Abstract group 800.1199: $D_{10}^2.C_2$

• Label: 800.1199
• Order: \( 2^{5} \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $40$
• Rank: $4$

Group information

Description:	$D_{10}^2.C_2$
Order:	\(800\)\(\medspace = 2^{5} \cdot 5^{2} \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$C_{10}^2.A_4.C_4\wr C_2.C_2$, of order \(76800\)\(\medspace = 2^{10} \cdot 3 \cdot 5^{2} \)
Composition factors:	$C_2$ x 5, $C_5$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	10
Elements	1	143	400	24	232	800
Conjugacy classes	1	15	16	4	20	56
Divisions	1	15	8	4	20	48
Autjugacy classes	1	4	3	2	3	13

Dimension	1	2	4	8
Irr. complex chars.	32	0	16	8	56
Irr. rational chars.	16	8	16	8	48

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$40$
Rank:	$4$
Inequivalent generating quadruples:	$1614480$

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:	$\langle a, b, c, d \mid a^{2}=b^{4}=c^{10}=d^{10}=[a,b]=[a,c]=[a,d]=1, c^{b}=c^{7}d^{4}, d^{b}=d^{7}, d^{c}=d^{9} \rangle$
Permutation group:	Degree $14$ $\langle(2,3,4,5)(6,7)(8,9)(11,12,13,14), (2,4)(3,5)(6,7)(8,9), (6,8)(7,9), (6,7)(8,9), (2,4)(3,5)(11,13)(12,14), (1,2,5,3,4), (10,11,14,12,13)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 2 & 2 & 4 & 2 \\ 1 & 4 & 1 & 4 \\ 3 & 0 & 2 & 2 \\ 1 & 2 & 2 & 3 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 1 & 4 & 2 & 4 \\ 3 & 0 & 0 & 2 \\ 1 & 0 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 3 & 0 & 0 & 0 \\ 1 & 3 & 3 & 0 \\ 3 & 2 & 3 & 0 \\ 2 & 2 & 4 & 4 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 4 & 1 & 0 \\ 3 & 2 & 1 & 2 \\ 0 & 3 & 4 & 2 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 3 & 2 & 2 & 2 \\ 4 & 4 & 0 & 1 \\ 4 & 4 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 1 & 1 & 2 & 4 \\ 3 & 0 & 2 & 2 \\ 1 & 0 & 1 & 3 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
Transitive group:	40T865				more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(D_5:F_5)$
Semidirect product:	$(D_5\times D_{10})$ $\,\rtimes\,$ $C_4$	$(C_2\times D_{10})$ $\,\rtimes\,$ $F_5$	$D_{10}$ $\,\rtimes\,$ $(C_2\times F_5)$	$(C_{10}\times D_{10})$ $\,\rtimes\,$ $C_4$	all 22
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{10}^2$ . $C_2$	$D_5^2$ . $C_2^3$	$(C_5:D_5)$ . $C_2^4$	$(D_5\times D_{10})$ . $C_2^2$	all 6

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 4020 subgroups in 526 conjugacy classes, 155 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_5:F_5$
Commutator:	$G' \simeq$ $C_5^2$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_{10}^2.C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{10}^2$	$G/\operatorname{Fit} \simeq$ $C_2\times C_4$
Radical:	$R \simeq$ $D_{10}^2.C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{10}^2$	$G/\operatorname{soc} \simeq$ $C_2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\times C_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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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

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Classes of subgroups up to conjugation

Order 800: $D_{10}^2.C_2$

Order 400: $D_{10}:F_5$ x 12, $D_{10}^2$, $C_{10}^2:C_4$, $C_{10}^2:C_4$

Order 200: $D_5:F_5$ x 16, $D_5\times D_{10}$ x 12, $C_{10}:F_5$ x 6, $C_{10}:F_5$ x 6, $C_{10}\times D_{10}$ x 2, $C_{10}:D_{10}$

Order 160: $C_2^3\times F_5$ x 2

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

Order 80: $C_2^2\times F_5$ x 30, $C_2^2\times D_{10}$ x 2

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 68, $C_2\times D_{10}$ x 30, $C_2^2\times C_{10}$ x 2

Order 32: $C_2^3\times C_4$

Order 25: $C_5^2$

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

Order 16: $C_2^2\times C_4$ x 14, $C_2^4$

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

Order 8: $C_2\times C_4$ x 28, $C_2^3$ x 15

Order 5: $C_5$ x 4

Order 4: $C_2^2$ x 35, $C_4$ x 8

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 800: $D_{10}^2.C_2$

Order 400: $D_{10}^2$, $C_{10}^2:C_4$, $C_{10}^2:C_4$, $D_{10}:F_5$

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

Order 160: $C_2^3\times F_5$

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

Order 80: $C_2^2\times F_5$ x 4, $C_2^2\times D_{10}$

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

Order 40: $C_2\times D_{10}$ x 5, $C_2\times F_5$ x 4, $C_2^2\times C_{10}$

Order 32: $C_2^3\times C_4$

Order 25: $C_5^2$

Order 20: $D_{10}$ x 7, $C_2\times C_{10}$ x 3, $F_5$ x 3

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

Order 10: $D_5$ x 5, $C_{10}$ x 3

Order 8: $C_2^3$ x 4, $C_2\times C_4$ x 3

Order 5: $C_5$ x 2

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

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 800: $D_{10}^2.C_2$ ($C_1$)

Order 400: $D_{10}:F_5$ ($C_2$) x 12, $D_{10}^2$ ($C_2$), $C_{10}^2:C_4$ ($C_2$), $C_{10}^2:C_4$ ($C_2$)

Order 200: $D_5:F_5$ ($C_2^2$) x 16, $D_5\times D_{10}$ ($C_2^2$) x 6, $D_5\times D_{10}$ ($C_4$) x 6, $C_{10}:F_5$ ($C_2^2$) x 6, $C_{10}:F_5$ ($C_2^2$) x 6, $C_{10}\times D_{10}$ ($C_4$) x 2, $C_{10}:D_{10}$ ($C_2^2$)

Order 100: $C_5\times D_{10}$ ($C_2\times C_4$) x 12, $D_5^2$ ($C_2\times C_4$) x 12, $D_5^2$ ($C_2^3$) x 4, $C_5:F_5$ ($C_2^3$) x 4, $C_5:F_5$ ($C_2^3$) x 4, $C_5:D_{10}$ ($C_2^3$) x 3, $C_5:D_{10}$ ($C_2\times C_4$) x 3, $C_{10}^2$ ($C_2\times C_4$)

Order 50: $C_5\times D_5$ ($C_2^2\times C_4$) x 8, $C_5\times C_{10}$ ($C_2^2\times C_4$) x 3, $C_5:D_5$ ($C_2^2\times C_4$) x 3, $C_5:D_5$ ($C_2^4$)

Order 40: $C_2\times D_{10}$ ($F_5$) x 2

Order 25: $C_5^2$ ($C_2^3\times C_4$)

Order 20: $D_{10}$ ($C_2\times F_5$) x 12, $C_2\times C_{10}$ ($C_2\times F_5$) x 2

Order 10: $D_5$ ($C_2^2\times F_5$) x 8, $C_{10}$ ($C_2^2\times F_5$) x 6

Order 5: $C_5$ ($C_2^3\times F_5$) x 2

Order 4: $C_2^2$ ($D_5:F_5$)

Order 2: $C_2$ ($D_{10}:F_5$) x 3

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 800: $D_{10}^2.C_2$ ($C_1$)

Order 400: $D_{10}^2$ ($C_2$), $C_{10}^2:C_4$ ($C_2$), $C_{10}^2:C_4$ ($C_2$), $D_{10}:F_5$ ($C_2$)

Order 200: $C_{10}:D_{10}$ ($C_2^2$), $C_{10}\times D_{10}$ ($C_4$), $D_5\times D_{10}$ ($C_2^2$), $D_5\times D_{10}$ ($C_4$), $C_{10}:F_5$ ($C_2^2$), $C_{10}:F_5$ ($C_2^2$), $D_5:F_5$ ($C_2^2$)

Order 100: $C_{10}^2$ ($C_2\times C_4$), $C_5:D_{10}$ ($C_2^3$), $C_5:D_{10}$ ($C_2\times C_4$), $C_5\times D_{10}$ ($C_2\times C_4$), $D_5^2$ ($C_2^3$), $D_5^2$ ($C_2\times C_4$), $C_5:F_5$ ($C_2^3$), $C_5:F_5$ ($C_2^3$)

Order 50: $C_5\times C_{10}$ ($C_2^2\times C_4$), $C_5:D_5$ ($C_2^4$), $C_5:D_5$ ($C_2^2\times C_4$), $C_5\times D_5$ ($C_2^2\times C_4$)

Order 40: $C_2\times D_{10}$ ($F_5$)

Order 25: $C_5^2$ ($C_2^3\times C_4$)

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

Order 10: $C_{10}$ ($C_2^2\times F_5$), $D_5$ ($C_2^2\times F_5$)

Order 5: $C_5$ ($C_2^3\times F_5$)

Order 4: $C_2^2$ ($D_5:F_5$)

Order 2: $C_2$ ($D_{10}:F_5$)

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

Series

Derived series

$D_{10}^2.C_2$

$\rhd$

$C_5^2$

$\rhd$

$C_1$

Chief series

$D_{10}^2.C_2$

$\rhd$

$C_{10}^2:C_4$

$\rhd$

$C_{10}:D_{10}$

$\rhd$

$C_5:D_{10}$

$\rhd$

$C_5\times C_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$D_{10}^2.C_2$

$\rhd$

$C_5^2$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 39 larger groups in the database.

This group is a maximal quotient of 56 larger groups in the database.

Character theory

Complex character table

See the $56 \times 56$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $48 \times 48$ rational character table.