Abstract group 49800.b: $C_{6225}:Q_8$

• Label: 49800.b
• Order: \( 2^{3} \cdot 3 \cdot 5^{2} \cdot 83 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 83 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \cdot 83 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \cdot 83 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 5^{3} \cdot 41 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 5 \cdot 41 \)
• Perm deg.: $119$
• Trans deg.: $49800$
• Rank: $2$

Group information

Description:	$C_{6225}:Q_8$
Order:	\(49800\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 83 \)
Exponent:	\(24900\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 83 \)
Automorphism group:	$C_{50}.C_{410}.C_2^5$, of order \(656000\)\(\medspace = 2^{7} \cdot 5^{3} \cdot 41 \)
Composition factors:	$C_2$ x 3, $C_3$, $C_5$ x 2, $C_{83}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	25	30	50	60	75	83	100	150	166	249	300	332	415	498	830	996	1245	1660	2075	2490	4150	4980	6225	8300	12450	24900
Elements	1	1	2	102	4	2	4	204	8	8	20	8	20	16	40	82	40	40	82	164	80	8364	328	164	328	16728	656	656	1640	656	1640	1312	3280	3280	3280	6560	49800
Conjugacy classes	1	1	2	3	2	2	2	6	4	4	10	4	10	8	20	82	20	20	82	164	40	246	164	164	164	492	328	328	820	328	820	656	1640	1640	1640	3280	13197
Divisions	1	1	1	3	1	1	1	3	1	1	1	1	1	1	1	1	1	1	1	1	1	3	1	1	1	3	1	1	1	1	1	1	1	1	1	1	44
Autjugacy classes	1	1	1	2	1	1	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	2	1	1	1	2	1	1	1	1	1	1	1	1	1	1	40

Minimal presentations

Permutation degree:	$119$
Transitive degree:	$49800$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Presentation:	$\langle a, b \mid b^{24900}=1, a^{2}=b^{11950}, b^{a}=b^{499} \rangle$
Permutation group:	Degree $119$ $\langle(1,2,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79,82,81,83,78,80,75,77,72,74,69,71,66,68,63,65,60,62,57,59,54,56,51,53,48,50,45,47,42,44,39,41,36,38,33,35,30,32,27,29,24,26,21,23,18,20,15,17,12,14,9,11,6,8,3,5) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 436 & 74 \\ 132 & 63 \end{array}\right), \left(\begin{array}{rr} 287 & 49 \\ 7 & 287 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{499})$
Direct product:	not computed
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_{4980}$ . $D_5$	$C_{996}$ . $D_{25}$	$C_{498}$ . $D_{50}$	$C_{8300}$ . $C_6$	all 32

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

Homology

Abelianization:	$C_{2} \times C_{498} \simeq C_{2}^{2} \times C_{3} \times C_{83}$
Schur multiplier:	$C_1$
Commutator length:	not computed

Subgroups

There are 408 subgroups in 72 conjugacy classes, 48 normal (40 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{498}$	$G/Z \simeq$ $D_{50}$
Commutator:	$G' \simeq$ $C_{50}$	$G/G' \simeq$ $C_2\times C_{498}$
Frattini:	$\Phi \simeq$ $C_{10}$	$G/\Phi \simeq$ $D_5\times C_{498}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{24900}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{6225}:Q_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{2490}$	$G/\operatorname{soc} \simeq$ $D_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8$
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

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

Order 49800: $C_{6225}:Q_8$

Order 24900: $C_{25}:C_{996}$ x 2, $C_{24900}$

Order 16600: $C_{2075}:Q_8$

Order 12450: $C_{12450}$

Order 9960: $C_{1245}:Q_8$

Order 8300: $C_{25}:C_{332}$ x 2, $C_{8300}$

Order 6225: $C_{6225}$

Order 4980: $C_5:C_{996}$ x 2, $C_{4980}$

Order 4150: $C_{4150}$

Order 3320: $C_{415}:Q_8$

Order 2490: $C_{2490}$

Order 2075: $C_{2075}$

Order 1992: $Q_8\times C_{249}$

Order 1660: $C_5:C_{332}$ x 2, $C_{1660}$

Order 1245: $C_{1245}$

Order 996: $C_{996}$ x 3

Order 830: $C_{830}$

Order 664: $Q_8\times C_{83}$

Order 600: $C_{75}:Q_8$

Order 498: $C_{498}$

Order 415: $C_{415}$

Order 332: $C_{332}$ x 3

Order 300: $C_{25}:C_{12}$ x 2, $C_{300}$

Order 249: $C_{249}$

Order 200: $C_{25}:Q_8$

Order 166: $C_{166}$

Order 150: $C_{150}$

Order 120: $C_{15}:Q_8$

Order 100: $C_{25}:C_4$ x 2, $C_{100}$

Order 83: $C_{83}$

Order 75: $C_{75}$

Order 60: $C_5:C_{12}$ x 2, $C_{60}$

Order 50: $C_{50}$

Order 40: $C_5:Q_8$

Order 30: $C_{30}$

Order 25: $C_{25}$

Order 24: $C_3\times Q_8$

Order 20: $C_5:C_4$ x 2, $C_{20}$

Order 15: $C_{15}$

Order 12: $C_{12}$ x 3

Order 10: $C_{10}$

Order 8: $Q_8$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 49800: $C_{6225}:Q_8$

Order 24900: $C_{24900}$

Order 16600: $C_{2075}:Q_8$

Order 12450: $C_{12450}$

Order 9960: $C_{1245}:Q_8$

Order 6225: $C_{6225}$

Order 4150: $C_{4150}$

Order 3320: $C_{415}:Q_8$

Order 2490: $C_{2490}$

Order 2075: $C_{2075}$

Order 1992: $Q_8\times C_{249}$

Order 1245: $C_{1245}$

Order 830: $C_{830}$

Order 664: $Q_8\times C_{83}$

Order 600: $C_{75}:Q_8$

Order 498: $C_{498}$

Order 415: $C_{415}$

Order 249: $C_{249}$

Order 200: $C_{25}:Q_8$

Order 166: $C_{166}$

Order 150: $C_{150}$

Order 120: $C_{15}:Q_8$

Order 83: $C_{83}$

Order 75: $C_{75}$

Order 50: $C_{50}$

Order 40: $C_5:Q_8$

Order 30: $C_{30}$

Order 25: $C_{25}$

Order 24: $C_3\times Q_8$

Order 15: $C_{15}$

Order 10: $C_{10}$

Order 8: $Q_8$

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 49800: $C_{6225}:Q_8$ ($C_1$)

Order 24900: $C_{25}:C_{996}$ ($C_2$) x 2, $C_{24900}$ ($C_2$)

Order 16600: $C_{2075}:Q_8$ ($C_3$)

Order 12450: $C_{12450}$ ($C_2^2$)

Order 8300: $C_{25}:C_{332}$ ($C_6$) x 2, $C_{8300}$ ($C_6$)

Order 6225: $C_{6225}$ ($Q_8$)

Order 4980: $C_{4980}$ ($D_5$)

Order 4150: $C_{4150}$ ($C_2\times C_6$)

Order 2490: $C_{2490}$ ($D_{10}$)

Order 2075: $C_{2075}$ ($C_3\times Q_8$)

Order 1660: $C_{1660}$ ($C_3\times D_5$)

Order 1245: $C_{1245}$ ($C_5:Q_8$)

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

Order 830: $C_{830}$ ($C_3\times D_{10}$)

Order 600: $C_{75}:Q_8$ ($C_{83}$)

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

Order 415: $C_{415}$ ($C_{15}:Q_8$)

Order 332: $C_{332}$ ($C_3\times D_{25}$)

Order 300: $C_{25}:C_{12}$ ($C_{166}$) x 2, $C_{300}$ ($C_{166}$)

Order 249: $C_{249}$ ($C_{25}:Q_8$)

Order 200: $C_{25}:Q_8$ ($C_{249}$)

Order 166: $C_{166}$ ($C_3\times D_{50}$)

Order 150: $C_{150}$ ($C_2\times C_{166}$)

Order 100: $C_{25}:C_4$ ($C_{498}$) x 2, $C_{100}$ ($C_{498}$)

Order 83: $C_{83}$ ($C_{75}:Q_8$)

Order 75: $C_{75}$ ($Q_8\times C_{83}$)

Order 60: $C_{60}$ ($D_5\times C_{83}$)

Order 50: $C_{50}$ ($C_2\times C_{498}$)

Order 30: $C_{30}$ ($D_5\times C_{166}$)

Order 25: $C_{25}$ ($Q_8\times C_{249}$)

Order 20: $C_{20}$ ($D_5\times C_{249}$)

Order 15: $C_{15}$ ($C_{415}:Q_8$)

Order 12: $C_{12}$ ($D_{25}\times C_{83}$)

Order 10: $C_{10}$ ($D_5\times C_{498}$)

Order 6: $C_6$ ($C_{83}\times D_{50}$)

Order 5: $C_5$ ($C_{1245}:Q_8$)

Order 4: $C_4$ ($D_{25}\times C_{249}$)

Order 3: $C_3$ ($C_{2075}:Q_8$)

Order 2: $C_2$ ($D_{50}\times C_{249}$)

Order 1: $C_1$ ($C_{6225}:Q_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 49800: $C_{6225}:Q_8$ ($C_1$)

Order 24900: $C_{24900}$ ($C_2$)

Order 16600: $C_{2075}:Q_8$ ($C_3$)

Order 12450: $C_{12450}$ ($C_2^2$)

Order 6225: $C_{6225}$ ($Q_8$)

Order 4150: $C_{4150}$ ($C_2\times C_6$)

Order 2490: $C_{2490}$ ($D_{10}$)

Order 2075: $C_{2075}$ ($C_3\times Q_8$)

Order 1245: $C_{1245}$ ($C_5:Q_8$)

Order 830: $C_{830}$ ($C_3\times D_{10}$)

Order 600: $C_{75}:Q_8$ ($C_{83}$)

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

Order 415: $C_{415}$ ($C_{15}:Q_8$)

Order 249: $C_{249}$ ($C_{25}:Q_8$)

Order 200: $C_{25}:Q_8$ ($C_{249}$)

Order 166: $C_{166}$ ($C_3\times D_{50}$)

Order 150: $C_{150}$ ($C_2\times C_{166}$)

Order 83: $C_{83}$ ($C_{75}:Q_8$)

Order 75: $C_{75}$ ($Q_8\times C_{83}$)

Order 50: $C_{50}$ ($C_2\times C_{498}$)

Order 30: $C_{30}$ ($D_5\times C_{166}$)

Order 25: $C_{25}$ ($Q_8\times C_{249}$)

Order 15: $C_{15}$ ($C_{415}:Q_8$)

Order 10: $C_{10}$ ($D_5\times C_{498}$)

Order 6: $C_6$ ($C_{83}\times D_{50}$)

Order 5: $C_5$ ($C_{1245}:Q_8$)

Order 3: $C_3$ ($C_{2075}:Q_8$)

Order 2: $C_2$ ($D_{50}\times C_{249}$)

Order 1: $C_1$ ($C_{6225}:Q_8$)

Series

Derived series

$C_{6225}:Q_8$

$\rhd$

$C_{50}$

$\rhd$

$C_1$

Chief series

$C_{6225}:Q_8$

$\rhd$

$C_{25}:C_{996}$

$\rhd$

$C_{12450}$

$\rhd$

$C_{6225}$

$\rhd$

$C_{2075}$

$\rhd$

$C_{415}$

$\rhd$

$C_{83}$

$\rhd$

$C_1$

Lower central series

$C_{6225}:Q_8$

$\rhd$

$C_{50}$

$\rhd$

$C_{25}$

Upper central series

$C_1$

$\lhd$

$C_{498}$

$\lhd$

$C_{996}$

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 $13197 \times 13197$ character table is not available for this group.

Rational character table

The $44 \times 44$ rational character table is not available for this group.