Abstract group 3578400.a: $C_{10}.\PGL(2,71)$

• Label: 3578400.a
• Order: \( 2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 71 \)
• Exponent: \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 71 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 5 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 71 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $149$
• Trans deg.: $720$
• Rank: $2$

Group information

Description:	$C_{10}.\PGL(2,71)$
Order:	\(3578400\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 71 \)
Exponent:	\(357840\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 71 \)
Automorphism group:	$C_2\times C_4\times \PSL(2,71).C_2$, of order \(2862720\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 71 \)
Composition factors:	$C_2$ x 2, $C_5$, $\PSL(2,71)$
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	9	10	12	14	15	16	18	20	24	30	35	36	40	45	48	60	70	71	72	80	90	120	142	144	180	240	355	360	710	720
Elements	1	5113	4970	4970	51124	4970	15336	9940	14910	173812	9940	46008	19880	19880	14910	19880	19880	19880	368064	29820	39760	59640	39760	39760	1104192	5040	59640	79520	59640	79520	5040	119280	119280	159040	20160	238560	20160	477120	3578400
Conjugacy classes	1	2	1	1	14	1	3	2	3	38	2	9	4	4	3	4	4	4	72	6	8	12	8	8	216	1	12	16	12	16	1	24	24	32	4	48	4	96	720
Divisions	1	2	1	1	4	1	1	1	1	10	1	2	1	1	1	1	1	1	4	1	1	1	1	1	10	1	1	1	1	1	1	1	1	1	1	1	1	1	64
Autjugacy classes	1	2	1	1	5	1	3	2	3	10	2	6	1	2	3	1	4	1	27	6	2	3	4	2	54	1	12	2	3	4	1	12	6	4	1	12	1	12	218

Dimension	1	4	70	71	72	140	144	210	216	280	284	288	420	432	560	840	864	1120	1680	1728	2240	3360	6720
Irr. complex chars.	10	0	355	10	345	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	720
Irr. rational chars.	2	2	3	2	1	2	2	2	2	5	2	10	1	1	3	3	4	2	2	10	1	1	1	64

Minimal presentations

Permutation degree:	$149$
Transitive degree:	$720$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	70	not computed	288
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $149$ $\langle(1,2,8,30,94,68,106,131,98,58,15,3,12,47,44,83,62,129,137,113,46,60,128,72,93,29,92,53,13,50,89,134,114,54,110,88,26,86,91,28,90,67,123,121,76,70,80,141,118,56,20,73,108,40,103,37,9,34,84,95,100,130,127,143,115,133,66,107,78,22,5) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 14 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{71})$
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:	$\SL(2,71)$ . $C_{10}$	$C_{10}$ . $\PGL(2,71)$	$(C_5\times \SL(2,71))$ . $C_2$	$(\SL(2,71):C_2)$ . $C_5$	all 6

Elements of the group are displayed as matrices in $\GL_{2}(\F_{71})$.

Homology

Abelianization:	$C_{10} \simeq C_{2} \times C_{5}$
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 1379192 subgroups in 234 conjugacy classes, 8 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $\PGL(2,71)$
Commutator:	$G' \simeq$ $\SL(2,71)$	$G/G' \simeq$ $C_{10}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_5.\PSL(2,71).C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{10}$	$G/\operatorname{Fit} \simeq$ $\PGL(2,71)$
Radical:	$R \simeq$ $C_{10}$	$G/R \simeq$ $\PGL(2,71)$
Socle:	$\operatorname{soc} \simeq$ $C_{10}$	$G/\operatorname{soc} \simeq$ $\PGL(2,71)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{32}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
71-Sylow subgroup:	$P_{ 71 } \simeq$ $C_{71}$

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 3578400: $C_{10}.\PGL(2,71)$

Order 1789200: $C_5\times \SL(2,71)$

Order 715680: $\SL(2,71):C_2$

Order 357840: $\SL(2,71)$

Order 49700: $C_{10}\times F_{71}$

Order 24850: $C_5\times F_{71}$ x 2, $C_{355}:C_{70}$

Order 12425: $C_{355}:C_{35}$

Order 9940: $C_2\times F_{71}$ x 5, $C_{142}:C_{70}$

Order 7100: $C_{710}:C_{10}$

Order 4970: $F_{71}$ x 10, $C_{142}:C_{35}$ x 5, $C_{355}:C_{14}$ x 2, $C_{142}:C_{35}$

Order 3550: $C_{355}:C_{10}$ x 2, $C_{355}:C_{10}$

Order 2485: $C_{71}:C_{35}$ x 5, $C_{355}:C_7$

Order 1988: $C_{142}:C_{14}$

Order 1775: $C_{355}:C_5$

Order 1440: $C_{144}:C_{10}$

Order 1420: $C_{142}:C_{10}$ x 5, $C_5\times D_{142}$

Order 1400: $D_{70}:C_{10}$

Order 994: $C_{71}:C_{14}$ x 2, $C_{142}:C_7$

Order 720: $C_5\times D_{72}$, $C_{45}:Q_{16}$, $C_{720}$

Order 710: $C_{71}:C_{10}$ x 10, $C_{142}:C_5$ x 5, $C_5\times D_{71}$ x 2, $C_{710}$

Order 700: $C_{10}\times C_{70}$, $C_5\times D_{70}$, $C_{35}:C_{20}$

Order 600: $C_5\times \SL(2,5)$

Order 497: $C_{71}:C_7$

Order 480: $C_{240}:C_2$

Order 360: $C_{360}$, $C_5\times D_{36}$, $C_{45}:Q_8$

Order 355: $C_{71}:C_5$ x 5, $C_{355}$

Order 350: $C_5\times C_{70}$ x 2, $C_5\times D_{35}$

Order 288: $C_{144}:C_2$

Order 284: $D_{142}$

Order 280: $C_{35}:D_4$, $C_{35}:D_4$

Order 240: $C_{15}:Q_{16}$, $C_5\times D_{24}$, $C_{240}$, $C_{10}.S_4$

Order 200: $C_{10}\wr C_2$

Order 180: $C_{180}$, $C_5\times D_{18}$, $C_9:C_{20}$

Order 175: $C_5\times C_{35}$

Order 160: $C_5\times \SD_{32}$

Order 144: $D_{72}$, $C_9:Q_{16}$, $C_{144}$

Order 142: $D_{71}$ x 2, $C_{142}$

Order 140: $C_2\times C_{70}$ x 4, $C_5\times D_{14}$, $C_{35}:C_4$, $C_7:C_{20}$, $D_{70}$

Order 120: $\SL(2,5)$, $C_{120}$, $C_5\times D_{12}$, $C_{15}:Q_8$, $C_5\times \SL(2,3)$

Order 100: $C_5:C_{20}$, $C_{10}^2$, $C_5\times D_{10}$

Order 96: $C_{48}:C_2$

Order 90: $C_{90}$, $C_5\times D_9$

Order 80: $C_5\times Q_{16}$, $C_5\times D_8$, $C_{80}$

Order 72: $D_{36}$, $C_9:Q_8$, $C_{72}$

Order 71: $C_{71}$

Order 70: $C_{70}$ x 10, $D_{35}$, $C_5\times D_7$

Order 60: $C_{60}$, $S_3\times C_{10}$, $C_3:C_{20}$

Order 56: $C_7:D_4$

Order 50: $C_5\times C_{10}$ x 2, $C_5\times D_5$

Order 48: $C_3:Q_{16}$, $D_{24}$, $C_2.S_4$, $C_{48}$

Order 45: $C_{45}$

Order 40: $C_5:D_4$, $C_{40}$, $C_5\times Q_8$, $C_5\times D_4$

Order 36: $D_{18}$, $C_{36}$, $C_9:C_4$

Order 35: $C_{35}$ x 4

Order 32: $\SD_{32}$

Order 30: $C_{30}$, $C_5\times S_3$

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

Order 25: $C_5^2$

Order 24: $D_{12}$, $C_3:Q_8$, $\SL(2,3)$, $C_{24}$

Order 20: $C_2\times C_{10}$ x 4, $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_{18}$, $D_9$

Order 16: $Q_{16}$, $D_8$, $C_{16}$

Order 15: $C_{15}$

Order 14: $C_{14}$ x 2, $D_7$

Order 12: $D_6$, $C_{12}$, $C_3:C_4$

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

Order 9: $C_9$

Order 8: $Q_8$, $D_4$, $C_8$

Order 7: $C_7$

Order 6: $C_6$, $S_3$

Order 5: $C_5$ x 4

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3578400: $C_{10}.\PGL(2,71)$

Order 1789200: $C_5\times \SL(2,71)$

Order 715680: $\SL(2,71):C_2$

Order 357840: $\SL(2,71)$

Order 49700: $C_{10}\times F_{71}$

Order 24850: $C_5\times F_{71}$

Order 12425: $C_{355}:C_{35}$

Order 7100: $C_{710}:C_{10}$

Order 1988: $C_{142}:C_{14}$

Order 1775: $C_{355}:C_5$

Order 1440: $C_{144}:C_{10}$

Order 1400: $D_{70}:C_{10}$

Order 600: $C_5\times \SL(2,5)$

Order 497: $C_{71}:C_7$

Order 480: $C_{240}:C_2$

Order 288: $C_{144}:C_2$

Order 284: $D_{142}$

Order 200: $C_{10}\wr C_2$

Order 175: $C_5\times C_{35}$

Order 160: $C_5\times \SD_{32}$

Order 96: $C_{48}:C_2$

Order 71: $C_{71}$

Order 56: $C_7:D_4$

Order 45: $C_{45}$

Order 32: $\SD_{32}$

Order 25: $C_5^2$

Order 15: $C_{15}$

Order 9: $C_9$

Order 7: $C_7$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3578400: $C_{10}.\PGL(2,71)$ ($C_1$)

Order 1789200: $C_5\times \SL(2,71)$ ($C_2$)

Order 715680: $\SL(2,71):C_2$ ($C_5$)

Order 357840: $\SL(2,71)$ ($C_{10}$)

Order 10: $C_{10}$ ($\PGL(2,71)$)

Order 5: $C_5$ ($\SL(2,71):C_2$)

Order 2: $C_2$ ($C_5.\PSL(2,71).C_2$)

Order 1: $C_1$ ($C_{10}.\PGL(2,71)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3578400: $C_{10}.\PGL(2,71)$ ($C_1$)

Order 1789200: $C_5\times \SL(2,71)$ ($C_2$)

Order 715680: $\SL(2,71):C_2$ ($C_5$)

Order 357840: $\SL(2,71)$ ($C_{10}$)

Order 10: $C_{10}$ ($\PGL(2,71)$)

Order 1: $C_1$ ($C_{10}.\PGL(2,71)$)

Series

Derived series	$C_{10}.\PGL(2,71)$	$\rhd$	$\SL(2,71)$
Chief series	$C_{10}.\PGL(2,71)$	$\rhd$	$C_5\times \SL(2,71)$	$\rhd$	$C_{10}$	$\rhd$	$C_5$	$\rhd$	$C_1$
Lower central series	$C_{10}.\PGL(2,71)$	$\rhd$	$\SL(2,71)$
Upper central series	$C_1$	$\lhd$	$C_{10}$

Character theory

Complex character table

The $720 \times 720$ character table is not available for this group.

Rational character table

See the $64 \times 64$ rational character table.