Abstract group 1600.9909: $D_{10}^2.C_2^2$

• Label: 1600.9909
• Order: \( 2^{6} \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $16$
• Trans deg.: $80$
• Rank: $4$

Group information

Description:	$D_{10}^2.C_2^2$
Order:	\(1600\)\(\medspace = 2^{6} \cdot 5^{2} \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$C_5:F_5.C_2^5.C_2^6$, of order \(204800\)\(\medspace = 2^{13} \cdot 5^{2} \)
Composition factors:	$C_2$ x 6, $C_5$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	10	20
Elements	1	143	944	24	232	256	1600
Conjugacy classes	1	15	24	4	20	24	88
Divisions	1	15	16	4	20	12	68
Autjugacy classes	1	6	6	2	5	3	23

Dimension	1	2	4	8	16
Irr. complex chars.	32	8	32	16	0	88
Irr. rational chars.	16	16	16	16	4	68

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$80$
Rank:	$4$
Inequivalent generating quadruples:	$9686880$

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:	$\langle a, b, c, d \mid a^{2}=b^{4}=c^{10}=d^{20}=[a,b]=[a,c]=[c,d]=1, d^{a}=c^{8}d^{9}, c^{b}=c^{3}, d^{b}=d^{3} \rangle$
Permutation group:	Degree $16$ $\langle(2,3,4,5)(6,7)(8,9)(13,14,15,16), (6,8)(7,9)(13,15)(14,16), (6,8)(7,9)(10,11) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(D_5^2.D_4)$
Semidirect product:	$D_5$ $\,\rtimes\,$ $(D_{10}.D_4)$	$(C_4\times D_{10})$ $\,\rtimes\,$ $F_5$	$C_4$ $\,\rtimes\,$ $(D_{10}:F_5)$	$(D_{10}.D_{10})$ $\,\rtimes\,$ $C_4$	all 37
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{10}^2$ . $C_2^2$	$(D_5\times D_{10})$ . $Q_8$	$(D_5\times D_{10})$ . $D_4$	$D_5^2$ . $(C_2\times Q_8)$	all 36

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

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 8748 subgroups in 928 conjugacy classes, 231 normal (21 characteristic).

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

Special subgroups

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

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

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Normal subgroups

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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 1600: $D_{10}^2.C_2^2$

Order 800: $D_5^2.D_4$ x 8, $D_{10}^2.C_2$ x 2, $C_{10}^2.(C_2\times C_4)$ x 2, $C_2.D_{10}^2$, $C_2\times C_{20}:F_5$, $C_2\times C_{20}:F_5$

Order 400: $D_{10}:F_5$ x 16, $C_4\times D_5^2$ x 8, $C_{10}.(C_2\times F_5)$ x 8, $C_{20}:F_5$ x 4, $C_{20}:F_5$ x 4, $C_{10}^2:C_4$ x 2, $C_{10}^2:C_4$ x 2, $C_{20}\times D_{10}$ x 2, $D_{10}.D_{10}$ x 2, $D_{10}^2$, $C_{20}:D_{10}$, $C_{10}^2.C_2^2$

Order 320: $D_{10}.C_2^4$ x 2

Order 200: $D_5:F_5$ x 16, $D_5\times D_{10}$ x 12, $C_{10}:F_5$ x 8, $C_{10}:F_5$ x 8, $D_5\times C_{20}$ x 8, $C_{10}.D_{10}$ x 8, $C_{20}:D_5$ x 4, $C_{10}.D_{10}$ x 4, $C_{10}\times D_{10}$ x 2, $C_{10}:C_{20}$ x 2, $C_{10}:D_{10}$, $C_{10}\times C_{20}$, $C_{10}.D_{10}$

Order 160: $D_{10}.D_4$ x 26, $C_2^3\times F_5$ x 4, $C_{20}:C_2^3$ x 2

Order 100: $D_5^2$ x 16, $C_5\times D_{10}$ x 12, $C_5:D_{10}$ x 6, $C_5:C_{20}$ x 4, $C_5:F_5$ x 4, $C_5:F_5$ x 4, $C_5\times C_{20}$ x 2, $C_5^2:C_4$ x 2, $C_{10}^2$

Order 80: $C_2^2\times F_5$ x 44, $C_{20}:C_4$ x 40, $C_4\times D_{10}$ x 26, $C_2^2\times D_{10}$ x 2, $C_2^2\times C_{20}$ x 2, $C_2^2.D_{10}$ x 2

Order 64: $C_2^3.D_4$

Order 50: $C_5\times D_5$ x 8, $C_5:D_5$ x 4, $C_5\times C_{10}$ x 3

Order 40: $C_2\times F_5$ x 80, $C_4\times D_5$ x 40, $C_2\times D_{10}$ x 30, $C_2\times C_{20}$ x 14, $C_{10}:C_4$ x 14, $C_2^2\times C_{10}$ x 2

Order 32: $C_2^2.D_4$ x 12, $C_2^3\times C_4$ x 3

Order 25: $C_5^2$

Order 20: $D_{10}$ x 68, $F_5$ x 24, $C_2\times C_{10}$ x 16, $C_{20}$ x 12, $C_5:C_4$ x 12

Order 16: $C_2^2\times C_4$ x 34, $C_4:C_4$ x 16, $C_2^4$

Order 10: $D_5$ x 24, $C_{10}$ x 20

Order 8: $C_2\times C_4$ x 60, $C_2^3$ x 15

Order 5: $C_5$ x 4

Order 4: $C_2^2$ x 35, $C_4$ x 16

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1600: $D_{10}^2.C_2^2$

Order 800: $D_{10}^2.C_2$, $C_2.D_{10}^2$, $C_2\times C_{20}:F_5$, $C_2\times C_{20}:F_5$, $C_{10}^2.(C_2\times C_4)$, $D_5^2.D_4$

Order 400: $D_{10}:F_5$ x 2, $C_4\times D_5^2$ x 2, $D_{10}^2$, $C_{10}^2:C_4$, $C_{10}^2:C_4$, $C_{20}:D_{10}$, $C_{20}\times D_{10}$, $C_{10}^2.C_2^2$, $D_{10}.D_{10}$, $C_{20}:F_5$, $C_{20}:F_5$, $C_{10}.(C_2\times F_5)$

Order 320: $D_{10}.C_2^4$

Order 200: $D_5\times D_{10}$ x 4, $C_{10}:F_5$ x 2, $C_{10}:F_5$ x 2, $C_{20}:D_5$ x 2, $C_{10}.D_{10}$ x 2, $C_{10}:D_{10}$, $C_{10}\times D_{10}$, $D_5:F_5$, $C_{10}\times C_{20}$, $C_{10}.D_{10}$, $C_{10}:C_{20}$, $D_5\times C_{20}$, $C_{10}.D_{10}$

Order 160: $D_{10}.D_4$ x 5, $C_2^3\times F_5$, $C_{20}:C_2^3$

Order 100: $C_5:D_{10}$ x 4, $D_5^2$ x 3, $C_5\times D_{10}$ x 2, $C_5\times C_{20}$, $C_5^2:C_4$, $C_5:C_{20}$, $C_{10}^2$, $C_5:F_5$, $C_5:F_5$

Order 80: $C_4\times D_{10}$ x 7, $C_2^2\times F_5$ x 5, $C_{20}:C_4$ x 4, $C_2^2\times D_{10}$, $C_2^2\times C_{20}$, $C_2^2.D_{10}$

Order 64: $C_2^3.D_4$

Order 50: $C_5:D_5$ x 3, $C_5\times C_{10}$ x 2, $C_5\times D_5$

Order 40: $C_4\times D_5$ x 8, $C_2\times D_{10}$ x 7, $C_2\times F_5$ x 7, $C_2\times C_{20}$ x 4, $C_{10}:C_4$ x 4, $C_2^2\times C_{10}$

Order 32: $C_2^2.D_4$ x 4, $C_2^3\times C_4$ x 2

Order 25: $C_5^2$

Order 20: $D_{10}$ x 13, $C_2\times C_{10}$ x 4, $F_5$ x 3, $C_{20}$ x 3, $C_5:C_4$ x 3

Order 16: $C_2^2\times C_4$ x 10, $C_4:C_4$ x 3, $C_2^4$

Order 10: $D_5$ x 7, $C_{10}$ x 5

Order 8: $C_2\times C_4$ x 14, $C_2^3$ x 6

Order 5: $C_5$ x 2

Order 4: $C_2^2$ x 10, $C_4$ x 5

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1600: $D_{10}^2.C_2^2$ ($C_1$)

Order 800: $D_5^2.D_4$ ($C_2$) x 8, $D_{10}^2.C_2$ ($C_2$) x 2, $C_{10}^2.(C_2\times C_4)$ ($C_2$) x 2, $C_2.D_{10}^2$ ($C_2$), $C_2\times C_{20}:F_5$ ($C_2$), $C_2\times C_{20}:F_5$ ($C_2$)

Order 400: $D_{10}:F_5$ ($C_2^2$) x 8, $C_{10}.(C_2\times F_5)$ ($C_2^2$) x 8, $C_4\times D_5^2$ ($C_2^2$) x 4, $C_4\times D_5^2$ ($C_4$) x 4, $C_{20}:F_5$ ($C_2^2$) x 4, $C_{20}:F_5$ ($C_2^2$) x 4, $C_{10}^2:C_4$ ($C_2^2$) x 2, $C_{10}^2:C_4$ ($C_2^2$) x 2, $C_{20}\times D_{10}$ ($C_4$) x 2, $D_{10}.D_{10}$ ($C_4$) x 2, $D_{10}^2$ ($C_2^2$), $C_{20}:D_{10}$ ($C_2^2$), $C_{10}^2.C_2^2$ ($C_2^2$)

Order 200: $D_5\times C_{20}$ ($C_2\times C_4$) x 8, $C_{10}.D_{10}$ ($C_2\times C_4$) x 8, $D_5\times D_{10}$ ($Q_8$) x 4, $D_5\times D_{10}$ ($D_4$) x 4, $C_{10}:F_5$ ($C_2^3$) x 4, $C_{10}:F_5$ ($C_2^3$) x 4, $C_{10}\times D_{10}$ ($C_2\times C_4$) x 2, $D_5\times D_{10}$ ($C_2^3$) x 2, $D_5\times D_{10}$ ($C_2\times C_4$) x 2, $C_{20}:D_5$ ($C_2^3$) x 2, $C_{20}:D_5$ ($C_2\times C_4$) x 2, $C_{10}:C_{20}$ ($C_2\times C_4$) x 2, $C_{10}.D_{10}$ ($C_2^3$) x 2, $C_{10}.D_{10}$ ($C_2\times C_4$) x 2, $C_{10}:D_{10}$ ($C_2^3$), $C_{10}\times C_{20}$ ($C_2\times C_4$), $C_{10}.D_{10}$ ($C_2\times C_4$)

Order 100: $C_5\times D_{10}$ ($C_4:C_4$) x 8, $D_5^2$ ($C_4:C_4$) x 8, $C_5:C_{20}$ ($C_2^2\times C_4$) x 4, $C_5\times D_{10}$ ($C_2^2\times C_4$) x 4, $D_5^2$ ($C_2\times Q_8$) x 4, $D_5^2$ ($C_2\times D_4$) x 4, $C_5\times C_{20}$ ($C_2^2\times C_4$) x 2, $C_5^2:C_4$ ($C_2^2\times C_4$) x 2, $C_5:D_{10}$ ($C_2\times Q_8$) x 2, $C_5:D_{10}$ ($C_2\times D_4$) x 2, $C_{10}^2$ ($C_2^2\times C_4$), $C_5:D_{10}$ ($C_2^4$), $C_5:D_{10}$ ($C_2^2\times C_4$)

Order 80: $C_4\times D_{10}$ ($F_5$) x 2

Order 50: $C_5\times D_5$ ($C_2^2.D_4$) x 8, $C_5\times C_{10}$ ($C_2^2.D_4$) x 2, $C_5:D_5$ ($C_2^2.D_4$) x 2, $C_5\times C_{10}$ ($C_2^3\times C_4$), $C_5:D_5$ ($C_2^2\times Q_8$), $C_5:D_5$ ($C_2^2\times D_4$)

Order 40: $C_4\times D_5$ ($C_2\times F_5$) x 8, $C_2\times C_{20}$ ($C_2\times F_5$) x 2, $C_{10}:C_4$ ($C_2\times F_5$) x 2, $C_2\times D_{10}$ ($C_2\times F_5$) x 2

Order 25: $C_5^2$ ($C_2^3.D_4$)

Order 20: $D_{10}$ ($C_{20}:C_4$) x 8, $D_{10}$ ($C_2^2\times F_5$) x 4, $C_{20}$ ($C_2^2\times F_5$) x 4, $C_5:C_4$ ($C_2^2\times F_5$) x 4, $C_2\times C_{10}$ ($C_2^2\times F_5$) x 2

Order 10: $D_5$ ($D_{10}.D_4$) x 8, $C_{10}$ ($D_{10}.D_4$) x 4, $C_{10}$ ($C_2^3\times F_5$) x 2

Order 8: $C_2\times C_4$ ($D_5:F_5$)

Order 5: $C_5$ ($D_{10}.C_2^4$) x 2

Order 4: $C_4$ ($D_{10}:F_5$) x 2, $C_2^2$ ($D_{10}:F_5$)

Order 2: $C_2$ ($D_5^2.D_4$) x 2, $C_2$ ($D_{10}^2.C_2$)

Order 1: $C_1$ ($D_{10}^2.C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1600: $D_{10}^2.C_2^2$ ($C_1$)

Order 800: $D_{10}^2.C_2$ ($C_2$), $C_2.D_{10}^2$ ($C_2$), $C_2\times C_{20}:F_5$ ($C_2$), $C_2\times C_{20}:F_5$ ($C_2$), $C_{10}^2.(C_2\times C_4)$ ($C_2$), $D_5^2.D_4$ ($C_2$)

Order 400: $D_{10}^2$ ($C_2^2$), $C_{10}^2:C_4$ ($C_2^2$), $C_{10}^2:C_4$ ($C_2^2$), $D_{10}:F_5$ ($C_2^2$), $C_{20}:D_{10}$ ($C_2^2$), $C_{20}\times D_{10}$ ($C_4$), $C_{10}^2.C_2^2$ ($C_2^2$), $D_{10}.D_{10}$ ($C_4$), $C_4\times D_5^2$ ($C_2^2$), $C_4\times D_5^2$ ($C_4$), $C_{20}:F_5$ ($C_2^2$), $C_{20}:F_5$ ($C_2^2$), $C_{10}.(C_2\times F_5)$ ($C_2^2$)

Order 200: $C_{10}:D_{10}$ ($C_2^3$), $C_{10}\times D_{10}$ ($C_2\times C_4$), $D_5\times D_{10}$ ($C_2^3$), $D_5\times D_{10}$ ($Q_8$), $D_5\times D_{10}$ ($D_4$), $D_5\times D_{10}$ ($C_2\times C_4$), $C_{10}:F_5$ ($C_2^3$), $C_{10}:F_5$ ($C_2^3$), $C_{10}\times C_{20}$ ($C_2\times C_4$), $C_{10}.D_{10}$ ($C_2\times C_4$), $C_{20}:D_5$ ($C_2^3$), $C_{20}:D_5$ ($C_2\times C_4$), $C_{10}:C_{20}$ ($C_2\times C_4$), $D_5\times C_{20}$ ($C_2\times C_4$), $C_{10}.D_{10}$ ($C_2^3$), $C_{10}.D_{10}$ ($C_2\times C_4$), $C_{10}.D_{10}$ ($C_2\times C_4$)

Order 100: $C_5\times C_{20}$ ($C_2^2\times C_4$), $C_5^2:C_4$ ($C_2^2\times C_4$), $C_5:C_{20}$ ($C_2^2\times C_4$), $C_{10}^2$ ($C_2^2\times C_4$), $C_5:D_{10}$ ($C_2^4$), $C_5:D_{10}$ ($C_2\times Q_8$), $C_5:D_{10}$ ($C_2\times D_4$), $C_5:D_{10}$ ($C_2^2\times C_4$), $C_5\times D_{10}$ ($C_4:C_4$), $C_5\times D_{10}$ ($C_2^2\times C_4$), $D_5^2$ ($C_4:C_4$), $D_5^2$ ($C_2\times Q_8$), $D_5^2$ ($C_2\times D_4$)

Order 80: $C_4\times D_{10}$ ($F_5$)

Order 50: $C_5\times C_{10}$ ($C_2^3\times C_4$), $C_5\times C_{10}$ ($C_2^2.D_4$), $C_5:D_5$ ($C_2^2\times Q_8$), $C_5:D_5$ ($C_2^2\times D_4$), $C_5:D_5$ ($C_2^2.D_4$), $C_5\times D_5$ ($C_2^2.D_4$)

Order 40: $C_2\times C_{20}$ ($C_2\times F_5$), $C_{10}:C_4$ ($C_2\times F_5$), $C_4\times D_5$ ($C_2\times F_5$), $C_2\times D_{10}$ ($C_2\times F_5$)

Order 25: $C_5^2$ ($C_2^3.D_4$)

Order 20: $C_2\times C_{10}$ ($C_2^2\times F_5$), $D_{10}$ ($C_2^2\times F_5$), $D_{10}$ ($C_{20}:C_4$), $C_{20}$ ($C_2^2\times F_5$), $C_5:C_4$ ($C_2^2\times F_5$)

Order 10: $C_{10}$ ($C_2^3\times F_5$), $C_{10}$ ($D_{10}.D_4$), $D_5$ ($D_{10}.D_4$)

Order 8: $C_2\times C_4$ ($D_5:F_5$)

Order 5: $C_5$ ($D_{10}.C_2^4$)

Order 4: $C_2^2$ ($D_{10}:F_5$), $C_4$ ($D_{10}:F_5$)

Order 2: $C_2$ ($D_{10}^2.C_2$), $C_2$ ($D_5^2.D_4$)

Order 1: $C_1$ ($D_{10}^2.C_2^2$)

Series

Derived series

$D_{10}^2.C_2^2$

$\rhd$

$C_5\times C_{10}$

$\rhd$

$C_1$

Chief series

$D_{10}^2.C_2^2$

$\rhd$

$D_{10}^2.C_2$

$\rhd$

$D_{10}:F_5$

$\rhd$

$D_5\times D_{10}$

$\rhd$

$C_5:D_{10}$

$\rhd$

$C_5:D_5$

$\rhd$

$C_5^2$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$D_{10}^2.C_2^2$

$\rhd$

$C_5\times C_{10}$

$\rhd$

$C_5^2$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $88 \times 88$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $68 \times 68$ rational character table.