Abstract group 168.39: $C_2\times C_{84}$

• Label: 168.39
• Order: \( 2^{3} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $16$
• Trans deg.: $168$
• Rank: $2$

Group information

Description:	$C_2\times C_{84}$
Order:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$C_{12}:C_2^3$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \)
Composition factors:	$C_2$ x 3, $C_3$, $C_7$
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.

Group statistics

Order	1	2	3	4	6	7	12	14	21	28	42	84
Elements	1	3	2	4	6	6	8	18	12	24	36	48	168
Conjugacy classes	1	3	2	4	6	6	8	18	12	24	36	48	168
Divisions	1	3	1	2	3	1	2	3	1	2	3	2	24
Autjugacy classes	1	2	1	1	2	1	1	2	1	1	2	1	16

Dimension	1	2	4	6	12	24
Irr. complex chars.	168	0	0	0	0	0	168
Irr. rational chars.	4	6	2	4	6	2	24

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$168$
Rank:	$2$
Inequivalent generating pairs:	$96$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	2	3	10

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{2}=b^{84}=1 \rangle$
Permutation group:	Degree $16$ $\langle(3,6,4,5), (1,2), (7,9,8), (10,16,15,14,13,12,11), (3,4)(5,6)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 258 & 0 \\ 0 & 273 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 265 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{337})$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \times\, $ $C_3$ $\, \times\, $ $C_7$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{42}$ . $C_2^2$	$C_2^2$ . $C_{42}$	$(C_2\times C_{42})$ . $C_2$	$(C_2\times C_{14})$ . $C_6$	all 8
Aut. group:	$\Aut(C_{203})$	$\Aut(C_{215})$	$\Aut(C_{245})$	$\Aut(C_{261})$	all 8

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

Homology

Primary decomposition:	$C_{2} \times C_{4} \times C_{3} \times C_{7}$
Schur multiplier:	$C_{2}$
Commutator length:	$0$

Subgroups

There are 32 subgroups, all normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{84}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2\times C_{84}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times C_{42}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{84}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2\times C_{84}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{42}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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 168: $C_2\times C_{84}$

Order 84: $C_{84}$ x 2, $C_2\times C_{42}$

Order 56: $C_2\times C_{28}$

Order 42: $C_{42}$ x 3

Order 28: $C_{28}$ x 2, $C_2\times C_{14}$

Order 24: $C_2\times C_{12}$

Order 21: $C_{21}$

Order 14: $C_{14}$ x 3

Order 12: $C_{12}$ x 2, $C_2\times C_6$

Order 8: $C_2\times C_4$

Order 7: $C_7$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 168: $C_2\times C_{84}$

Order 84: $C_{84}$, $C_2\times C_{42}$

Order 56: $C_2\times C_{28}$

Order 42: $C_{42}$ x 2

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

Order 24: $C_2\times C_{12}$

Order 21: $C_{21}$

Order 14: $C_{14}$ x 2

Order 12: $C_2\times C_6$, $C_{12}$

Order 8: $C_2\times C_4$

Order 7: $C_7$

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 168: $C_2\times C_{84}$ ($C_1$)

Order 84: $C_{84}$ ($C_2$) x 2, $C_2\times C_{42}$ ($C_2$)

Order 56: $C_2\times C_{28}$ ($C_3$)

Order 42: $C_{42}$ ($C_4$) x 2, $C_{42}$ ($C_2^2$)

Order 28: $C_{28}$ ($C_6$) x 2, $C_2\times C_{14}$ ($C_6$)

Order 24: $C_2\times C_{12}$ ($C_7$)

Order 21: $C_{21}$ ($C_2\times C_4$)

Order 14: $C_{14}$ ($C_{12}$) x 2, $C_{14}$ ($C_2\times C_6$)

Order 12: $C_{12}$ ($C_{14}$) x 2, $C_2\times C_6$ ($C_{14}$)

Order 8: $C_2\times C_4$ ($C_{21}$)

Order 7: $C_7$ ($C_2\times C_{12}$)

Order 6: $C_6$ ($C_{28}$) x 2, $C_6$ ($C_2\times C_{14}$)

Order 4: $C_4$ ($C_{42}$) x 2, $C_2^2$ ($C_{42}$)

Order 3: $C_3$ ($C_2\times C_{28}$)

Order 2: $C_2$ ($C_{84}$) x 2, $C_2$ ($C_2\times C_{42}$)

Order 1: $C_1$ ($C_2\times C_{84}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 168: $C_2\times C_{84}$ ($C_1$)

Order 84: $C_{84}$ ($C_2$), $C_2\times C_{42}$ ($C_2$)

Order 56: $C_2\times C_{28}$ ($C_3$)

Order 42: $C_{42}$ ($C_2^2$), $C_{42}$ ($C_4$)

Order 28: $C_2\times C_{14}$ ($C_6$), $C_{28}$ ($C_6$)

Order 24: $C_2\times C_{12}$ ($C_7$)

Order 21: $C_{21}$ ($C_2\times C_4$)

Order 14: $C_{14}$ ($C_2\times C_6$), $C_{14}$ ($C_{12}$)

Order 12: $C_2\times C_6$ ($C_{14}$), $C_{12}$ ($C_{14}$)

Order 8: $C_2\times C_4$ ($C_{21}$)

Order 7: $C_7$ ($C_2\times C_{12}$)

Order 6: $C_6$ ($C_2\times C_{14}$), $C_6$ ($C_{28}$)

Order 4: $C_2^2$ ($C_{42}$), $C_4$ ($C_{42}$)

Order 3: $C_3$ ($C_2\times C_{28}$)

Order 2: $C_2$ ($C_{84}$), $C_2$ ($C_2\times C_{42}$)

Order 1: $C_1$ ($C_2\times C_{84}$)

Series

Derived series

$C_2\times C_{84}$

$\rhd$

$C_1$

Chief series

$C_2\times C_{84}$

$\rhd$

$C_2\times C_{42}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_2\times C_{84}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{84}$

Supergroups

This group is a maximal subgroup of 92 larger groups in the database.

This group is a maximal quotient of 53 larger groups in the database.

Character theory

Complex character table

See the $168 \times 168$ 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 $24 \times 24$ rational character table.