Abstract group 400.39: $D_4\times D_{25}$

• Label: 400.39
• Order: \( 2^{4} \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 5^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 5^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 5 \)
• Perm deg.: $29$
• Trans deg.: $100$
• Rank: $3$

Group information

Description:	$D_4\times D_{25}$
Order:	\(400\)\(\medspace = 2^{4} \cdot 5^{2} \)
Exponent:	\(100\)\(\medspace = 2^{2} \cdot 5^{2} \)
Automorphism group:	$C_{50}.(C_2^3\times C_{20})$, of order \(8000\)\(\medspace = 2^{6} \cdot 5^{3} \)
Composition factors:	$C_2$ x 4, $C_5$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	4	5	10	20	25	50	100
Elements	1	155	52	4	20	8	20	100	40	400
Conjugacy classes	1	7	2	2	6	2	10	30	10	70
Divisions	1	7	2	1	3	1	1	3	1	20
Autjugacy classes	1	4	2	1	2	1	1	2	1	15

Dimension	1	2	4	8	20	40
Irr. complex chars.	8	50	12	0	0	0	70
Irr. rational chars.	8	2	4	1	4	1	20

Minimal presentations

Permutation degree:	$29$
Transitive degree:	$100$
Rank:	$3$
Inequivalent generating triples:	$2520$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	4	40
Arbitrary	4	4	22

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{100}=[a,b]=1, c^{a}=c^{49}, c^{b}=c^{51} \rangle$
Permutation group:	Degree $29$ $\langle(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,12)(14,20)(17,21)(18,22)(19,24)(23,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 26 & 0 \\ 0 & 26 \end{array}\right), \left(\begin{array}{rr} 1 & 50 \\ 50 & 26 \end{array}\right), \left(\begin{array}{rr} 46 & 18 \\ 30 & 46 \end{array}\right), \left(\begin{array}{rr} 26 & 0 \\ 25 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 49 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/75\Z)$
Direct product:	$D_4$ $\, \times\, $ $D_{25}$
Semidirect product:	$C_4$ $\,\rtimes\,$ $D_{50}$	$D_{100}$ $\,\rtimes\,$ $C_2$	$D_{50}$ $\,\rtimes\,$ $C_2^2$ (2)	$C_2^2$ $\,\rtimes\,$ $D_{50}$ (2)	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{20}$ . $D_{10}$	$D_{50}$ . $C_2^2$	$C_{50}$ . $C_2^3$	$(C_5\times D_4)$ . $D_5$	all 8

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 805 subgroups in 81 conjugacy classes, 31 normal (17 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2\times D_{50}$
Commutator:	$G' \simeq$ $C_{50}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_{10}$	$G/\Phi \simeq$ $C_2\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_{25}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_4\times D_{25}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{10}$	$G/\operatorname{soc} \simeq$ $C_2\times D_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$

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

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Classes of subgroups up to conjugation

Order 400: $D_4\times D_{25}$

Order 200: $C_{25}:D_4$ x 2, $C_2\times D_{50}$ x 2, $D_{100}$, $C_4\times D_{25}$, $D_4\times C_{25}$

Order 100: $D_{50}$ x 7, $C_2\times C_{50}$ x 2, $C_{100}$, $C_{25}:C_4$

Order 80: $D_4\times D_5$

Order 50: $D_{25}$ x 4, $C_{50}$ x 3

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

Order 25: $C_{25}$

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

Order 16: $C_2\times D_4$

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

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

Order 5: $C_5$

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: $D_4\times D_{25}$

Order 200: $C_{25}:D_4$, $D_{100}$, $C_4\times D_{25}$, $C_2\times D_{50}$, $D_4\times C_{25}$

Order 100: $D_{50}$ x 3, $C_2\times C_{50}$, $C_{100}$, $C_{25}:C_4$

Order 80: $D_4\times D_5$

Order 50: $C_{50}$ x 2, $D_{25}$ x 2

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

Order 25: $C_{25}$

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

Order 16: $C_2\times D_4$

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

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

Order 5: $C_5$

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: $D_4\times D_{25}$ ($C_1$)

Order 200: $C_{25}:D_4$ ($C_2$) x 2, $C_2\times D_{50}$ ($C_2$) x 2, $D_{100}$ ($C_2$), $C_4\times D_{25}$ ($C_2$), $D_4\times C_{25}$ ($C_2$)

Order 100: $D_{50}$ ($C_2^2$) x 3, $C_2\times C_{50}$ ($C_2^2$) x 2, $C_{100}$ ($C_2^2$), $C_{25}:C_4$ ($C_2^2$)

Order 50: $D_{25}$ ($D_4$) x 2, $C_{50}$ ($C_2^3$)

Order 40: $C_5\times D_4$ ($D_5$)

Order 25: $C_{25}$ ($C_2\times D_4$)

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

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

Order 8: $D_4$ ($D_{25}$)

Order 5: $C_5$ ($D_4\times D_5$)

Order 4: $C_2^2$ ($D_{50}$) x 2, $C_4$ ($D_{50}$)

Order 2: $C_2$ ($C_2\times D_{50}$)

Order 1: $C_1$ ($D_4\times D_{25}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 400: $D_4\times D_{25}$ ($C_1$)

Order 200: $C_{25}:D_4$ ($C_2$), $D_{100}$ ($C_2$), $C_4\times D_{25}$ ($C_2$), $C_2\times D_{50}$ ($C_2$), $D_4\times C_{25}$ ($C_2$)

Order 100: $D_{50}$ ($C_2^2$) x 2, $C_2\times C_{50}$ ($C_2^2$), $C_{100}$ ($C_2^2$), $C_{25}:C_4$ ($C_2^2$)

Order 50: $C_{50}$ ($C_2^3$), $D_{25}$ ($D_4$)

Order 40: $C_5\times D_4$ ($D_5$)

Order 25: $C_{25}$ ($C_2\times D_4$)

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

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

Order 8: $D_4$ ($D_{25}$)

Order 5: $C_5$ ($D_4\times D_5$)

Order 4: $C_2^2$ ($D_{50}$), $C_4$ ($D_{50}$)

Order 2: $C_2$ ($C_2\times D_{50}$)

Order 1: $C_1$ ($D_4\times D_{25}$)

Series

Derived series

$D_4\times D_{25}$

$\rhd$

$C_{50}$

$\rhd$

$C_1$

Chief series

$D_4\times D_{25}$

$\rhd$

$C_2\times D_{50}$

$\rhd$

$D_{50}$

$\rhd$

$C_{50}$

$\rhd$

$C_{25}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$D_4\times D_{25}$

$\rhd$

$C_{50}$

$\rhd$

$C_{25}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$D_4$

Supergroups

This group is a maximal subgroup of 25 larger groups in the database.

This group is a maximal quotient of 42 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 $20 \times 20$ rational character table.