Abstract group 12450.a: $D_{25}\times C_{249}$

• Label: 12450.a
• Order: \( 2 \cdot 3 \cdot 5^{2} \cdot 83 \)
• Exponent: \( 2 \cdot 3 \cdot 5^{2} \cdot 83 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \cdot 83 \)
• $\card{Z(G)}$: \( 3 \cdot 83 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 5^{3} \cdot 41 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 5 \cdot 41 \)
• Perm deg.: $111$
• Trans deg.: $6225$
• Rank: $2$

Group information

Description:	$D_{25}\times C_{249}$
Order:	\(12450\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \cdot 83 \)
Exponent:	\(12450\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \cdot 83 \)
Automorphism group:	$C_2\times C_{82}\times D_{25}.C_{10}$, of order \(82000\)\(\medspace = 2^{4} \cdot 5^{3} \cdot 41 \)
Composition factors:	$C_2$, $C_3$, $C_5$ x 2, $C_{83}$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.

Group statistics

Order	1	2	3	5	6	15	25	75	83	166	249	415	498	1245	2075	6225
Elements	1	25	2	4	50	8	20	40	82	2050	164	328	4100	656	1640	3280	12450
Conjugacy classes	1	1	2	2	2	4	10	20	82	82	164	164	164	328	820	1640	3486
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	16
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	16

Minimal presentations

Permutation degree:	$111$
Transitive degree:	$6225$
Rank:	$2$
Inequivalent generating pairs:	$1008$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{6225}=1, b^{a}=b^{499} \rangle$
Permutation group:	Degree $111$ $\langle(1,2,3,5,7,9,11,13,15,17,19,21,23,25,24,22,20,18,16,14,12,10,8,6,4)(26,27,28) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 155 & 3 \\ 143 & 155 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 498 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{499})$
Direct product:	$C_3$ $\, \times\, $ $C_{83}$ $\, \times\, $ $D_{25}$
Semidirect product:	$C_{6225}$ $\,\rtimes\,$ $C_2$	$C_{2075}$ $\,\rtimes\,$ $C_6$	$C_{75}$ $\,\rtimes\,$ $C_{166}$	$C_{25}$ $\,\rtimes\,$ $C_{498}$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{1245}$ . $D_5$	$C_{415}$ . $(C_3\times D_5)$	$C_{15}$ . $(D_5\times C_{83})$	$C_5$ . $(D_5\times C_{249})$	more information

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

Homology

Abelianization:	$C_{498} \simeq C_{2} \times C_{3} \times C_{83}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 136 subgroups in 24 conjugacy classes, 16 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{249}$	$G/Z \simeq$ $D_{25}$
Commutator:	$G' \simeq$ $C_{25}$	$G/G' \simeq$ $C_{498}$
Frattini:	$\Phi \simeq$ $C_5$	$G/\Phi \simeq$ $D_5\times C_{249}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{6225}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{25}\times C_{249}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{1245}$	$G/\operatorname{soc} \simeq$ $D_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$
83-Sylow subgroup:	$P_{ 83 } \simeq$ $C_{83}$

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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Classes of subgroups up to conjugation

Order 12450: $D_{25}\times C_{249}$

Order 6225: $C_{6225}$

Order 4150: $D_{25}\times C_{83}$

Order 2490: $D_5\times C_{249}$

Order 2075: $C_{2075}$

Order 1245: $C_{1245}$

Order 830: $D_5\times C_{83}$

Order 498: $C_{498}$

Order 415: $C_{415}$

Order 249: $C_{249}$

Order 166: $C_{166}$

Order 150: $C_3\times D_{25}$

Order 83: $C_{83}$

Order 75: $C_{75}$

Order 50: $D_{25}$

Order 30: $C_3\times D_5$

Order 25: $C_{25}$

Order 15: $C_{15}$

Order 10: $D_5$

Order 6: $C_6$

Order 5: $C_5$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 12450: $D_{25}\times C_{249}$

Order 6225: $C_{6225}$

Order 4150: $D_{25}\times C_{83}$

Order 2490: $D_5\times C_{249}$

Order 2075: $C_{2075}$

Order 1245: $C_{1245}$

Order 830: $D_5\times C_{83}$

Order 498: $C_{498}$

Order 415: $C_{415}$

Order 249: $C_{249}$

Order 166: $C_{166}$

Order 150: $C_3\times D_{25}$

Order 83: $C_{83}$

Order 75: $C_{75}$

Order 50: $D_{25}$

Order 30: $C_3\times D_5$

Order 25: $C_{25}$

Order 15: $C_{15}$

Order 10: $D_5$

Order 6: $C_6$

Order 5: $C_5$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 12450: $D_{25}\times C_{249}$ ($C_1$)

Order 6225: $C_{6225}$ ($C_2$)

Order 4150: $D_{25}\times C_{83}$ ($C_3$)

Order 2075: $C_{2075}$ ($C_6$)

Order 1245: $C_{1245}$ ($D_5$)

Order 415: $C_{415}$ ($C_3\times D_5$)

Order 249: $C_{249}$ ($D_{25}$)

Order 150: $C_3\times D_{25}$ ($C_{83}$)

Order 83: $C_{83}$ ($C_3\times D_{25}$)

Order 75: $C_{75}$ ($C_{166}$)

Order 50: $D_{25}$ ($C_{249}$)

Order 25: $C_{25}$ ($C_{498}$)

Order 15: $C_{15}$ ($D_5\times C_{83}$)

Order 5: $C_5$ ($D_5\times C_{249}$)

Order 3: $C_3$ ($D_{25}\times C_{83}$)

Order 1: $C_1$ ($D_{25}\times C_{249}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 12450: $D_{25}\times C_{249}$ ($C_1$)

Order 6225: $C_{6225}$ ($C_2$)

Order 4150: $D_{25}\times C_{83}$ ($C_3$)

Order 2075: $C_{2075}$ ($C_6$)

Order 1245: $C_{1245}$ ($D_5$)

Order 415: $C_{415}$ ($C_3\times D_5$)

Order 249: $C_{249}$ ($D_{25}$)

Order 150: $C_3\times D_{25}$ ($C_{83}$)

Order 83: $C_{83}$ ($C_3\times D_{25}$)

Order 75: $C_{75}$ ($C_{166}$)

Order 50: $D_{25}$ ($C_{249}$)

Order 25: $C_{25}$ ($C_{498}$)

Order 15: $C_{15}$ ($D_5\times C_{83}$)

Order 5: $C_5$ ($D_5\times C_{249}$)

Order 3: $C_3$ ($D_{25}\times C_{83}$)

Order 1: $C_1$ ($D_{25}\times C_{249}$)

Series

Derived series

$D_{25}\times C_{249}$

$\rhd$

$C_{25}$

$\rhd$

$C_1$

Chief series

$D_{25}\times C_{249}$

$\rhd$

$C_{6225}$

$\rhd$

$C_{2075}$

$\rhd$

$C_{415}$

$\rhd$

$C_{83}$

$\rhd$

$C_1$

Lower central series

$D_{25}\times C_{249}$

$\rhd$

$C_{25}$

Upper central series

$C_1$

$\lhd$

$C_{249}$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

The $3486 \times 3486$ character table is not available for this group.

Rational character table

The $16 \times 16$ rational character table is not available for this group.