Abstract group 1000.178: $D_5^2:C_{10}$

• Label: 1000.178
• Order: \( 2^{3} \cdot 5^{3} \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \)
• $\card{Z(G)}$: \( 5 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $15$
• Trans deg.: $20$
• Rank: $2$

Group information

Description:	$D_5^2:C_{10}$
Order:	\(1000\)\(\medspace = 2^{3} \cdot 5^{3} \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$C_4\times F_5\wr C_2$, of order \(3200\)\(\medspace = 2^{7} \cdot 5^{2} \)
Composition factors:	$C_2$ x 3, $C_5$ x 3
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	4	5	10	20
Elements	1	45	50	124	580	200	1000
Conjugacy classes	1	3	1	29	32	4	70
Divisions	1	3	1	9	9	1	24
Autjugacy classes	1	2	1	5	4	1	14

Dimension	1	2	4	8	16	32
Irr. complex chars.	20	5	40	5	0	0	70
Irr. rational chars.	4	1	4	6	8	1	24

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$20$
Rank:	$2$
Inequivalent generating pairs:	$108$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	16
Arbitrary	4	6	12

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{10}=c^{10}=d^{5}=[a,b]=1, c^{a}=b^{5}c^{7}d^{2}, d^{a}=c^{2}, c^{b}=c^{9}d^{3}, d^{b}=d^{4}, d^{c}=d^{4} \rangle$
Permutation group:	Degree $15$ $\langle(1,2)(3,5)(4,8)(6,10)(7,9), (1,2)(3,4)(5,7)(6,9)(8,10), (11,12,13,14,15), (3,6)(4,9)(5,10)(7,8), (2,4,5,10,9), (1,3,7,8,6)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 3 & 2 & 2 & 2 \\ 1 & 1 & 3 & 4 \\ 0 & 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 4 & 1 & 4 & 1 \\ 2 & 1 & 1 & 3 \\ 4 & 1 & 2 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 3 & 0 & 4 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 3 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 4 & 1 & 1 & 2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 3 & 1 & 1 & 3 \end{array}\right), \left(\begin{array}{rrrr} 4 & 1 & 4 & 0 \\ 3 & 1 & 4 & 2 \\ 2 & 3 & 0 & 3 \\ 1 & 0 & 2 & 3 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 2 & 4 & 1 & 3 \\ 4 & 2 & 1 & 1 \\ 0 & 1 & 3 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
Transitive group:	20T185	40T969	40T981		more information
Direct product:	$C_5$ $\, \times\, $ $(D_5\wr C_2)$
Semidirect product:	$C_5^3$ $\,\rtimes\,$ $D_4$	$D_5^2$ $\,\rtimes\,$ $C_{10}$ (2)	$(C_5:F_5)$ $\,\rtimes\,$ $C_{10}$	$(C_5\times D_5^2)$ $\,\rtimes\,$ $C_2$ (2)	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_5^2:C_{10})$ . $C_2^2$	$(C_5:D_5)$ . $(C_2\times C_{10})$			more information

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

Homology

Abelianization:	$C_{2} \times C_{10} \simeq C_{2}^{2} \times C_{5}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 636 subgroups in 76 conjugacy classes, 14 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_5$	$G/Z \simeq$ $D_5\wr C_2$
Commutator:	$G' \simeq$ $C_5:D_5$	$G/G' \simeq$ $C_2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_5^2:C_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_5^3$	$G/\operatorname{Fit} \simeq$ $D_4$
Radical:	$R \simeq$ $D_5^2:C_{10}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5^3$	$G/\operatorname{soc} \simeq$ $D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^3$

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 1000: $D_5^2:C_{10}$

Order 500: $C_5\times D_5^2$ x 2, $C_5^2:C_{20}$

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

Order 200: $D_5\wr C_2$

Order 125: $C_5^3$

Order 100: $C_5\times D_{10}$ x 2, $D_5^2$ x 2, $C_5\times F_5$, $C_5:F_5$

Order 50: $C_5\times D_5$ x 11, $C_5\times C_{10}$ x 2, $C_5:D_5$

Order 40: $C_5\times D_4$

Order 25: $C_5^2$ x 9

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

Order 10: $C_{10}$ x 9, $D_5$ x 5

Order 8: $D_4$

Order 5: $C_5$ x 9

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1000: $D_5^2:C_{10}$

Order 500: $C_5\times D_5^2$, $C_5^2:C_{20}$

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

Order 200: $D_5\wr C_2$

Order 125: $C_5^3$

Order 100: $C_5\times F_5$, $C_5\times D_{10}$, $D_5^2$, $C_5:F_5$

Order 50: $C_5\times D_5$ x 5, $C_5\times C_{10}$, $C_5:D_5$

Order 40: $C_5\times D_4$

Order 25: $C_5^2$ x 5

Order 20: $C_2\times C_{10}$, $D_{10}$, $F_5$, $C_{20}$

Order 10: $C_{10}$ x 4, $D_5$ x 3

Order 8: $D_4$

Order 5: $C_5$ x 5

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1000: $D_5^2:C_{10}$ ($C_1$)

Order 500: $C_5\times D_5^2$ ($C_2$) x 2, $C_5^2:C_{20}$ ($C_2$)

Order 250: $C_5^2:C_{10}$ ($C_2^2$)

Order 200: $D_5\wr C_2$ ($C_5$)

Order 125: $C_5^3$ ($D_4$)

Order 100: $D_5^2$ ($C_{10}$) x 2, $C_5:F_5$ ($C_{10}$)

Order 50: $C_5:D_5$ ($C_2\times C_{10}$)

Order 25: $C_5^2$ ($C_5\times D_4$)

Order 5: $C_5$ ($D_5\wr C_2$)

Order 1: $C_1$ ($D_5^2:C_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1000: $D_5^2:C_{10}$ ($C_1$)

Order 500: $C_5\times D_5^2$ ($C_2$), $C_5^2:C_{20}$ ($C_2$)

Order 250: $C_5^2:C_{10}$ ($C_2^2$)

Order 200: $D_5\wr C_2$ ($C_5$)

Order 125: $C_5^3$ ($D_4$)

Order 100: $D_5^2$ ($C_{10}$), $C_5:F_5$ ($C_{10}$)

Order 50: $C_5:D_5$ ($C_2\times C_{10}$)

Order 25: $C_5^2$ ($C_5\times D_4$)

Order 5: $C_5$ ($D_5\wr C_2$)

Order 1: $C_1$ ($D_5^2:C_{10}$)

Series

Derived series

$D_5^2:C_{10}$

$\rhd$

$C_5:D_5$

$\rhd$

$C_5^2$

$\rhd$

$C_1$

Chief series

$D_5^2:C_{10}$

$\rhd$

$C_5\times D_5^2$

$\rhd$

$C_5^2:C_{10}$

$\rhd$

$C_5:D_5$

$\rhd$

$C_5^2$

$\rhd$

$C_1$

Lower central series

$D_5^2:C_{10}$

$\rhd$

$C_5:D_5$

$\rhd$

$C_5^2$

Upper central series

$C_1$

$\lhd$

$C_5$

Supergroups

This group is a maximal subgroup of 11 larger groups in the database.

This group is a maximal quotient of 12 larger groups in the database.

Character theory

Complex character table

See the $70 \times 70$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $24 \times 24$ rational character table.