Abstract group 24900.c: $C_{25}:C_{996}$

• Label: 24900.c
• Order: \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 83 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 83 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \cdot 83 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \cdot 83 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 5^{3} \cdot 41 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 5 \cdot 41 \)
• Perm deg.: $115$
• Trans deg.: $24900$
• Rank: $2$

Group information

Description:	$C_{25}:C_{996}$
Order:	\(24900\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 83 \)
Exponent:	\(24900\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 83 \)
Automorphism group:	$C_2^2\times C_{82}\times D_{25}.C_{10}$, of order \(164000\)\(\medspace = 2^{5} \cdot 5^{3} \cdot 41 \)
Composition factors:	$C_2$ x 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	4	5	6	10	12	15	25	30	50	75	83	150	166	249	332	415	498	830	996	1245	2075	2490	4150	6225	12450
Elements	1	1	2	50	4	2	4	100	8	20	8	20	40	82	40	82	164	4100	328	164	328	8200	656	1640	656	1640	3280	3280	24900
Conjugacy classes	1	1	2	2	2	2	2	4	4	10	4	10	20	82	20	82	164	164	164	164	164	328	328	820	328	820	1640	1640	6972
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	28
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	28

Minimal presentations

Permutation degree:	$115$
Transitive degree:	$24900$
Rank:	$2$
Inequivalent generating pairs:	$2016$

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:	$\langle a, b \mid a^{996}=b^{25}=1, b^{a}=b^{24} \rangle$
Permutation group:	Degree $115$ $\langle(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 436 & 74 \\ 132 & 63 \end{array}\right), \left(\begin{array}{rr} 377 & 182 \\ 26 & 377 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{499})$
Direct product:	$C_3$ $\, \times\, $ $C_{83}$ $\, \times\, $ $(C_{25}:C_4)$
Semidirect product:	$C_{6225}$ $\,\rtimes\,$ $C_4$	$C_{75}$ $\,\rtimes\,$ $C_{332}$	$C_{25}$ $\,\rtimes\,$ $C_{996}$	$C_{2075}$ $\,\rtimes\,$ $C_{12}$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{2490}$ . $D_5$	$C_{498}$ . $D_{25}$	$C_{4150}$ . $C_6$	$C_{50}$ . $C_{498}$	all 16

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

Homology

Abelianization:	$C_{996} \simeq C_{4} \times C_{3} \times C_{83}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 148 subgroups in 36 conjugacy classes, 28 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{498}$	$G/Z \simeq$ $D_{25}$
Commutator:	$G' \simeq$ $C_{25}$	$G/G' \simeq$ $C_{996}$
Frattini:	$\Phi \simeq$ $C_{10}$	$G/\Phi \simeq$ $D_5\times C_{249}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12450}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{25}:C_{996}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{2490}$	$G/\operatorname{soc} \simeq$ $D_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
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 24900: $C_{25}:C_{996}$

Order 12450: $C_{12450}$

Order 8300: $C_{25}:C_{332}$

Order 6225: $C_{6225}$

Order 4980: $C_5:C_{996}$

Order 4150: $C_{4150}$

Order 2490: $C_{2490}$

Order 2075: $C_{2075}$

Order 1660: $C_5:C_{332}$

Order 1245: $C_{1245}$

Order 996: $C_{996}$

Order 830: $C_{830}$

Order 498: $C_{498}$

Order 415: $C_{415}$

Order 332: $C_{332}$

Order 300: $C_{25}:C_{12}$

Order 249: $C_{249}$

Order 166: $C_{166}$

Order 150: $C_{150}$

Order 100: $C_{25}:C_4$

Order 83: $C_{83}$

Order 75: $C_{75}$

Order 60: $C_5:C_{12}$

Order 50: $C_{50}$

Order 30: $C_{30}$

Order 25: $C_{25}$

Order 20: $C_5:C_4$

Order 15: $C_{15}$

Order 12: $C_{12}$

Order 10: $C_{10}$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 24900: $C_{25}:C_{996}$

Order 12450: $C_{12450}$

Order 8300: $C_{25}:C_{332}$

Order 6225: $C_{6225}$

Order 4980: $C_5:C_{996}$

Order 4150: $C_{4150}$

Order 2490: $C_{2490}$

Order 2075: $C_{2075}$

Order 1660: $C_5:C_{332}$

Order 1245: $C_{1245}$

Order 996: $C_{996}$

Order 830: $C_{830}$

Order 498: $C_{498}$

Order 415: $C_{415}$

Order 332: $C_{332}$

Order 300: $C_{25}:C_{12}$

Order 249: $C_{249}$

Order 166: $C_{166}$

Order 150: $C_{150}$

Order 100: $C_{25}:C_4$

Order 83: $C_{83}$

Order 75: $C_{75}$

Order 60: $C_5:C_{12}$

Order 50: $C_{50}$

Order 30: $C_{30}$

Order 25: $C_{25}$

Order 20: $C_5:C_4$

Order 15: $C_{15}$

Order 12: $C_{12}$

Order 10: $C_{10}$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 24900: $C_{25}:C_{996}$ ($C_1$)

Order 12450: $C_{12450}$ ($C_2$)

Order 8300: $C_{25}:C_{332}$ ($C_3$)

Order 6225: $C_{6225}$ ($C_4$)

Order 4150: $C_{4150}$ ($C_6$)

Order 2490: $C_{2490}$ ($D_5$)

Order 2075: $C_{2075}$ ($C_{12}$)

Order 1245: $C_{1245}$ ($C_5:C_4$)

Order 830: $C_{830}$ ($C_3\times D_5$)

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

Order 415: $C_{415}$ ($C_5:C_{12}$)

Order 300: $C_{25}:C_{12}$ ($C_{83}$)

Order 249: $C_{249}$ ($C_{25}:C_4$)

Order 166: $C_{166}$ ($C_3\times D_{25}$)

Order 150: $C_{150}$ ($C_{166}$)

Order 100: $C_{25}:C_4$ ($C_{249}$)

Order 83: $C_{83}$ ($C_{25}:C_{12}$)

Order 75: $C_{75}$ ($C_{332}$)

Order 50: $C_{50}$ ($C_{498}$)

Order 30: $C_{30}$ ($D_5\times C_{83}$)

Order 25: $C_{25}$ ($C_{996}$)

Order 15: $C_{15}$ ($C_5:C_{332}$)

Order 10: $C_{10}$ ($D_5\times C_{249}$)

Order 6: $C_6$ ($D_{25}\times C_{83}$)

Order 5: $C_5$ ($C_5:C_{996}$)

Order 3: $C_3$ ($C_{25}:C_{332}$)

Order 2: $C_2$ ($D_{25}\times C_{249}$)

Order 1: $C_1$ ($C_{25}:C_{996}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 24900: $C_{25}:C_{996}$ ($C_1$)

Order 12450: $C_{12450}$ ($C_2$)

Order 8300: $C_{25}:C_{332}$ ($C_3$)

Order 6225: $C_{6225}$ ($C_4$)

Order 4150: $C_{4150}$ ($C_6$)

Order 2490: $C_{2490}$ ($D_5$)

Order 2075: $C_{2075}$ ($C_{12}$)

Order 1245: $C_{1245}$ ($C_5:C_4$)

Order 830: $C_{830}$ ($C_3\times D_5$)

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

Order 415: $C_{415}$ ($C_5:C_{12}$)

Order 300: $C_{25}:C_{12}$ ($C_{83}$)

Order 249: $C_{249}$ ($C_{25}:C_4$)

Order 166: $C_{166}$ ($C_3\times D_{25}$)

Order 150: $C_{150}$ ($C_{166}$)

Order 100: $C_{25}:C_4$ ($C_{249}$)

Order 83: $C_{83}$ ($C_{25}:C_{12}$)

Order 75: $C_{75}$ ($C_{332}$)

Order 50: $C_{50}$ ($C_{498}$)

Order 30: $C_{30}$ ($D_5\times C_{83}$)

Order 25: $C_{25}$ ($C_{996}$)

Order 15: $C_{15}$ ($C_5:C_{332}$)

Order 10: $C_{10}$ ($D_5\times C_{249}$)

Order 6: $C_6$ ($D_{25}\times C_{83}$)

Order 5: $C_5$ ($C_5:C_{996}$)

Order 3: $C_3$ ($C_{25}:C_{332}$)

Order 2: $C_2$ ($D_{25}\times C_{249}$)

Order 1: $C_1$ ($C_{25}:C_{996}$)

Series

Derived series

$C_{25}:C_{996}$

$\rhd$

$C_{25}$

$\rhd$

$C_1$

Chief series

$C_{25}:C_{996}$

$\rhd$

$C_{12450}$

$\rhd$

$C_{6225}$

$\rhd$

$C_{2075}$

$\rhd$

$C_{25}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{25}:C_{996}$

$\rhd$

$C_{25}$

Upper central series

$C_1$

$\lhd$

$C_{498}$

Supergroups

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

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $6972 \times 6972$ character table is not available for this group.

Rational character table

The $28 \times 28$ rational character table is not available for this group.