Abstract group 1302.10: $C_{217}:C_6$

• Label: 1302.10
• Order: \( 2 \cdot 3 \cdot 7 \cdot 31 \)
• Exponent: \( 2 \cdot 3 \cdot 7 \cdot 31 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \cdot 5 \)
• Perm deg.: $38$
• Trans deg.: $217$
• Rank: $2$

Group information

Description:	$C_{217}:C_6$
Order:	\(1302\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 31 \)
Exponent:	\(1302\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 31 \)
Automorphism group:	$C_{31}:(C_{30}\times F_7)$, of order \(39060\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \)
Composition factors:	$C_2$, $C_3$, $C_7$, $C_{31}$
Derived length:	$2$

This group is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).

Group statistics

Order	1	2	3	6	7	31	62	217
Elements	1	7	434	434	6	30	210	180	1302
Conjugacy classes	1	1	2	2	1	10	10	30	57
Divisions	1	1	1	1	1	1	1	1	8
Autjugacy classes	1	1	2	2	1	1	1	1	10

Dimension	1	2	3	6	30	180
Irr. complex chars.	6	0	20	31	0	0	57
Irr. rational chars.	2	2	0	1	2	1	8

Minimal presentations

Permutation degree:	$38$
Transitive degree:	$217$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	180
Arbitrary	6	12	36

Constructions

Presentation:	$\langle a, b \mid a^{6}=b^{217}=1, b^{a}=b^{129} \rangle$
Permutation group:	Degree $38$ $\langle(33,34)(35,37)(36,38), (2,3,5)(4,7,12)(6,10,9)(8,14,21)(11,15,18)(13,19,16) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{31}$ $\,\rtimes\,$ $F_7$	$C_{217}$ $\,\rtimes\,$ $C_6$	$(D_7\times C_{31})$ $\,\rtimes\,$ $C_3$	$D_7$ $\,\rtimes\,$ $(C_{31}:C_3)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 532 subgroups in 16 conjugacy classes, 8 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_{217}:C_6$
Commutator:	$G' \simeq$ $C_{217}$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{217}:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{217}$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $C_{217}:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{217}$	$G/\operatorname{soc} \simeq$ $C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
31-Sylow subgroup:	$P_{ 31 } \simeq$ $C_{31}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1302: $C_{217}:C_6$

Order 651: $C_{217}:C_3$

Order 434: $D_7\times C_{31}$

Order 217: $C_{217}$

Order 186: $C_{31}:C_6$

Order 93: $C_{31}:C_3$

Order 62: $C_{62}$

Order 42: $F_7$

Order 31: $C_{31}$

Order 21: $C_7:C_3$

Order 14: $D_7$

Order 7: $C_7$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1302: $C_{217}:C_6$

Order 651: $C_{217}:C_3$

Order 434: $D_7\times C_{31}$

Order 217: $C_{217}$

Order 186: $C_{31}:C_6$

Order 93: $C_{31}:C_3$

Order 62: $C_{62}$

Order 42: $F_7$

Order 31: $C_{31}$

Order 21: $C_7:C_3$

Order 14: $D_7$

Order 7: $C_7$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1302: $C_{217}:C_6$ ($C_1$)

Order 651: $C_{217}:C_3$ ($C_2$)

Order 434: $D_7\times C_{31}$ ($C_3$)

Order 217: $C_{217}$ ($C_6$)

Order 31: $C_{31}$ ($F_7$)

Order 14: $D_7$ ($C_{31}:C_3$)

Order 7: $C_7$ ($C_{31}:C_6$)

Order 1: $C_1$ ($C_{217}:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1302: $C_{217}:C_6$ ($C_1$)

Order 651: $C_{217}:C_3$ ($C_2$)

Order 434: $D_7\times C_{31}$ ($C_3$)

Order 217: $C_{217}$ ($C_6$)

Order 31: $C_{31}$ ($F_7$)

Order 14: $D_7$ ($C_{31}:C_3$)

Order 7: $C_7$ ($C_{31}:C_6$)

Order 1: $C_1$ ($C_{217}:C_6$)

Series

Derived series	$C_{217}:C_6$	$\rhd$	$C_{217}$	$\rhd$	$C_1$
Chief series	$C_{217}:C_6$	$\rhd$	$C_{217}:C_3$	$\rhd$	$C_{217}$	$\rhd$	$C_{31}$	$\rhd$	$C_1$
Lower central series	$C_{217}:C_6$	$\rhd$	$C_{217}$
Upper central series	$C_1$

Character theory

Complex character table

See the $57 \times 57$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	6A	7A	31A	62A	217A
	Size	1	7	434	434	6	30	210	180
	2 P	1A	1A	3A	3A	7A	31A	31A	217A
	3 P	1A	2A	1A	2A	7A	31A	62A	217A
	7 P	1A	2A	3A	6A	7A	31A	62A	217A
	31 P	1A	2A	3A	6A	1A	31A	62A	31A
1302.10.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1302.10.1b		$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$
1302.10.1c		$2$	$2$	$-1$	$-1$	$2$	$2$	$2$	$2$
1302.10.1d		$2$	$-2$	$-1$	$1$	$2$	$2$	$-2$	$2$
1302.10.3a		$30$	$30$	$0$	$0$	$30$	$-1$	$-1$	$-1$
1302.10.3b		$30$	$-30$	$0$	$0$	$30$	$-1$	$1$	$-1$
1302.10.6a		$6$	$0$	$0$	$0$	$-1$	$6$	$0$	$-1$
1302.10.6b		$180$	$0$	$0$	$0$	$-30$	$-6$	$0$	$1$