Abstract group 705600.b: $\SL(2,49).C_6$

• Label: 705600.b
• Order: \( 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \)
• Exponent: \( 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $803$
• Trans deg.: $2400$
• Rank: $2$

Group information

Description:	$\SL(2,49).C_6$
Order:	\(705600\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \)
Exponent:	\(8400\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Automorphism group:	$C_2^2\times \PSL(2,49).C_2$, of order \(470400\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Composition factors:	$C_2$ x 2, $C_3$, $\PSL(2,49)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	10	12	14	15	16	21	24	25	28	30	42	48	50	75	84	150
Elements	1	351	7352	2800	4704	66852	2400	19600	4704	79100	19200	9408	39200	4800	68600	23520	16800	9408	38400	137200	23520	47040	33600	47040	705600
Conjugacy classes	1	2	5	2	1	10	2	3	1	10	4	2	4	4	12	5	2	2	8	20	5	10	4	10	129
Divisions	1	2	3	2	1	6	2	2	1	6	3	1	2	2	4	1	1	1	3	4	1	1	1	1	52
Autjugacy classes	1	2	3	2	1	6	2	3	1	6	3	1	4	2	7	5	1	1	3	12	5	5	1	5	82

Dimension	1	2	24	25	48	49	50	96	98	100	192	200	400	480	800	960
Irr. complex chars.	6	0	12	12	0	6	30	36	0	27	0	0	0	0	0	0	129
Irr. rational chars.	2	2	0	4	2	2	10	4	2	9	2	5	3	2	1	2	52

Minimal presentations

Permutation degree:	$803$
Transitive degree:	$2400$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	48	96
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $803$ $\langle(1,2,7,15,5)(3,11,24,28,12)(4,9,20,30,13)(6,17,37,38,18)(8,22,41,39,19) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 4 & 3 & 1 & 3 \\ 2 & 5 & 0 & 1 \\ 3 & 6 & 2 & 4 \\ 3 & 3 & 5 & 3 \end{array}\right), \left(\begin{array}{rrrr} 2 & 6 & 2 & 5 \\ 6 & 2 & 1 & 0 \\ 6 & 6 & 2 & 1 \\ 0 & 2 & 4 & 6 \end{array}\right), \left(\begin{array}{rrrr} 6 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{7})$
Direct product:	$C_3$ $\, \times\, $ $(\SL(2,49):C_2)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$\SL(2,49)$ . $C_6$	$C_6$ . $\POMinus(4,7)$	$(C_3\times \SL(2,49))$ . $C_2$	$(\SL(2,49):C_2)$ . $C_3$	all 6

Elements of the group are displayed as matrices in $\GL_{4}(\F_{7})$.

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 678680 subgroups in 393 conjugacy classes, 8 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $\POMinus(4,7)$
Commutator:	$G' \simeq$ $\SL(2,49)$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_3.\PSL(2,49).C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_6$	$G/\operatorname{Fit} \simeq$ $\POMinus(4,7)$
Radical:	$R \simeq$ $C_6$	$G/R \simeq$ $\POMinus(4,7)$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $\POMinus(4,7)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{32}:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_{25}$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$

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 705600: $\SL(2,49).C_6$

Order 352800: $C_3\times \SL(2,49)$

Order 235200: $\SL(2,49):C_2$

Order 117600: $\SL(2,49)$

Order 14112: $C_7^2:(C_3^2\times \SD_{32})$

Order 7056: $C_7^2:(C_3^2\times D_8)$, $C_7^2:(C_3^2\times Q_{16})$, $C_3\times C_7^2:C_{48}$

Order 4704: $C_7^2:(C_3\times \SD_{32})$ x 3, $C_3\times C_7^2:\SD_{32}$

Order 4032: $\GL(2,7):C_2$, $C_{12}.\PGL(2,7)$

Order 3528: $C_3\times C_7^2:C_{24}$, $C_7^2:(Q_8\times C_3^2)$, $C_3\times D_{14}:F_7$

Order 2352: $C_7^2:C_{48}$ x 3, $C_7^2:(C_3\times D_8)$ x 3, $C_7^2:(C_3\times Q_{16})$ x 3, $C_3\times C_7^2:Q_{16}$, $C_3\times C_7^2:D_8$, $C_7^2:C_{48}$

Order 2016: $C_6.\PGL(2,7)$ x 3, $\SL(2,7):C_6$, $C_6\times \SL(2,7)$, $\GL(2,7)$

Order 1764: $C_{42}:F_7$, $C_{42}.F_7$, $C_{42}.F_7$

Order 1568: $C_7^2:\SD_{32}$

Order 1344: $C_2^2.\PGL(2,7)$, $C_4.\PGL(2,7)$

Order 1176: $C_7^2:(C_3\times Q_8)$ x 3, $D_{14}:F_7$ x 3, $C_7^2:C_{24}$ x 3, $C_{42}.D_{14}$, $C_{42}.D_{14}$, $C_7^2:C_{24}$

Order 1008: $C_3\times \SL(2,7)$ x 2

Order 882: $C_3\times C_7^2:C_6$, $C_{21}:F_7$

Order 784: $C_7^2:Q_{16}$, $C_7^2:D_8$, $C_7^2:C_{16}$

Order 720: $\Unitary(2,5)$, $C_6.S_5$

Order 672: $C_2.\PGL(2,7)$ x 3, $C_2\times \SL(2,7)$, $\SL(2,7):C_2$, $\SL(2,7):C_2$

Order 600: $C_{25}:C_{24}$

Order 588: $C_{14}:F_7$ x 3, $C_7^2:C_{12}$ x 3, $C_7^2:C_{12}$ x 3, $C_{21}\times D_{14}$, $C_7^2:C_{12}$, $C_7:C_{84}$

Order 576: $C_{48}.D_6$

Order 504: $C_{14}.C_6^2$, $C_{84}.C_6$

Order 441: $C_7^2:C_3^2$

Order 392: $C_7^2:Q_8$, $D_{14}:D_7$, $C_7^2:C_8$

Order 360: $C_3\times \SL(2,5)$ x 2

Order 336: $\SL(2,7)$ x 2

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

Order 294: $C_7^2:C_6$ x 3, $C_7:F_7$ x 3, $C_7\times C_{42}$, $D_7\times C_{21}$

Order 288: $\GL(2,3):C_6$, $C_2\times C_6.S_4$, $C_{24}.D_6$, $C_{24}.D_6$, $C_3^2\times \SD_{32}$, $C_3^2:Q_{32}$, $C_3^2:\SD_{32}$, $C_3^2:Q_{32}$, $C_{48}:C_6$

Order 252: $C_{21}:C_{12}$ x 3, $C_{42}:C_6$, $C_6\times F_7$, $C_{21}:C_{12}$

Order 240: $\SL(2,5):C_2$, $C_2.S_5$

Order 200: $C_{25}:C_8$

Order 196: $C_7^2:C_4$, $C_7:C_{28}$, $C_7\times D_{14}$

Order 192: $Q_{32}:C_6$, $\SD_{32}:S_3$

Order 168: $C_{28}.C_6$ x 3, $D_{14}:C_6$ x 3, $C_{21}:D_4$, $C_{21}:Q_8$

Order 150: $C_{150}$

Order 147: $C_7^2:C_3$ x 3, $C_7\times C_{21}$

Order 144: $C_6.S_4$ x 3, $C_3^2:Q_{16}$ x 2, $C_3^2:Q_{16}$, $C_{12}.D_6$, $S_3\times C_{24}$, $C_3\times C_{48}$, $C_3:C_{48}$, $C_{12}.D_6$, $C_{12}.D_6$, $\SL(2,3):C_6$, $C_6\times \SL(2,3)$, $C_3\times \GL(2,3)$, $C_3^2\times Q_{16}$, $C_3^2\times D_8$

Order 126: $C_{21}:C_6$ x 3, $C_3\times F_7$

Order 120: $\SL(2,5)$ x 2, $C_5:C_{24}$

Order 100: $C_{25}:C_4$

Order 98: $C_7\times C_{14}$, $C_7\times D_7$

Order 96: $C_3\times \SD_{32}$ x 4, $C_3\times Q_{32}$ x 2, $C_3:Q_{32}$, $C_3\times \OD_{32}$, $C_{48}:C_2$, $C_3:Q_{32}$, $C_3:\SD_{32}$, $\GL(2,3):C_2$, $C_2^2.S_4$, $D_8:C_6$, $C_6\times Q_{16}$, $S_3\times Q_{16}$, $D_8:S_3$

Order 84: $C_7:C_{12}$ x 9, $C_{14}:C_6$ x 3, $C_2\times F_7$ x 3, $C_7:C_{12}$ x 3, $C_7:C_{12}$ x 3, $C_{84}$, $C_2\times C_{42}$, $C_3\times D_{14}$

Order 75: $C_{75}$

Order 72: $C_3^2:Q_8$ x 3, $S_3\times C_{12}$ x 2, $C_3\times \SL(2,3)$ x 2, $Q_8\times C_3^2$, $D_4\times C_3^2$, $C_6\wr C_2$, $C_6:C_{12}$, $C_3\times C_{24}$, $C_3:C_{24}$

Order 64: $Q_{32}:C_2$

Order 63: $C_{21}:C_3$ x 2

Order 60: $C_5:C_{12}$

Order 56: $C_7:D_4$, $C_7:Q_8$

Order 50: $C_{50}$

Order 49: $C_7^2$

Order 48: $C_3\times Q_{16}$ x 6, $C_{48}$ x 4, $C_2.S_4$ x 3, $C_3\times D_8$ x 3, $C_3:Q_{16}$ x 2, $\SL(2,3):C_2$ x 2, $C_2\times \SL(2,3)$ x 2, $D_4:C_6$, $C_6\times Q_8$, $S_3\times Q_8$, $S_3\times C_8$, $D_4:S_3$, $\GL(2,3)$, $C_3\times \SD_{16}$, $C_2\times C_{24}$, $C_3:Q_{16}$, $C_3:\SD_{16}$, $C_3:C_{16}$

Order 42: $C_7:C_6$ x 9, $C_{42}$ x 3, $F_7$ x 3, $C_3\times D_7$

Order 40: $C_5:C_8$

Order 36: $C_3:C_{12}$ x 3, $C_3\times C_{12}$ x 2, $C_6^2$, $C_6\times S_3$

Order 32: $Q_{32}$ x 2, $\SD_{32}$ x 2, $D_8:C_2$, $C_2\times Q_{16}$, $\OD_{32}$

Order 30: $C_{30}$

Order 28: $C_7:C_4$ x 3, $C_2\times C_{14}$, $D_{14}$, $C_{28}$

Order 25: $C_{25}$

Order 24: $C_3\times Q_8$ x 5, $\SL(2,3)$ x 4, $C_{24}$ x 4, $C_3:Q_8$ x 3, $C_3\times D_4$ x 3, $C_4\times S_3$ x 2, $C_2\times C_{12}$, $C_3:D_4$, $C_6:C_4$, $C_3:C_8$

Order 21: $C_7:C_3$ x 6, $C_{21}$ x 2

Order 20: $C_5:C_4$

Order 18: $C_3\times C_6$ x 2, $C_3\times S_3$

Order 16: $Q_{16}$ x 4, $C_{16}$ x 2, $\SD_{16}$, $D_8$, $C_2\times C_8$, $D_4:C_2$, $C_2\times Q_8$

Order 15: $C_{15}$

Order 14: $C_{14}$ x 3, $D_7$

Order 12: $C_{12}$ x 6, $C_2\times C_6$ x 3, $C_3:C_4$ x 3, $D_6$

Order 10: $C_{10}$

Order 9: $C_3^2$

Order 8: $Q_8$ x 3, $C_8$ x 2, $D_4$, $C_2\times C_4$

Order 7: $C_7$ x 2

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

Order 5: $C_5$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 705600: $\SL(2,49).C_6$

Order 352800: $C_3\times \SL(2,49)$

Order 235200: $\SL(2,49):C_2$

Order 117600: $\SL(2,49)$

Order 14112: $C_7^2:(C_3^2\times \SD_{32})$

Order 7056: $C_7^2:(C_3^2\times D_8)$

Order 1568: $C_7^2:\SD_{32}$

Order 600: $C_{25}:C_{24}$

Order 576: $C_{48}.D_6$

Order 441: $C_7^2:C_3^2$

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

Order 200: $C_{25}:C_8$

Order 150: $C_{150}$

Order 100: $C_{25}:C_4$

Order 75: $C_{75}$

Order 64: $Q_{32}:C_2$

Order 60: $C_5:C_{12}$

Order 50: $C_{50}$

Order 49: $C_7^2$

Order 40: $C_5:C_8$

Order 30: $C_{30}$

Order 25: $C_{25}$

Order 20: $C_5:C_4$

Order 15: $C_{15}$

Order 10: $C_{10}$

Order 9: $C_3^2$

Order 5: $C_5$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 705600: $\SL(2,49).C_6$ ($C_1$)

Order 352800: $C_3\times \SL(2,49)$ ($C_2$)

Order 235200: $\SL(2,49):C_2$ ($C_3$)

Order 117600: $\SL(2,49)$ ($C_6$)

Order 6: $C_6$ ($\POMinus(4,7)$)

Order 3: $C_3$ ($\SL(2,49):C_2$)

Order 2: $C_2$ ($C_3.\PSL(2,49).C_2$)

Order 1: $C_1$ ($\SL(2,49).C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 705600: $\SL(2,49).C_6$ ($C_1$)

Order 352800: $C_3\times \SL(2,49)$ ($C_2$)

Order 235200: $\SL(2,49):C_2$ ($C_3$)

Order 117600: $\SL(2,49)$ ($C_6$)

Order 6: $C_6$ ($\POMinus(4,7)$)

Order 1: $C_1$ ($\SL(2,49).C_6$)

Series

Derived series	$\SL(2,49).C_6$	$\rhd$	$\SL(2,49)$
Chief series	$\SL(2,49).C_6$	$\rhd$	$C_3\times \SL(2,49)$	$\rhd$	$C_6$	$\rhd$	$C_3$	$\rhd$	$C_1$
Lower central series	$\SL(2,49).C_6$	$\rhd$	$\SL(2,49)$
Upper central series	$C_1$	$\lhd$	$C_6$

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $129 \times 129$ 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 $52 \times 52$ rational character table.