Abstract group 384.5396: $C_2\times C_4\times C_{48}$

• Label: 384.5396
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{13} \)
• Perm deg.: $25$
• Trans deg.: $384$
• Rank: $3$

Group information

Description:	$C_2\times C_4\times C_{48}$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_2.C_2^6.C_2^6$, of order \(8192\)\(\medspace = 2^{13} \)
Composition factors:	$C_2$ x 7, $C_3$
Nilpotency class:	$1$
Derived length:	$1$

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

Group statistics

Order	1	2	3	4	6	8	12	16	24	48
Elements	1	7	2	24	14	32	48	64	64	128	384
Conjugacy classes	1	7	2	24	14	32	48	64	64	128	384
Divisions	1	7	1	12	7	8	12	8	8	8	72
Autjugacy classes	1	3	1	4	3	3	4	1	3	1	24

Dimension	1	2	4	8	16
Irr. complex chars.	384	0	0	0	0	384
Irr. rational chars.	8	20	20	16	8	72

Minimal presentations

Permutation degree:	$25$
Transitive degree:	$384$
Rank:	$3$
Inequivalent generating triples:	$2184$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	not computed	not computed

Constructions

Presentation:	Abelian group $\langle a, b, c \mid a^{2}=b^{4}=c^{48}=1 \rangle$
Permutation group:	Degree $25$ $\langle(7,22,14,18,10,20,12,16,8,21,13,17,9,19,11,15), (3,6,4,5), (1,2), (23,25,24) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 24 \\ 8 & 25 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 16 & 17 \end{array}\right), \left(\begin{array}{rr} 5 & 6 \\ 2 & 3 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 28 & 11 \\ 25 & 3 \end{array}\right), \left(\begin{array}{rr} 13 & 8 \\ 24 & 21 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
	$\left\langle \left(\begin{array}{rr} 5 & 4 \\ 12 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 7 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 0 & 15 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 8 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 21 & 24 \\ 16 & 25 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/28\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \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:	$C_{24}$ . $C_4^2$	$C_4^2$ . $C_{24}$	$(C_4\times C_{24})$ . $C_4$	$(C_2\times C_{24})$ . $C_8$	all 60
Aut. group:	$\Aut(C_{595})$	$\Aut(C_{663})$	$\Aut(C_{765})$	$\Aut(C_{832})$	all 8

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

Homology

Primary decomposition:	$C_{2} \times C_{4} \times C_{16} \times C_{3}$
Schur multiplier:	$C_{2}^{2} \times C_{4}$
Commutator length:	$0$

Subgroups

There are 216 subgroups, all normal (32 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4\times C_{48}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2\times C_4\times C_{48}$
Frattini:	$\Phi \simeq$ $C_2\times C_8$	$G/\Phi \simeq$ $C_2^2\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_4\times C_{48}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2\times C_4\times C_{48}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4\times C_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

Normal subgroups

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Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 384: $C_2\times C_4\times C_{48}$

Order 192: $C_4\times C_{48}$ x 4, $C_2^2\times C_{48}$ x 2, $C_2\times C_4\times C_{24}$

Order 128: $C_2\times C_4\times C_{16}$

Order 96: $C_2\times C_{48}$ x 12, $C_4\times C_{24}$ x 4, $C_2^2\times C_{24}$ x 2, $C_2\times C_4\times C_{12}$

Order 64: $C_4\times C_{16}$ x 4, $C_2^2\times C_{16}$ x 2, $C_2\times C_4\times C_8$

Order 48: $C_2\times C_{24}$ x 12, $C_{48}$ x 8, $C_4\times C_{12}$ x 4, $C_2^2\times C_{12}$ x 3

Order 32: $C_2\times C_{16}$ x 12, $C_4\times C_8$ x 4, $C_2^2\times C_8$ x 2, $C_2\times C_4^2$

Order 24: $C_2\times C_{12}$ x 18, $C_{24}$ x 8, $C_2^2\times C_6$

Order 16: $C_2\times C_8$ x 12, $C_{16}$ x 8, $C_4^2$ x 4, $C_2^2\times C_4$ x 3

Order 12: $C_{12}$ x 12, $C_2\times C_6$ x 7

Order 8: $C_2\times C_4$ x 18, $C_8$ x 8, $C_2^3$

Order 6: $C_6$ x 7

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $C_2\times C_4\times C_{48}$

Order 192: $C_2^2\times C_{48}$, $C_2\times C_4\times C_{24}$, $C_4\times C_{48}$

Order 128: $C_2\times C_4\times C_{16}$

Order 96: $C_2\times C_{48}$ x 2, $C_4\times C_{24}$ x 2, $C_2^2\times C_{24}$ x 2, $C_2\times C_4\times C_{12}$

Order 64: $C_2\times C_4\times C_8$, $C_4\times C_{16}$, $C_2^2\times C_{16}$

Order 48: $C_2\times C_{24}$ x 5, $C_2^2\times C_{12}$ x 2, $C_4\times C_{12}$ x 2, $C_{48}$

Order 32: $C_2^2\times C_8$ x 2, $C_4\times C_8$ x 2, $C_2\times C_{16}$ x 2, $C_2\times C_4^2$

Order 24: $C_2\times C_{12}$ x 6, $C_{24}$ x 3, $C_2^2\times C_6$

Order 16: $C_2\times C_8$ x 5, $C_4^2$ x 2, $C_2^2\times C_4$ x 2, $C_{16}$

Order 12: $C_{12}$ x 4, $C_2\times C_6$ x 3

Order 8: $C_2\times C_4$ x 6, $C_8$ x 3, $C_2^3$

Order 6: $C_6$ x 3

Order 4: $C_4$ x 4, $C_2^2$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $C_2\times C_4\times C_{48}$ ($C_1$)

Order 192: $C_4\times C_{48}$ ($C_2$) x 4, $C_2^2\times C_{48}$ ($C_2$) x 2, $C_2\times C_4\times C_{24}$ ($C_2$)

Order 128: $C_2\times C_4\times C_{16}$ ($C_3$)

Order 96: $C_2\times C_{48}$ ($C_4$) x 8, $C_2\times C_{48}$ ($C_2^2$) x 4, $C_4\times C_{24}$ ($C_2^2$) x 2, $C_4\times C_{24}$ ($C_4$) x 2, $C_2^2\times C_{24}$ ($C_2^2$), $C_2^2\times C_{24}$ ($C_4$), $C_2\times C_4\times C_{12}$ ($C_4$)

Order 64: $C_4\times C_{16}$ ($C_6$) x 4, $C_2^2\times C_{16}$ ($C_6$) x 2, $C_2\times C_4\times C_8$ ($C_6$)

Order 48: $C_{48}$ ($C_2\times C_4$) x 8, $C_2\times C_{24}$ ($C_2\times C_4$) x 7, $C_2\times C_{24}$ ($C_8$) x 4, $C_2^2\times C_{12}$ ($C_8$) x 2, $C_4\times C_{12}$ ($C_2\times C_4$) x 2, $C_4\times C_{12}$ ($C_8$) x 2, $C_2^2\times C_{12}$ ($C_2\times C_4$), $C_2\times C_{24}$ ($C_2^3$)

Order 32: $C_2\times C_{16}$ ($C_{12}$) x 8, $C_2\times C_{16}$ ($C_2\times C_6$) x 4, $C_4\times C_8$ ($C_2\times C_6$) x 2, $C_4\times C_8$ ($C_{12}$) x 2, $C_2^2\times C_8$ ($C_2\times C_6$), $C_2^2\times C_8$ ($C_{12}$), $C_2\times C_4^2$ ($C_{12}$)

Order 24: $C_2\times C_{12}$ ($C_{16}$) x 8, $C_2\times C_{12}$ ($C_2\times C_8$) x 7, $C_{24}$ ($C_2\times C_8$) x 4, $C_2\times C_{12}$ ($C_4^2$) x 2, $C_{24}$ ($C_4^2$) x 2, $C_{24}$ ($C_2^2\times C_4$) x 2, $C_2\times C_{12}$ ($C_2^2\times C_4$), $C_2^2\times C_6$ ($C_2\times C_8$)

Order 16: $C_{16}$ ($C_2\times C_{12}$) x 8, $C_2\times C_8$ ($C_2\times C_{12}$) x 7, $C_2\times C_8$ ($C_{24}$) x 4, $C_4^2$ ($C_2\times C_{12}$) x 2, $C_4^2$ ($C_{24}$) x 2, $C_2^2\times C_4$ ($C_{24}$) x 2, $C_2\times C_8$ ($C_2^2\times C_6$), $C_2^2\times C_4$ ($C_2\times C_{12}$)

Order 12: $C_{12}$ ($C_2\times C_{16}$) x 8, $C_2\times C_6$ ($C_2\times C_{16}$) x 4, $C_2\times C_6$ ($C_4\times C_8$) x 2, $C_{12}$ ($C_4\times C_8$) x 2, $C_2\times C_6$ ($C_2^2\times C_8$), $C_{12}$ ($C_2^2\times C_8$), $C_{12}$ ($C_2\times C_4^2$)

Order 8: $C_2\times C_4$ ($C_{48}$) x 8, $C_2\times C_4$ ($C_2\times C_{24}$) x 7, $C_8$ ($C_2\times C_{24}$) x 4, $C_2\times C_4$ ($C_4\times C_{12}$) x 2, $C_8$ ($C_2^2\times C_{12}$) x 2, $C_8$ ($C_4\times C_{12}$) x 2, $C_2^3$ ($C_2\times C_{24}$), $C_2\times C_4$ ($C_2^2\times C_{12}$)

Order 6: $C_6$ ($C_4\times C_{16}$) x 4, $C_6$ ($C_2^2\times C_{16}$) x 2, $C_6$ ($C_2\times C_4\times C_8$)

Order 4: $C_4$ ($C_2\times C_{48}$) x 8, $C_2^2$ ($C_2\times C_{48}$) x 4, $C_2^2$ ($C_4\times C_{24}$) x 2, $C_4$ ($C_4\times C_{24}$) x 2, $C_2^2$ ($C_2^2\times C_{24}$), $C_4$ ($C_2^2\times C_{24}$), $C_4$ ($C_2\times C_4\times C_{12}$)

Order 3: $C_3$ ($C_2\times C_4\times C_{16}$)

Order 2: $C_2$ ($C_4\times C_{48}$) x 4, $C_2$ ($C_2^2\times C_{48}$) x 2, $C_2$ ($C_2\times C_4\times C_{24}$)

Order 1: $C_1$ ($C_2\times C_4\times C_{48}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $C_2\times C_4\times C_{48}$ ($C_1$)

Order 192: $C_2^2\times C_{48}$ ($C_2$), $C_2\times C_4\times C_{24}$ ($C_2$), $C_4\times C_{48}$ ($C_2$)

Order 128: $C_2\times C_4\times C_{16}$ ($C_3$)

Order 96: $C_2\times C_{48}$ ($C_2^2$), $C_2\times C_{48}$ ($C_4$), $C_4\times C_{24}$ ($C_2^2$), $C_4\times C_{24}$ ($C_4$), $C_2^2\times C_{24}$ ($C_2^2$), $C_2^2\times C_{24}$ ($C_4$), $C_2\times C_4\times C_{12}$ ($C_4$)

Order 64: $C_2\times C_4\times C_8$ ($C_6$), $C_4\times C_{16}$ ($C_6$), $C_2^2\times C_{16}$ ($C_6$)

Order 48: $C_2\times C_{24}$ ($C_2\times C_4$) x 3, $C_2^2\times C_{12}$ ($C_2\times C_4$), $C_2^2\times C_{12}$ ($C_8$), $C_2\times C_{24}$ ($C_2^3$), $C_2\times C_{24}$ ($C_8$), $C_4\times C_{12}$ ($C_2\times C_4$), $C_4\times C_{12}$ ($C_8$), $C_{48}$ ($C_2\times C_4$)

Order 32: $C_2^2\times C_8$ ($C_2\times C_6$), $C_2^2\times C_8$ ($C_{12}$), $C_4\times C_8$ ($C_2\times C_6$), $C_4\times C_8$ ($C_{12}$), $C_2\times C_4^2$ ($C_{12}$), $C_2\times C_{16}$ ($C_2\times C_6$), $C_2\times C_{16}$ ($C_{12}$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_8$) x 3, $C_2\times C_{12}$ ($C_4^2$), $C_2\times C_{12}$ ($C_2^2\times C_4$), $C_2\times C_{12}$ ($C_{16}$), $C_{24}$ ($C_2\times C_8$), $C_{24}$ ($C_4^2$), $C_{24}$ ($C_2^2\times C_4$), $C_2^2\times C_6$ ($C_2\times C_8$)

Order 16: $C_2\times C_8$ ($C_2\times C_{12}$) x 3, $C_2\times C_8$ ($C_{24}$), $C_2\times C_8$ ($C_2^2\times C_6$), $C_4^2$ ($C_2\times C_{12}$), $C_4^2$ ($C_{24}$), $C_2^2\times C_4$ ($C_2\times C_{12}$), $C_2^2\times C_4$ ($C_{24}$), $C_{16}$ ($C_2\times C_{12}$)

Order 12: $C_2\times C_6$ ($C_2^2\times C_8$), $C_2\times C_6$ ($C_4\times C_8$), $C_2\times C_6$ ($C_2\times C_{16}$), $C_{12}$ ($C_2^2\times C_8$), $C_{12}$ ($C_4\times C_8$), $C_{12}$ ($C_2\times C_4^2$), $C_{12}$ ($C_2\times C_{16}$)

Order 8: $C_2\times C_4$ ($C_2\times C_{24}$) x 3, $C_2^3$ ($C_2\times C_{24}$), $C_2\times C_4$ ($C_2^2\times C_{12}$), $C_2\times C_4$ ($C_4\times C_{12}$), $C_2\times C_4$ ($C_{48}$), $C_8$ ($C_2^2\times C_{12}$), $C_8$ ($C_2\times C_{24}$), $C_8$ ($C_4\times C_{12}$)

Order 6: $C_6$ ($C_2\times C_4\times C_8$), $C_6$ ($C_4\times C_{16}$), $C_6$ ($C_2^2\times C_{16}$)

Order 4: $C_2^2$ ($C_2\times C_{48}$), $C_2^2$ ($C_4\times C_{24}$), $C_2^2$ ($C_2^2\times C_{24}$), $C_4$ ($C_2\times C_{48}$), $C_4$ ($C_4\times C_{24}$), $C_4$ ($C_2^2\times C_{24}$), $C_4$ ($C_2\times C_4\times C_{12}$)

Order 3: $C_3$ ($C_2\times C_4\times C_{16}$)

Order 2: $C_2$ ($C_2^2\times C_{48}$), $C_2$ ($C_2\times C_4\times C_{24}$), $C_2$ ($C_4\times C_{48}$)

Order 1: $C_1$ ($C_2\times C_4\times C_{48}$)

Series

Derived series

$C_2\times C_4\times C_{48}$

$\rhd$

$C_1$

Chief series

$C_2\times C_4\times C_{48}$

$\rhd$

$C_2\times C_4\times C_{24}$

$\rhd$

$C_2^2\times C_{24}$

$\rhd$

$C_2\times C_{24}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2\times C_4\times C_{48}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4\times C_{48}$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $384 \times 384$ 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 $72 \times 72$ rational character table.