Abstract group 19600.c: $C_{280}:C_{70}$

• Label: 19600.c
• Order: \( 2^{4} \cdot 5^{2} \cdot 7^{2} \)
• Exponent: \( 2^{3} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \cdot 7 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 5 \cdot 7 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3^{2} \)
• Perm deg.: $32$
• Trans deg.: $560$
• Rank: $2$

Group information

Description:	$C_{280}:C_{70}$
Order:	\(19600\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 7^{2} \)
Exponent:	\(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \)
Automorphism group:	$C_2^4\times C_{12}\times F_5\times F_7$, of order \(161280\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_5$ x 2, $C_7$ x 2
Derived length:	$2$

This group is nonabelian and metacyclic (hence solvable, supersolvable, monomial, and metabelian).

Group statistics

Order	1	2	4	5	7	8	10	14	20	28	35	40	56	70	140	280
Elements	1	71	72	24	48	144	304	468	328	516	1152	656	1032	2832	3984	7968	19600
Conjugacy classes	1	2	3	14	27	4	18	33	32	60	588	56	108	612	1200	2352	5110
Divisions	1	2	2	4	5	2	5	6	5	6	26	5	6	27	27	27	156
Autjugacy classes	1	2	2	3	3	2	4	4	4	4	9	4	4	10	10	10	76

Minimal presentations

Permutation degree:	$32$
Transitive degree:	$560$
Rank:	$2$
Inequivalent generating pairs:	$576$

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

Constructions

Presentation:	$\langle a, b \mid b^{280}=1, a^{70}=b^{70}, b^{a}=b^{69} \rangle$
Permutation group:	Degree $32$ $\langle(14,16)(15,17)(19,21)(24,25)(27,29)(28,31)(30,32), (1,2,3,4,5,6,7)(8,9,10,11,12) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 27 & 0 \\ 0 & 94 \end{array}\right), \left(\begin{array}{rr} 81 & 0 \\ 0 & 170 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{281})$
Direct product:	$C_5$ $\, \times\, $ $C_7$ $\, \times\, $ $(C_{280}:C_2)$
Semidirect product:	$C_{280}$ $\,\rtimes\,$ $D_{35}$	$C_{280}$ $\,\rtimes\,$ $C_{70}$	$C_{35}^2$ $\,\rtimes\,$ $\OD_{16}$	$(C_7\times C_{280})$ $\,\rtimes\,$ $D_5$	all 32
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{70}$ . $C_{140}$	$C_{140}$ . $D_{70}$	$(C_{35}\times D_{70})$ . $C_4$	$(C_7\times D_{70})$ . $C_{20}$	all 44

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

Homology

Abelianization:	$C_{2} \times C_{140} \simeq C_{2} \times C_{4} \times C_{5} \times C_{7}$
Schur multiplier:	$C_1$
Commutator length:	not computed

Subgroups

There are 1664 subgroups in 288 conjugacy classes, 84 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{140}$	$G/Z \simeq$ $D_{70}$
Commutator:	$G' \simeq$ $C_{70}$	$G/G' \simeq$ $C_2\times C_{140}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_{35}\times D_{70}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{35}\times C_{280}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{280}:C_{70}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{35}\times C_{70}$	$G/\operatorname{soc} \simeq$ $C_2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{16}$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$

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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Subgroup information

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Classes of subgroups up to conjugation

Order 19600: $C_{280}:C_{70}$

Order 9800: $C_{35}:C_{280}$, $C_{35}\times C_{280}$, $D_{35}\times C_{140}$

Order 4900: $C_{35}\times C_{140}$, $C_{35}\times D_{70}$, $C_{35}:C_{140}$

Order 3920: $C_{56}:C_{70}$, $C_{280}:C_{14}$

Order 2800: $C_{40}:C_{70}$, $C_{280}:C_{10}$

Order 2450: $C_{35}\times D_{35}$, $C_{35}\times C_{70}$

Order 1960: $C_7\times C_{280}$ x 4, $D_7\times C_{140}$, $C_{35}:C_{56}$, $C_7:C_{280}$, $C_{28}\times D_{35}$

Order 1400: $C_5\times C_{280}$ x 5, $C_{20}\times D_{35}$, $D_5\times C_{140}$, $C_{35}:C_{40}$, $C_5:C_{280}$

Order 1225: $C_{35}^2$

Order 980: $C_7\times C_{140}$ x 4, $C_7\times D_{70}$, $D_7\times C_{70}$, $C_{35}:C_{28}$, $C_7:C_{140}$

Order 784: $C_{56}:C_{14}$

Order 700: $C_5\times C_{140}$ x 5, $C_5\times D_{70}$, $D_5\times C_{70}$, $C_{35}:C_{20}$, $C_5:C_{140}$

Order 560: $\OD_{16}\times C_{35}$, $C_{280}:C_2$, $C_{280}:C_2$, $C_{280}:C_2$

Order 490: $C_7\times C_{70}$ x 4, $C_7\times D_{35}$, $D_7\times C_{35}$

Order 400: $C_{40}:C_{10}$

Order 392: $D_7\times C_{28}$, $C_7\times C_{56}$, $C_7:C_{56}$

Order 350: $C_5\times C_{70}$ x 5, $C_5\times D_{35}$, $D_5\times C_{35}$

Order 280: $C_{280}$ x 27, $C_{35}:C_8$, $C_2\times C_{140}$, $C_4\times D_{35}$, $D_5\times C_{28}$, $C_7:C_{40}$, $D_7\times C_{20}$, $C_5:C_{56}$

Order 245: $C_7\times C_{35}$ x 4

Order 200: $D_5\times C_{20}$, $C_5\times C_{40}$, $C_5:C_{40}$

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

Order 175: $C_5\times C_{35}$ x 5

Order 140: $C_{140}$ x 27, $C_7\times D_{10}$, $C_5\times D_{14}$, $C_{35}:C_4$, $C_7:C_{20}$, $C_2\times C_{70}$, $D_{70}$, $C_5:C_{28}$

Order 112: $C_{56}:C_2$, $C_7\times \OD_{16}$

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

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

Order 80: $C_{40}:C_2$, $C_5\times \OD_{16}$

Order 70: $C_{70}$ x 27, $D_{35}$, $C_5\times D_7$, $C_7\times D_5$

Order 56: $C_{56}$ x 6, $C_2\times C_{28}$, $C_4\times D_7$, $C_7:C_8$

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

Order 49: $C_7^2$

Order 40: $C_{40}$ x 5, $C_2\times C_{20}$, $C_4\times D_5$, $C_5:C_8$

Order 35: $C_{35}$ x 26

Order 28: $C_{28}$ x 6, $C_2\times C_{14}$, $D_{14}$, $C_7:C_4$

Order 25: $C_5^2$

Order 20: $C_{20}$ x 5, $C_2\times C_{10}$, $D_{10}$, $C_5:C_4$

Order 16: $\OD_{16}$

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

Order 10: $C_{10}$ x 5, $D_5$

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

Order 7: $C_7$ x 5

Order 5: $C_5$ x 4

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 19600: $C_{280}:C_{70}$

Order 9800: $C_{35}:C_{280}$, $C_{35}\times C_{280}$, $D_{35}\times C_{140}$

Order 4900: $C_{35}\times C_{140}$, $C_{35}\times D_{70}$, $C_{35}:C_{140}$

Order 3920: $C_{56}:C_{70}$, $C_{280}:C_{14}$

Order 2800: $C_{40}:C_{70}$, $C_{280}:C_{10}$

Order 2450: $C_{35}\times D_{35}$, $C_{35}\times C_{70}$

Order 1960: $C_7\times C_{280}$ x 3, $D_7\times C_{140}$, $C_{35}:C_{56}$, $C_7:C_{280}$, $C_{28}\times D_{35}$

Order 1400: $C_5\times C_{280}$ x 3, $C_{20}\times D_{35}$, $D_5\times C_{140}$, $C_{35}:C_{40}$, $C_5:C_{280}$

Order 1225: $C_{35}^2$

Order 980: $C_7\times C_{140}$ x 3, $C_7\times D_{70}$, $D_7\times C_{70}$, $C_{35}:C_{28}$, $C_7:C_{140}$

Order 784: $C_{56}:C_{14}$

Order 700: $C_5\times C_{140}$ x 3, $C_5\times D_{70}$, $D_5\times C_{70}$, $C_{35}:C_{20}$, $C_5:C_{140}$

Order 560: $\OD_{16}\times C_{35}$, $C_{280}:C_2$, $C_{280}:C_2$, $C_{280}:C_2$

Order 490: $C_7\times C_{70}$ x 3, $C_7\times D_{35}$, $D_7\times C_{35}$

Order 400: $C_{40}:C_{10}$

Order 392: $D_7\times C_{28}$, $C_7\times C_{56}$, $C_7:C_{56}$

Order 350: $C_5\times C_{70}$ x 3, $C_5\times D_{35}$, $D_5\times C_{35}$

Order 280: $C_{280}$ x 10, $C_{35}:C_8$, $C_2\times C_{140}$, $C_4\times D_{35}$, $D_5\times C_{28}$, $C_7:C_{40}$, $D_7\times C_{20}$, $C_5:C_{56}$

Order 245: $C_7\times C_{35}$ x 3

Order 200: $D_5\times C_{20}$, $C_5\times C_{40}$, $C_5:C_{40}$

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

Order 175: $C_5\times C_{35}$ x 3

Order 140: $C_{140}$ x 10, $C_7\times D_{10}$, $C_5\times D_{14}$, $C_{35}:C_4$, $C_7:C_{20}$, $C_2\times C_{70}$, $D_{70}$, $C_5:C_{28}$

Order 112: $C_{56}:C_2$, $C_7\times \OD_{16}$

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

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

Order 80: $C_{40}:C_2$, $C_5\times \OD_{16}$

Order 70: $C_{70}$ x 10, $D_{35}$, $C_5\times D_7$, $C_7\times D_5$

Order 56: $C_{56}$ x 4, $C_2\times C_{28}$, $C_4\times D_7$, $C_7:C_8$

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

Order 49: $C_7^2$

Order 40: $C_{40}$ x 4, $C_2\times C_{20}$, $C_4\times D_5$, $C_5:C_8$

Order 35: $C_{35}$ x 9

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

Order 25: $C_5^2$

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

Order 16: $\OD_{16}$

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

Order 10: $C_{10}$ x 4, $D_5$

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

Order 7: $C_7$ x 3

Order 5: $C_5$ x 3

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 19600: $C_{280}:C_{70}$ ($C_1$)

Order 9800: $C_{35}:C_{280}$ ($C_2$), $C_{35}\times C_{280}$ ($C_2$), $D_{35}\times C_{140}$ ($C_2$)

Order 4900: $C_{35}\times C_{140}$ ($C_2^2$), $C_{35}\times D_{70}$ ($C_4$), $C_{35}:C_{140}$ ($C_4$)

Order 3920: $C_{280}:C_{14}$ ($C_5$)

Order 2800: $C_{280}:C_{10}$ ($C_7$)

Order 2450: $C_{35}\times C_{70}$ ($C_2\times C_4$)

Order 1960: $C_7\times C_{280}$ ($C_{10}$), $C_7\times C_{280}$ ($D_5$), $C_{35}:C_{56}$ ($C_{10}$), $C_{28}\times D_{35}$ ($C_{10}$)

Order 1400: $C_{20}\times D_{35}$ ($C_{14}$), $C_5\times C_{280}$ ($C_{14}$), $C_5\times C_{280}$ ($D_7$), $C_{35}:C_{40}$ ($C_{14}$)

Order 1225: $C_{35}^2$ ($\OD_{16}$)

Order 980: $C_7\times D_{70}$ ($C_{20}$), $C_7\times C_{140}$ ($C_2\times C_{10}$), $C_7\times C_{140}$ ($D_{10}$), $C_{35}:C_{28}$ ($C_{20}$)

Order 700: $C_5\times D_{70}$ ($C_{28}$), $C_5\times C_{140}$ ($C_2\times C_{14}$), $C_5\times C_{140}$ ($D_{14}$), $C_{35}:C_{20}$ ($C_{28}$)

Order 560: $C_{280}:C_2$ ($C_{35}$)

Order 490: $C_7\times C_{70}$ ($C_2\times C_{20}$), $C_7\times C_{70}$ ($C_4\times D_5$)

Order 392: $C_7\times C_{56}$ ($C_5\times D_5$)

Order 350: $C_5\times C_{70}$ ($C_2\times C_{28}$), $C_5\times C_{70}$ ($C_4\times D_7$)

Order 280: $C_{280}$ ($C_{70}$), $C_{280}$ ($D_{35}$), $C_{280}$ ($C_5\times D_7$), $C_{280}$ ($C_7\times D_5$), $C_{35}:C_8$ ($C_{70}$), $C_4\times D_{35}$ ($C_{70}$)

Order 245: $C_7\times C_{35}$ ($C_{40}:C_2$), $C_7\times C_{35}$ ($C_5\times \OD_{16}$)

Order 200: $C_5\times C_{40}$ ($C_7\times D_7$)

Order 196: $C_7\times C_{28}$ ($C_5\times D_{10}$)

Order 175: $C_5\times C_{35}$ ($C_{56}:C_2$), $C_5\times C_{35}$ ($C_7\times \OD_{16}$)

Order 140: $C_{140}$ ($C_7\times D_{10}$), $C_{140}$ ($C_5\times D_{14}$), $C_{140}$ ($C_2\times C_{70}$), $C_{140}$ ($D_{70}$), $C_{35}:C_4$ ($C_{140}$), $D_{70}$ ($C_{140}$)

Order 100: $C_5\times C_{20}$ ($C_7\times D_{14}$)

Order 98: $C_7\times C_{14}$ ($D_5\times C_{20}$)

Order 70: $C_{70}$ ($C_2\times C_{140}$), $C_{70}$ ($C_4\times D_{35}$), $C_{70}$ ($D_5\times C_{28}$), $C_{70}$ ($D_7\times C_{20}$)

Order 56: $C_{56}$ ($C_5\times D_{35}$), $C_{56}$ ($D_5\times C_{35}$)

Order 50: $C_5\times C_{10}$ ($D_7\times C_{28}$)

Order 49: $C_7^2$ ($C_{40}:C_{10}$)

Order 40: $C_{40}$ ($C_7\times D_{35}$), $C_{40}$ ($D_7\times C_{35}$)

Order 35: $C_{35}$ ($\OD_{16}\times C_{35}$), $C_{35}$ ($C_{280}:C_2$), $C_{35}$ ($C_{280}:C_2$), $C_{35}$ ($C_{280}:C_2$)

Order 28: $C_{28}$ ($C_5\times D_{70}$), $C_{28}$ ($D_5\times C_{70}$)

Order 25: $C_5^2$ ($C_{56}:C_{14}$)

Order 20: $C_{20}$ ($C_7\times D_{70}$), $C_{20}$ ($D_7\times C_{70}$)

Order 14: $C_{14}$ ($C_{20}\times D_{35}$), $C_{14}$ ($D_5\times C_{140}$)

Order 10: $C_{10}$ ($D_7\times C_{140}$), $C_{10}$ ($C_{28}\times D_{35}$)

Order 8: $C_8$ ($C_{35}\times D_{35}$)

Order 7: $C_7$ ($C_{40}:C_{70}$), $C_7$ ($C_{280}:C_{10}$)

Order 5: $C_5$ ($C_{56}:C_{70}$), $C_5$ ($C_{280}:C_{14}$)

Order 4: $C_4$ ($C_{35}\times D_{70}$)

Order 2: $C_2$ ($D_{35}\times C_{140}$)

Order 1: $C_1$ ($C_{280}:C_{70}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 19600: $C_{280}:C_{70}$ ($C_1$)

Order 9800: $C_{35}:C_{280}$ ($C_2$), $C_{35}\times C_{280}$ ($C_2$), $D_{35}\times C_{140}$ ($C_2$)

Order 4900: $C_{35}\times C_{140}$ ($C_2^2$), $C_{35}\times D_{70}$ ($C_4$), $C_{35}:C_{140}$ ($C_4$)

Order 3920: $C_{280}:C_{14}$ ($C_5$)

Order 2800: $C_{280}:C_{10}$ ($C_7$)

Order 2450: $C_{35}\times C_{70}$ ($C_2\times C_4$)

Order 1960: $C_7\times C_{280}$ ($C_{10}$), $C_7\times C_{280}$ ($D_5$), $C_{35}:C_{56}$ ($C_{10}$), $C_{28}\times D_{35}$ ($C_{10}$)

Order 1400: $C_{20}\times D_{35}$ ($C_{14}$), $C_5\times C_{280}$ ($C_{14}$), $C_5\times C_{280}$ ($D_7$), $C_{35}:C_{40}$ ($C_{14}$)

Order 1225: $C_{35}^2$ ($\OD_{16}$)

Order 980: $C_7\times D_{70}$ ($C_{20}$), $C_7\times C_{140}$ ($C_2\times C_{10}$), $C_7\times C_{140}$ ($D_{10}$), $C_{35}:C_{28}$ ($C_{20}$)

Order 700: $C_5\times D_{70}$ ($C_{28}$), $C_5\times C_{140}$ ($C_2\times C_{14}$), $C_5\times C_{140}$ ($D_{14}$), $C_{35}:C_{20}$ ($C_{28}$)

Order 560: $C_{280}:C_2$ ($C_{35}$)

Order 490: $C_7\times C_{70}$ ($C_2\times C_{20}$), $C_7\times C_{70}$ ($C_4\times D_5$)

Order 392: $C_7\times C_{56}$ ($C_5\times D_5$)

Order 350: $C_5\times C_{70}$ ($C_2\times C_{28}$), $C_5\times C_{70}$ ($C_4\times D_7$)

Order 280: $C_{280}$ ($C_{70}$), $C_{280}$ ($D_{35}$), $C_{280}$ ($C_5\times D_7$), $C_{280}$ ($C_7\times D_5$), $C_{35}:C_8$ ($C_{70}$), $C_4\times D_{35}$ ($C_{70}$)

Order 245: $C_7\times C_{35}$ ($C_{40}:C_2$), $C_7\times C_{35}$ ($C_5\times \OD_{16}$)

Order 200: $C_5\times C_{40}$ ($C_7\times D_7$)

Order 196: $C_7\times C_{28}$ ($C_5\times D_{10}$)

Order 175: $C_5\times C_{35}$ ($C_{56}:C_2$), $C_5\times C_{35}$ ($C_7\times \OD_{16}$)

Order 140: $C_{140}$ ($C_7\times D_{10}$), $C_{140}$ ($C_5\times D_{14}$), $C_{140}$ ($C_2\times C_{70}$), $C_{140}$ ($D_{70}$), $C_{35}:C_4$ ($C_{140}$), $D_{70}$ ($C_{140}$)

Order 100: $C_5\times C_{20}$ ($C_7\times D_{14}$)

Order 98: $C_7\times C_{14}$ ($D_5\times C_{20}$)

Order 70: $C_{70}$ ($C_2\times C_{140}$), $C_{70}$ ($C_4\times D_{35}$), $C_{70}$ ($D_5\times C_{28}$), $C_{70}$ ($D_7\times C_{20}$)

Order 56: $C_{56}$ ($C_5\times D_{35}$), $C_{56}$ ($D_5\times C_{35}$)

Order 50: $C_5\times C_{10}$ ($D_7\times C_{28}$)

Order 49: $C_7^2$ ($C_{40}:C_{10}$)

Order 40: $C_{40}$ ($C_7\times D_{35}$), $C_{40}$ ($D_7\times C_{35}$)

Order 35: $C_{35}$ ($\OD_{16}\times C_{35}$), $C_{35}$ ($C_{280}:C_2$), $C_{35}$ ($C_{280}:C_2$), $C_{35}$ ($C_{280}:C_2$)

Order 28: $C_{28}$ ($C_5\times D_{70}$), $C_{28}$ ($D_5\times C_{70}$)

Order 25: $C_5^2$ ($C_{56}:C_{14}$)

Order 20: $C_{20}$ ($C_7\times D_{70}$), $C_{20}$ ($D_7\times C_{70}$)

Order 14: $C_{14}$ ($C_{20}\times D_{35}$), $C_{14}$ ($D_5\times C_{140}$)

Order 10: $C_{10}$ ($D_7\times C_{140}$), $C_{10}$ ($C_{28}\times D_{35}$)

Order 8: $C_8$ ($C_{35}\times D_{35}$)

Order 7: $C_7$ ($C_{40}:C_{70}$), $C_7$ ($C_{280}:C_{10}$)

Order 5: $C_5$ ($C_{56}:C_{70}$), $C_5$ ($C_{280}:C_{14}$)

Order 4: $C_4$ ($C_{35}\times D_{70}$)

Order 2: $C_2$ ($D_{35}\times C_{140}$)

Order 1: $C_1$ ($C_{280}:C_{70}$)

Series

Derived series

$C_{280}:C_{70}$

$\rhd$

$C_{70}$

$\rhd$

$C_1$

Chief series

$C_{280}:C_{70}$

$\rhd$

$C_{35}\times C_{280}$

$\rhd$

$C_7\times C_{280}$

$\rhd$

$C_{280}$

$\rhd$

$C_{140}$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{280}:C_{70}$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

Upper central series

$C_1$

$\lhd$

$C_{140}$

$\lhd$

$C_{280}$

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

The $5110 \times 5110$ character table is not available for this group.

Rational character table

The $156 \times 156$ rational character table is not available for this group.