Abstract group 470400.b: $\SL(2,49).C_2^2$

• Label: 470400.b
• Order: \( 2^{7} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
• Exponent: \( 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $800$
• Trans deg.: $800$
• Rank: $2$

Group information

Description:	$\SL(2,49).C_2^2$
Order:	\(470400\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Exponent:	\(8400\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Automorphism group:	$D_4.\PSL(2,49).C_2$, of order \(940800\)\(\medspace = 2^{8} \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Composition factors:	$C_2$ x 3, $\PSL(2,49)$
Derived length:	$1$

This group is nonabelian and nonsolvable. Whether it is almost simple has not been computed.

Group statistics

Order	1	2	3	4	5	6	7	8	10	12	14	16	20	24	25	28	48	50	100
Elements	1	3151	2450	3152	4704	46550	2400	39200	4704	49000	36000	78400	9408	19600	23520	38400	39200	23520	47040	470400
Conjugacy classes	1	4	1	4	1	5	2	5	1	4	4	10	2	4	5	4	8	5	10	80
Divisions	1	4	1	4	1	4	2	4	1	4	4	4	1	2	1	4	2	1	1	46
Autjugacy classes	1	3	1	3	1	3	2	4	1	3	3	6	1	3	5	3	6	5	5	59

Dimension	1	25	48	49	50	96	100	192	200	400	480	800	960
Irr. complex chars.	4	8	2	4	20	24	18	0	0	0	0	0	0	80
Irr. rational chars.	4	8	2	4	12	2	6	1	2	1	2	1	1	46

Minimal presentations

Permutation degree:	$800$
Transitive degree:	$800$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Permutation group:	Degree $800$ $\langle(1,2,9,70,316,630,261,525,588,121,231,649,7,14,75,432,576,119,78,377,443,475,366,154) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 6 & 3 & 5 & 3 \\ 5 & 4 & 4 & 5 \\ 4 & 2 & 4 & 4 \\ 6 & 4 & 2 & 2 \end{array}\right), \left(\begin{array}{rrrr} 6 & 2 & 1 & 2 \\ 1 & 6 & 1 & 1 \\ 5 & 6 & 0 & 5 \\ 3 & 5 & 6 & 0 \end{array}\right), \left(\begin{array}{rrrr} 3 & 1 & 5 & 0 \\ 6 & 4 & 0 & 2 \\ 0 & 0 & 4 & 1 \\ 0 & 0 & 6 & 3 \end{array}\right), \left(\begin{array}{rrrr} 6 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrrr} 1 & 6 & 6 & 1 \\ 5 & 3 & 4 & 6 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 2 & 3 \end{array}\right), \left(\begin{array}{rrrr} 6 & 0 & 0 & 3 \\ 0 & 1 & 6 & 0 \\ 0 & 1 & 0 & 0 \\ 6 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 5 & 6 & 0 & 0 \\ 2 & 4 & 2 & 6 \\ 2 & 2 & 5 & 1 \\ 4 & 0 & 3 & 6 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{7})$
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:	$\SL(2,49)$ . $C_2^2$	$C_4$ . $\POMinus(4,7)$	$(\SL(2,49):C_2)$ . $C_2$ (2)	$(\SL(2,49):C_2)$ . $C_2$	all 5

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

Homology

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

Subgroups

There are 1297487 subgroups in 597 conjugacy classes, 8 normal (6 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $\PSL(2,49):C_2^2$
Commutator:	$G' \simeq$ $\SL(2,49)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $\PSL(2,49):C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_4$	$G/\operatorname{Fit} \simeq$ $\POMinus(4,7)$
Radical:	$R \simeq$ $C_4$	$G/R \simeq$ $\POMinus(4,7)$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $\PSL(2,49):C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_{16}:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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 470400: $\SL(2,49).C_2^2$

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

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

Order 9408: $C_7^2:(C_6\times \SD_{32})$

Order 4704: $C_7^2:(C_3\times \SD_{32})$ x 4, $C_7^2:(C_2\times C_{48})$, $C_7^2:(C_6\times D_8)$, $C_7^2:(C_6\times Q_{16})$

Order 3136: $(C_7\times C_{28}).D_8$

Order 2688: $Q_8.\PGL(2,7)$, $D_4.\PGL(2,7)$

Order 2352: $C_7^2:(C_3\times D_8)$ x 3, $C_7^2:(C_3\times Q_{16})$ x 3, $C_7^2:C_{48}$ x 2, $C_7^2:(C_2\times C_{24})$, $D_{28}:F_7$, $C_7:(Q_8\times F_7)$

Order 1568: $C_7^2:\SD_{32}$ x 4, $C_7^2:(C_2\times Q_{16})$, $C_7^2:(C_2\times D_8)$, $C_7^2:(C_2\times C_{16})$

Order 1344: $C_4.\PGL(2,7)$ x 4, $C_4.\PGL(2,7)$ x 3, $C_2^2.\PGL(2,7)$ x 2, $\SL(2,7):C_2^2$ x 2, $\SL(2,7):C_2^2$, $D_4.\PSL(2,7)$, $\SL(2,7):C_2^2$

Order 1176: $C_7^2:(C_3\times Q_8)$ x 2, $D_{14}:F_7$ x 2, $C_7^2:C_{24}$ x 2, $C_7:(C_4\times F_7)$, $D_{14}:F_7$, $C_{28}:F_7$, $C_{28}:F_7$, $C_{28}.F_7$

Order 784: $C_7^2:Q_{16}$ x 3, $C_7^2:D_8$ x 3, $C_7^2:C_{16}$ x 2, $C_{28}:D_{14}$, $C_{28}.D_{14}$, $C_7^2:(C_2\times C_8)$

Order 672: $\SL(2,7):C_2$ x 4, $C_2.\PGL(2,7)$ x 4, $\SL(2,7):C_2$ x 4, $C_2\times \SL(2,7)$ x 2

Order 588: $C_{14}:F_7$ x 2, $D_7:F_7$ x 2, $C_7^2:C_{12}$ x 2, $C_{14}:F_7$, $C_7^2:C_{12}$, $C_7^2:C_{12}$

Order 480: $C_4.S_5$, $C_4.S_5$

Order 400: $C_{100}.C_4$

Order 392: $C_7^2:Q_8$ x 2, $D_{14}:D_7$ x 2, $C_7^2:C_8$ x 2, $D_7\times D_{14}$, $C_{28}:D_7$, $C_7\times D_{28}$, $C_7^2:Q_8$, $C_{14}.D_{14}$

Order 384: $D_{48}:C_2^2$

Order 336: $\SL(2,7)$ x 2, $Q_8\times F_7$, $D_4\times F_7$

Order 294: $C_7:F_7$ x 2, $C_7:F_7$ x 2, $C_7^2:C_6$

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

Order 200: $C_{25}:C_8$ x 2, $C_4\times D_{25}$

Order 196: $D_7^2$ x 2, $C_7:C_{28}$ x 2, $C_7\times D_{14}$ x 2, $C_7\times C_{28}$, $C_7^2:C_4$, $C_7:D_{14}$

Order 192: $\SD_{32}:S_3$ x 2, $\SD_{32}:S_3$ x 2, $C_{16}:D_6$ x 2, $S_3\times \SD_{32}$ x 2, $C_6\times \SD_{32}$, $C_{12}.D_8$, $D_{24}:C_2^2$, $D_{48}:C_2$, $C_{16}.D_6$, $D_4.S_4$, $\GL(2,3):C_2^2$, $D_{12}.D_4$, $D_8:D_6$

Order 168: $C_4\times F_7$ x 4, $C_{28}.C_6$ x 3, $C_2^2\times F_7$ x 2, $D_{14}:C_6$ x 2, $C_{28}:C_6$, $C_{28}.C_6$, $C_{28}:C_6$

Order 147: $C_7^2:C_3$

Order 128: $D_{16}:C_2^2$

Order 120: $\SL(2,5)$ x 2

Order 112: $Q_8\times D_7$, $D_4\times D_7$

Order 100: $D_{50}$, $C_{100}$, $C_{25}:C_4$

Order 98: $C_7:D_7$ x 2, $C_7\times D_7$ x 2, $C_7\times C_{14}$

Order 96: $C_3\times \SD_{32}$ x 4, $\GL(2,3):C_2$ x 4, $\GL(2,3):C_2$ x 3, $C_{48}:C_2$ x 2, $C_{48}:C_2$ x 2, $S_3\times C_{16}$ x 2, $C_3:Q_{32}$ x 2, $Q_{16}:S_3$ x 2, $C_3:\SD_{32}$ x 2, $C_3:D_{16}$ x 2, $\GL(2,3):C_2$ x 2, $C_2^2.S_4$ x 2, $D_{24}:C_2$ x 2, $Q_{16}:S_3$ x 2, $S_3\times Q_{16}$ x 2, $D_8:S_3$ x 2, $C_8:D_6$ x 2, $S_3\times D_8$ x 2, $D_{24}:C_2$ x 2, $C_3:Q_{32}$, $D_{48}$, $C_2\times C_{48}$, $Q_8.D_6$, $D_4:D_6$, $D_4.A_4$, $\SL(2,3):C_2^2$, $C_4.S_4$, $C_3:\OD_{32}$, $C_6\times Q_{16}$, $C_6\times D_8$, $C_{12}.D_4$, $D_{12}:C_2^2$, $C_8.D_6$

Order 84: $C_2\times F_7$ x 8, $C_7:C_{12}$ x 4, $C_7:C_{12}$ x 4, $C_{14}:C_6$ x 2

Order 80: $C_5:\OD_{16}$

Order 64: $Q_{32}:C_2$ x 3, $D_{16}:C_2$ x 3, $D_{16}:C_2$ x 3, $C_2\times \SD_{32}$ x 3, $D_4.D_4$, $D_8:C_2^2$, $\OD_{32}:C_2$

Order 56: $C_4\times D_7$ x 4, $C_7:Q_8$ x 3, $C_7:D_4$ x 2, $C_2\times D_{14}$ x 2, $C_7\times D_4$, $D_{28}$, $C_7\times Q_8$

Order 50: $C_{50}$, $D_{25}$

Order 49: $C_7^2$

Order 48: $D_{12}:C_2$ x 5, $\SL(2,3):C_2$ x 4, $\GL(2,3)$ x 4, $C_2.S_4$ x 4, $D_{12}:C_2$ x 3, $S_3\times Q_8$ x 3, $D_4:S_3$ x 3, $S_3\times D_4$ x 3, $C_3\times Q_{16}$ x 3, $C_3\times D_8$ x 3, $C_3:Q_{16}$ x 2, $D_{24}$ x 2, $C_{24}:C_2$ x 2, $C_{24}:C_2$ x 2, $C_6:D_4$ x 2, $S_3\times C_8$ x 2, $C_2\times \SL(2,3)$ x 2, $C_{48}$ x 2, $C_3:Q_{16}$ x 2, $Q_8:S_3$ x 2, $C_3:\SD_{16}$ x 2, $C_3:D_8$ x 2, $C_3:C_{16}$ x 2, $C_6\times Q_8$, $C_6\times D_4$, $C_2\times C_{24}$, $C_3:\OD_{16}$

Order 42: $F_7$ x 6, $C_7:C_6$ x 4

Order 40: $C_5:C_8$ x 2, $C_4\times D_5$

Order 32: $\SD_{32}$ x 10, $D_8:C_2$ x 6, $Q_{16}:C_2$ x 3, $D_8:C_2$ x 3, $C_2\times Q_{16}$ x 3, $C_2\times D_8$ x 3, $Q_{32}$ x 3, $D_{16}$ x 3, $\OD_{32}$ x 3, $C_2\times C_{16}$ x 3, $C_4.C_2^3$, $D_4:C_2^2$, $\OD_{16}:C_2$

Order 28: $D_{14}$ x 8, $C_{28}$ x 4, $C_7:C_4$ x 4, $C_2\times C_{14}$ x 2

Order 25: $C_{25}$

Order 24: $C_3:D_4$ x 8, $C_4\times S_3$ x 8, $D_{12}$ x 5, $C_3:Q_8$ x 5, $C_3\times Q_8$ x 3, $C_3\times D_4$ x 3, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2, $\SL(2,3)$ x 2, $C_{24}$ x 2, $C_2\times D_6$ x 2, $C_3:C_8$ x 2, $C_2^2\times C_6$

Order 21: $C_7:C_3$ x 2

Order 20: $D_{10}$, $C_{20}$, $C_5:C_4$

Order 16: $Q_{16}$ x 7, $D_8$ x 7, $D_4:C_2$ x 7, $\SD_{16}$ x 6, $C_{16}$ x 4, $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $C_2\times Q_8$ x 3, $C_2\times D_4$ x 3

Order 14: $D_7$ x 6, $C_{14}$ x 4

Order 12: $D_6$ x 6, $C_2\times C_6$ x 4, $C_{12}$ x 4, $C_3:C_4$ x 4

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

Order 8: $D_4$ x 7, $Q_8$ x 5, $C_2\times C_4$ x 4, $C_8$ x 4, $C_2^3$

Order 7: $C_7$ x 2

Order 6: $C_6$ x 4, $S_3$ x 4

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 470400: $\SL(2,49).C_2^2$

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

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

Order 9408: $C_7^2:(C_6\times \SD_{32})$

Order 3136: $(C_7\times C_{28}).D_8$

Order 400: $C_{100}.C_4$

Order 384: $D_{48}:C_2^2$

Order 147: $C_7^2:C_3$

Order 128: $D_{16}:C_2^2$

Order 80: $C_5:\OD_{16}$

Order 49: $C_7^2$

Order 25: $C_{25}$

Order 5: $C_5$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 470400: $\SL(2,49).C_2^2$ ($C_1$)

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

Order 117600: $\SL(2,49)$ ($C_2^2$)

Order 4: $C_4$ ($\POMinus(4,7)$)

Order 2: $C_2$ ($\PSL(2,49):C_2^2$)

Order 1: $C_1$ ($\SL(2,49).C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 470400: $\SL(2,49).C_2^2$ ($C_1$)

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

Order 117600: $\SL(2,49)$ ($C_2^2$)

Order 1: $C_1$ ($\SL(2,49).C_2^2$)

Series

Derived series	$\SL(2,49).C_2^2$	$\rhd$	$\SL(2,49)$
Chief series	$\SL(2,49).C_2^2$	$\rhd$	$\SL(2,49):C_2$	$\rhd$	$C_4$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$\SL(2,49).C_2^2$	$\rhd$	$\SL(2,49)$
Upper central series	$C_1$	$\lhd$	$C_2$	$\lhd$	$C_4$

Supergroups

This group is a maximal subgroup of 1 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 $80 \times 80$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $46 \times 46$ rational character table.