Abstract group 400.185: $C_{20}:D_{10}$

• Label: 400.185
• Order: \( 2^{4} \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $14$
• Trans deg.: $40$
• Rank: $3$

Group information

Description:	$C_{20}:D_{10}$
Order:	\(400\)\(\medspace = 2^{4} \cdot 5^{2} \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	$D_4\times C_{10}:C_4^2$, of order \(1280\)\(\medspace = 2^{8} \cdot 5 \)
Composition factors:	$C_2$ x 4, $C_5$ 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	35	12	24	240	88	400
Conjugacy classes	1	7	2	14	58	18	100
Divisions	1	7	2	4	16	5	35
Autjugacy classes	1	4	2	3	8	4	22

Dimension	1	2	4	8	16
Irr. complex chars.	40	50	10	0	0	100
Irr. rational chars.	8	2	12	11	2	35

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$40$
Rank:	$3$
Inequivalent generating triples:	$15624$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	16
Arbitrary	4	6	10

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{10}=c^{20}=[a,b]=1, c^{a}=c^{9}, c^{b}=c^{11} \rangle$
Permutation group:	Degree $14$ $\langle(1,2)(3,4)(11,12)(13,14), (3,4), (1,3)(2,4), (5,6,7,8,9), (1,2)(3,4), (10,11,13,14,12)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 0 & 26 \\ 25 & 0 \end{array}\right), \left(\begin{array}{rr} 22 & 24 \\ 6 & 22 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/33\Z)$
Transitive group:	40T305				more information
Direct product:	$C_5$ $\, \times\, $ $D_4$ $\, \times\, $ $D_5$
Semidirect product:	$D_{20}$ $\,\rtimes\,$ $C_{10}$	$C_{20}$ $\,\rtimes\,$ $D_{10}$	$C_{10}^2$ $\,\rtimes\,$ $C_2^2$ (2)	$(D_5\times C_{20})$ $\,\rtimes\,$ $C_2$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{10}$ . $(C_2\times C_{10})$	$C_{10}$ . $(C_2\times D_{10})$	$C_2$ . $(C_{10}\times D_{10})$	$(C_5\times D_{10})$ . $C_2^2$	all 6

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

Homology

Abelianization:	$C_{2}^{2} \times C_{10} \simeq C_{2}^{3} \times C_{5}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 380 subgroups in 124 conjugacy classes, 50 normal (26 characteristic).

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

Special subgroups

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

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 400: $C_{20}:D_{10}$

Order 200: $C_{10}\times D_{10}$ x 2, $C_{10}\wr C_2$ x 2, $D_4\times C_5^2$, $C_5\times D_{20}$, $D_5\times C_{20}$

Order 100: $C_5\times D_{10}$ x 7, $C_{10}^2$ x 2, $C_5\times C_{20}$, $C_5:C_{20}$

Order 80: $D_4\times C_{10}$, $D_4\times D_5$

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

Order 40: $C_5\times D_4$ x 7, $C_5:D_4$ x 2, $C_2^2\times C_{10}$ x 2, $C_2\times D_{10}$ x 2, $C_2\times C_{20}$, $D_{20}$, $C_4\times D_5$

Order 25: $C_5^2$

Order 20: $C_2\times C_{10}$ x 15, $D_{10}$ x 7, $C_{20}$ x 5, $C_5:C_4$

Order 16: $C_2\times D_4$

Order 10: $C_{10}$ x 16, $D_5$ x 4

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

Order 5: $C_5$ x 4

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 400: $C_{20}:D_{10}$

Order 200: $C_{10}\times D_{10}$, $D_4\times C_5^2$, $C_{10}\wr C_2$, $C_5\times D_{20}$, $D_5\times C_{20}$

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

Order 80: $D_4\times C_{10}$, $D_4\times D_5$

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

Order 40: $C_5\times D_4$ x 5, $C_2\times C_{20}$, $C_5:D_4$, $D_{20}$, $C_4\times D_5$, $C_2^2\times C_{10}$, $C_2\times D_{10}$

Order 25: $C_5^2$

Order 20: $C_2\times C_{10}$ x 6, $C_{20}$ x 4, $D_{10}$ x 3, $C_5:C_4$

Order 16: $C_2\times D_4$

Order 10: $C_{10}$ x 8, $D_5$ x 2

Order 8: $D_4$ x 3, $C_2^3$, $C_2\times C_4$

Order 5: $C_5$ x 3

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

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 400: $C_{20}:D_{10}$ ($C_1$)

Order 200: $C_{10}\times D_{10}$ ($C_2$) x 2, $C_{10}\wr C_2$ ($C_2$) x 2, $D_4\times C_5^2$ ($C_2$), $C_5\times D_{20}$ ($C_2$), $D_5\times C_{20}$ ($C_2$)

Order 100: $C_5\times D_{10}$ ($C_2^2$) x 3, $C_{10}^2$ ($C_2^2$) x 2, $C_5\times C_{20}$ ($C_2^2$), $C_5:C_{20}$ ($C_2^2$)

Order 80: $D_4\times D_5$ ($C_5$)

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

Order 40: $C_5:D_4$ ($C_{10}$) x 2, $C_2\times D_{10}$ ($C_{10}$) x 2, $D_{20}$ ($C_{10}$), $C_4\times D_5$ ($C_{10}$), $C_5\times D_4$ ($C_{10}$), $C_5\times D_4$ ($D_5$)

Order 25: $C_5^2$ ($C_2\times D_4$)

Order 20: $D_{10}$ ($C_2\times C_{10}$) x 3, $C_2\times C_{10}$ ($C_2\times C_{10}$) x 2, $C_2\times C_{10}$ ($D_{10}$) x 2, $C_{20}$ ($C_2\times C_{10}$), $C_{20}$ ($D_{10}$), $C_5:C_4$ ($C_2\times C_{10}$)

Order 10: $D_5$ ($C_5\times D_4$) x 2, $C_{10}$ ($C_2^2\times C_{10}$), $C_{10}$ ($C_2\times D_{10}$)

Order 8: $D_4$ ($C_5\times D_5$)

Order 5: $C_5$ ($D_4\times C_{10}$), $C_5$ ($D_4\times D_5$)

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

Order 2: $C_2$ ($C_{10}\times D_{10}$)

Order 1: $C_1$ ($C_{20}:D_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 400: $C_{20}:D_{10}$ ($C_1$)

Order 200: $C_{10}\times D_{10}$ ($C_2$), $D_4\times C_5^2$ ($C_2$), $C_{10}\wr C_2$ ($C_2$), $C_5\times D_{20}$ ($C_2$), $D_5\times C_{20}$ ($C_2$)

Order 100: $C_5\times D_{10}$ ($C_2^2$) x 2, $C_5\times C_{20}$ ($C_2^2$), $C_5:C_{20}$ ($C_2^2$), $C_{10}^2$ ($C_2^2$)

Order 80: $D_4\times D_5$ ($C_5$)

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

Order 40: $C_5:D_4$ ($C_{10}$), $D_{20}$ ($C_{10}$), $C_4\times D_5$ ($C_{10}$), $C_2\times D_{10}$ ($C_{10}$), $C_5\times D_4$ ($C_{10}$), $C_5\times D_4$ ($D_5$)

Order 25: $C_5^2$ ($C_2\times D_4$)

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

Order 10: $C_{10}$ ($C_2^2\times C_{10}$), $C_{10}$ ($C_2\times D_{10}$), $D_5$ ($C_5\times D_4$)

Order 8: $D_4$ ($C_5\times D_5$)

Order 5: $C_5$ ($D_4\times C_{10}$), $C_5$ ($D_4\times D_5$)

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

Order 2: $C_2$ ($C_{10}\times D_{10}$)

Order 1: $C_1$ ($C_{20}:D_{10}$)

Series

Derived series

$C_{20}:D_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_1$

Chief series

$C_{20}:D_{10}$

$\rhd$

$D_5\times C_{20}$

$\rhd$

$C_5\times C_{20}$

$\rhd$

$C_{20}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{20}:D_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_{10}$

$\lhd$

$C_5\times D_4$

Supergroups

This group is a maximal subgroup of 44 larger groups in the database.

This group is a maximal quotient of 48 larger groups in the database.

Character theory

Complex character table

See the $100 \times 100$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $35 \times 35$ rational character table.