ($C_2$), $C_5\times D_5^2$ ($C_2$)

Order 250: $C_5^3:C_2$ ($C_2^2$), $C_5^2:C_{10}$ ($C_2^2$), $D_5\times C_5^2$ ($C_2^2$)

Order 125: $C_5^3$ ($C_2^3$)

Order 100: $D_5^2$ ($D_5$)

Order 50: $C_5:D_5$ ($D_{10}$), $C_5\times D_5$ ($D_{10}$)

Order 25: $C_5^2$ ($C_2\times D_{10}$)

Order 10: $D_5$ ($D_5^2$)

Order 5: $C_5$ ($D_5\times D_{10}$)

Order 1: $C_1$ ($D_5^3$)

Series

Derived series

$D_5^3$

$\rhd$

$C_5^3$

$\rhd$

$C_1$

Chief series

$D_5^3$

$\rhd$

$C_5\times D_5^2$

$\rhd$

$D_5\times C_5^2$

$\rhd$

$C_5\times D_5$

$\rhd$

$C_5^2$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$D_5^3$

$\rhd$

$C_5^3$

Upper central series

$C_1$

Supergroups

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

This group is a maximal quotient of 24 larger groups in the database.

Character theory

Complex character table

See the $64 \times 64$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $36 \times 36$ rational character table.