Abstract group 96.59: $C_2\times C_{48}$

• Label: 96.59
• Order: \( 2^{5} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{6} \)
• Perm deg.: $21$
• Trans deg.: $96$
• Rank: $2$

Group information

Description:	$C_2\times C_{48}$
Order:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$D_4:C_2^3$, of order \(64\)\(\medspace = 2^{6} \)
Composition factors:	$C_2$ x 5, $C_3$
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	8	12	16	24	48
Elements	1	3	2	4	6	8	8	16	16	32	96
Conjugacy classes	1	3	2	4	6	8	8	16	16	32	96
Divisions	1	3	1	2	3	2	2	2	2	2	20
Autjugacy classes	1	2	1	2	2	2	2	1	2	1	16

Dimension	1	2	4	8	16
Irr. complex chars.	96	0	0	0	0	96
Irr. rational chars.	4	6	4	4	2	20

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$96$
Rank:	$2$
Inequivalent generating pairs:	$48$

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 $\langle a, b \mid a^{2}=b^{48}=1 \rangle$
Permutation group:	Degree $21$ $\langle(3,18,10,14,6,16,8,12,4,17,9,13,5,15,7,11), (1,2), (19,21,20), (3,10,6,8,4,9,5,7) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 48 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 22 & 0 \\ 0 & 75 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{97})$
Direct product:	$C_2$ $\, \times\, $ $C_{16}$ $\, \times\, $ $C_3$
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:	$C_{24}$ . $C_4$	$C_{12}$ . $C_8$	$C_8$ . $C_{12}$	$C_4$ . $C_{24}$	all 16
Aut. group:	$\Aut(C_{119})$	$\Aut(C_{153})$	$\Aut(C_{238})$	$\Aut(C_{306})$

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

Homology

Primary decomposition:	$C_{2} \times C_{16} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$0$

Subgroups

There are 28 subgroups, all normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{48}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2\times C_{48}$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $C_2\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{48}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2\times C_{48}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 96: $C_2\times C_{48}$

Order 48: $C_{48}$ x 2, $C_2\times C_{24}$

Order 32: $C_2\times C_{16}$

Order 24: $C_{24}$ x 2, $C_2\times C_{12}$

Order 16: $C_{16}$ x 2, $C_2\times C_8$

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

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

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 96: $C_2\times C_{48}$

Order 48: $C_2\times C_{24}$, $C_{48}$

Order 32: $C_2\times C_{16}$

Order 24: $C_{24}$ x 2, $C_2\times C_{12}$

Order 16: $C_2\times C_8$, $C_{16}$

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

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

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 96: $C_2\times C_{48}$ ($C_1$)

Order 48: $C_{48}$ ($C_2$) x 2, $C_2\times C_{24}$ ($C_2$)

Order 32: $C_2\times C_{16}$ ($C_3$)

Order 24: $C_2\times C_{12}$ ($C_4$), $C_{24}$ ($C_2^2$), $C_{24}$ ($C_4$)

Order 16: $C_{16}$ ($C_6$) x 2, $C_2\times C_8$ ($C_6$)

Order 12: $C_2\times C_6$ ($C_8$), $C_{12}$ ($C_2\times C_4$), $C_{12}$ ($C_8$)

Order 8: $C_2\times C_4$ ($C_{12}$), $C_8$ ($C_2\times C_6$), $C_8$ ($C_{12}$)

Order 6: $C_6$ ($C_{16}$) x 2, $C_6$ ($C_2\times C_8$)

Order 4: $C_2^2$ ($C_{24}$), $C_4$ ($C_2\times C_{12}$), $C_4$ ($C_{24}$)

Order 3: $C_3$ ($C_2\times C_{16}$)

Order 2: $C_2$ ($C_{48}$) x 2, $C_2$ ($C_2\times C_{24}$)

Order 1: $C_1$ ($C_2\times C_{48}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 96: $C_2\times C_{48}$ ($C_1$)

Order 48: $C_2\times C_{24}$ ($C_2$), $C_{48}$ ($C_2$)

Order 32: $C_2\times C_{16}$ ($C_3$)

Order 24: $C_2\times C_{12}$ ($C_4$), $C_{24}$ ($C_2^2$), $C_{24}$ ($C_4$)

Order 16: $C_2\times C_8$ ($C_6$), $C_{16}$ ($C_6$)

Order 12: $C_2\times C_6$ ($C_8$), $C_{12}$ ($C_2\times C_4$), $C_{12}$ ($C_8$)

Order 8: $C_2\times C_4$ ($C_{12}$), $C_8$ ($C_2\times C_6$), $C_8$ ($C_{12}$)

Order 6: $C_6$ ($C_2\times C_8$), $C_6$ ($C_{16}$)

Order 4: $C_2^2$ ($C_{24}$), $C_4$ ($C_2\times C_{12}$), $C_4$ ($C_{24}$)

Order 3: $C_3$ ($C_2\times C_{16}$)

Order 2: $C_2$ ($C_2\times C_{24}$), $C_2$ ($C_{48}$)

Order 1: $C_1$ ($C_2\times C_{48}$)

Series

Derived series

$C_2\times C_{48}$

$\rhd$

$C_1$

Chief series

$C_2\times C_{48}$

$\rhd$

$C_2\times C_{24}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2\times C_{48}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{48}$

Supergroups

This group is a maximal subgroup of 143 larger groups in the database.

This group is a maximal quotient of 105 larger groups in the database.

Character theory

Complex character table

See the $96 \times 96$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.