Abstract group 453840.a: $\SL(2,61):C_2$

• Label: 453840.a
• Order: \( 2^{4} \cdot 3 \cdot 5 \cdot 31 \cdot 61 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 31 \cdot 61 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 5 \cdot 31 \cdot 61 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $248$
• Trans deg.: $248$
• Rank: $2$

Group information

Description:	$\SL(2,61):C_2$
Order:	\(453840\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 31 \cdot 61 \)
Exponent:	\(113460\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 31 \cdot 61 \)
Automorphism group:	$C_2\times \PSL(2,61).C_2$, of order \(453840\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 31 \cdot 61 \)
Composition factors:	$C_2$ x 2, $\PSL(2,61)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	31	60	61	62	122	124	244
Elements	1	3783	3782	3784	7564	11346	22692	15128	15128	30256	45384	54900	60512	3720	54900	3720	109800	7440	453840
Conjugacy classes	1	2	1	3	2	3	6	4	4	8	12	15	16	2	15	2	30	4	130
Divisions	1	2	1	2	1	2	2	2	1	2	2	1	2	1	1	1	1	1	26
Autjugacy classes	1	2	1	2	2	2	4	3	4	6	8	15	12	1	15	1	15	1	95

Dimension	1	30	31	60	61	62	120	124	248	496	900	992	1800
Irr. complex chars.	2	4	4	60	2	58	0	0	0	0	0	0	0	130
Irr. rational chars.	2	0	0	0	2	6	1	5	5	1	2	1	1	26

Minimal presentations

Permutation degree:	$248$
Transitive degree:	$248$
Rank:	$2$
Inequivalent generating pairs:	$334374$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	30	60	120
Arbitrary	30	60	120

Constructions

Permutation group:	Degree $248$ $\langle(2,5,13,25,49,95,114,170,164,123,150,94,130,93,48,74,131,91,47,24,38,75,87,44,78,112,166,217,223,232,238,208,198,146,90,46,76,85,43,22,40,79,133,83,42,80,135,82,136,177,165,109,162,172,119,97,51,27,15,7,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 60 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{61})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$\SL(2,61)$ $\,\rtimes\,$ $C_2$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $\PSL(2,61)$	$C_2$ . $(C_2\times \PSL(2,61))$			more information

Elements of the group are displayed as matrices in $\GL_{2}(\F_{61})$.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 400502 subgroups in 118 conjugacy classes, 5 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $\PSL(2,61)$
Commutator:	$G' \simeq$ $\SL(2,61)$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times \PSL(2,61)$
Fitting:	$\operatorname{Fit} \simeq$ $C_4$	$G/\operatorname{Fit} \simeq$ $\PSL(2,61)$
Radical:	$R \simeq$ $C_4$	$G/R \simeq$ $\PSL(2,61)$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_2\times \PSL(2,61)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
31-Sylow subgroup:	$P_{ 31 } \simeq$ $C_{31}$
61-Sylow subgroup:	$P_{ 61 } \simeq$ $C_{61}$

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 453840: $\SL(2,61):C_2$

Order 226920: $\SL(2,61)$

Order 7320: $C_{244}:C_{30}$

Order 3660: $C_{244}:C_{15}$, $C_{61}:C_{60}$, $C_{122}:C_{30}$

Order 2440: $C_{244}:C_{10}$

Order 1830: $C_{61}:C_{30}$ x 2, $C_{122}:C_{15}$

Order 1464: $C_{244}:C_6$

Order 1220: $C_{122}:C_{10}$, $C_{244}:C_5$, $C_{61}:C_{20}$

Order 915: $C_{61}:C_{15}$

Order 732: $C_{122}:C_6$, $C_{244}:C_3$, $C_{61}:C_{12}$

Order 610: $C_{61}:C_{10}$ x 2, $C_{122}:C_5$

Order 488: $C_4\times D_{61}$

Order 366: $C_{61}:C_6$ x 2, $C_{122}:C_3$

Order 305: $C_{61}:C_5$

Order 248: $C_4\times D_{31}$

Order 244: $D_{122}$, $C_{244}$, $C_{61}:C_4$

Order 240: $\SL(2,5):C_2$ x 2, $D_{60}:C_2$

Order 183: $C_{61}:C_3$

Order 124: $D_{62}$, $C_{124}$, $C_{31}:C_4$

Order 122: $D_{61}$ x 2, $C_{122}$

Order 120: $\SL(2,5)$ x 2, $C_{15}:D_4$ x 2, $C_4\times D_{15}$ x 2, $C_2\times C_{60}$, $D_{60}$, $C_{15}:Q_8$

Order 80: $D_{20}:C_2$

Order 62: $D_{31}$ x 2, $C_{62}$

Order 61: $C_{61}$

Order 60: $C_{60}$ x 2, $C_{15}:C_4$ x 2, $D_{30}$ x 2, $C_2\times C_{30}$

Order 48: $D_{12}:C_2$, $\SL(2,3):C_2$

Order 40: $C_5:D_4$ x 2, $C_4\times D_5$ x 2, $C_2\times C_{20}$, $D_{20}$, $C_5:Q_8$

Order 31: $C_{31}$

Order 30: $C_{30}$ x 2, $D_{15}$ x 2

Order 24: $C_3:D_4$ x 2, $C_4\times S_3$ x 2, $C_2\times C_{12}$, $D_{12}$, $C_3:Q_8$, $\SL(2,3)$

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

Order 16: $D_4:C_2$

Order 15: $C_{15}$

Order 12: $D_6$ x 2, $C_{12}$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

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

Order 8: $Q_8$, $D_4$, $C_2\times C_4$

Order 6: $C_6$ x 2, $S_3$ x 2

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 453840: $\SL(2,61):C_2$

Order 226920: $\SL(2,61)$

Order 7320: $C_{244}:C_{30}$

Order 3660: $C_{244}:C_{15}$, $C_{61}:C_{60}$, $C_{122}:C_{30}$

Order 2440: $C_{244}:C_{10}$

Order 1830: $C_{122}:C_{15}$, $C_{61}:C_{30}$

Order 1464: $C_{244}:C_6$

Order 1220: $C_{122}:C_{10}$, $C_{244}:C_5$, $C_{61}:C_{20}$

Order 915: $C_{61}:C_{15}$

Order 732: $C_{122}:C_6$, $C_{244}:C_3$, $C_{61}:C_{12}$

Order 610: $C_{122}:C_5$, $C_{61}:C_{10}$

Order 488: $C_4\times D_{61}$

Order 366: $C_{122}:C_3$, $C_{61}:C_6$

Order 305: $C_{61}:C_5$

Order 248: $C_4\times D_{31}$

Order 244: $D_{122}$, $C_{244}$, $C_{61}:C_4$

Order 240: $\SL(2,5):C_2$, $D_{60}:C_2$

Order 183: $C_{61}:C_3$

Order 124: $D_{62}$, $C_{124}$, $C_{31}:C_4$

Order 122: $C_{122}$, $D_{61}$

Order 120: $\SL(2,5)$, $C_2\times C_{60}$, $C_{15}:D_4$, $D_{60}$, $C_4\times D_{15}$, $C_{15}:Q_8$

Order 80: $D_{20}:C_2$

Order 62: $C_{62}$, $D_{31}$

Order 61: $C_{61}$

Order 60: $C_{60}$ x 2, $C_{15}:C_4$, $C_2\times C_{30}$, $D_{30}$

Order 48: $D_{12}:C_2$, $\SL(2,3):C_2$

Order 40: $C_2\times C_{20}$, $C_5:D_4$, $D_{20}$, $C_4\times D_5$, $C_5:Q_8$

Order 31: $C_{31}$

Order 30: $C_{30}$ x 2, $D_{15}$

Order 24: $C_2\times C_{12}$, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $\SL(2,3)$

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

Order 16: $D_4:C_2$

Order 15: $C_{15}$

Order 12: $C_{12}$ x 2, $C_2\times C_6$, $D_6$, $C_3:C_4$

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

Order 8: $Q_8$, $D_4$, $C_2\times C_4$

Order 6: $C_6$ x 2, $S_3$

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 453840: $\SL(2,61):C_2$ ($C_1$)

Order 226920: $\SL(2,61)$ ($C_2$)

Order 4: $C_4$ ($\PSL(2,61)$)

Order 2: $C_2$ ($C_2\times \PSL(2,61)$)

Order 1: $C_1$ ($\SL(2,61):C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 453840: $\SL(2,61):C_2$ ($C_1$)

Order 226920: $\SL(2,61)$ ($C_2$)

Order 4: $C_4$ ($\PSL(2,61)$)

Order 2: $C_2$ ($C_2\times \PSL(2,61)$)

Order 1: $C_1$ ($\SL(2,61):C_2$)

Series

Derived series	$\SL(2,61):C_2$	$\rhd$	$\SL(2,61)$
Chief series	$\SL(2,61):C_2$	$\rhd$	$C_4$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$\SL(2,61):C_2$	$\rhd$	$\SL(2,61)$
Upper central series	$C_1$	$\lhd$	$C_4$

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

See the $130 \times 130$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $26 \times 26$ rational character table.