Abstract group 384.5615: $C_{16}.S_4$

• Label: 384.5615
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $64$
• Trans deg.: $64$
• Rank: $2$

Group information

Description:	$C_{16}.S_4$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_2^5.D_6$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \)
Composition factors:	$C_2$ x 7, $C_3$
Derived length:	$4$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	6	8	12	16	24	48
Elements	1	19	8	44	8	64	16	128	32	64	384
Conjugacy classes	1	3	1	6	1	10	2	16	4	8	52
Divisions	1	3	1	4	1	5	1	4	1	1	22
Autjugacy classes	1	3	1	4	1	5	1	4	1	1	22

Dimension	1	2	3	4	6	8	12	16	32
Irr. complex chars.	16	8	16	12	0	0	0	0	0	52
Irr. rational chars.	4	4	4	3	2	1	2	1	1	22

Minimal presentations

Permutation degree:	$64$
Transitive degree:	$64$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	16
Arbitrary	4	8	16

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=d^{4}=1, b^{24}=d^{2}, c^{2}=d^{2}, b^{a}=b^{17}d^{2}, c^{a}=d, d^{a}=c, c^{b}=d^{3}, d^{b}=cd^{3}, d^{c}=d^{3} \rangle$
Permutation group:	Degree $64$ $\langle(3,6)(4,5)(7,8)(11,14)(12,13)(15,16)(19,22)(20,21)(23,24)(27,30)(28,29) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_8.S_4)$ $\,\rtimes\,$ $C_2$	$(C_{16}.A_4)$ $\,\rtimes\,$ $C_2$	$(\OD_{32}:C_2)$ $\,\rtimes\,$ $S_3$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{16}$ . $S_4$	$\GL(2,3)$ . $C_8$	$Q_8$ . $(S_3\times C_8)$	$(C_2.S_4)$ . $C_8$	all 15

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

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 289 subgroups in 85 conjugacy classes, 20 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_8$	$G/Z \simeq$ $C_2\times S_4$
Commutator:	$G' \simeq$ $\SL(2,3)$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $C_2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $\OD_{32}:C_2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_{16}.S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_8\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{16}.D_4$
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

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Subgroup information

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

Classes of subgroups up to conjugation

Order 384: $C_{16}.S_4$

Order 192: $\GL(2,3):C_4$, $C_{16}.A_4$, $C_8.S_4$

Order 128: $C_{16}.D_4$

Order 96: $C_8.A_4$, $\Unitary(2,3)$, $C_{48}:C_2$, $\GL(2,3):C_2$

Order 64: $\OD_{32}:C_2$ x 2, $\OD_{32}:C_2$ x 2, $C_8.C_8$, $C_{16}:C_4$, $D_8:C_4$

Order 48: $S_3\times C_8$, $\SL(2,3):C_2$, $\GL(2,3)$, $C_2.S_4$, $C_{48}$, $C_3:C_{16}$

Order 32: $\OD_{32}$ x 3, $C_2\times C_{16}$ x 3, $\OD_{16}:C_2$ x 2, $C_4\wr C_2$ x 2, $D_8:C_2$, $C_4\times C_8$, $C_8.C_4$

Order 24: $C_4\times S_3$, $\SL(2,3)$, $C_{24}$, $C_3:C_8$

Order 16: $C_{16}$ x 4, $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $\SD_{16}$ x 2, $D_4:C_2$ x 2, $Q_{16}$, $D_8$, $C_4^2$

Order 12: $D_6$, $C_{12}$, $C_3:C_4$

Order 8: $C_8$ x 5, $D_4$ x 3, $C_2\times C_4$ x 3, $Q_8$ x 2

Order 6: $C_6$, $S_3$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $C_{16}.S_4$

Order 192: $\GL(2,3):C_4$, $C_{16}.A_4$, $C_8.S_4$

Order 128: $C_{16}.D_4$

Order 96: $C_8.A_4$, $\Unitary(2,3)$, $C_{48}:C_2$, $\GL(2,3):C_2$

Order 64: $\OD_{32}:C_2$ x 2, $\OD_{32}:C_2$ x 2, $C_8.C_8$, $C_{16}:C_4$, $D_8:C_4$

Order 48: $S_3\times C_8$, $\SL(2,3):C_2$, $\GL(2,3)$, $C_2.S_4$, $C_{48}$, $C_3:C_{16}$

Order 32: $\OD_{32}$ x 3, $C_2\times C_{16}$ x 3, $\OD_{16}:C_2$ x 2, $C_4\wr C_2$ x 2, $D_8:C_2$, $C_4\times C_8$, $C_8.C_4$

Order 24: $C_4\times S_3$, $\SL(2,3)$, $C_{24}$, $C_3:C_8$

Order 16: $C_{16}$ x 4, $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $\SD_{16}$ x 2, $D_4:C_2$ x 2, $Q_{16}$, $D_8$, $C_4^2$

Order 12: $D_6$, $C_{12}$, $C_3:C_4$

Order 8: $C_8$ x 5, $D_4$ x 3, $C_2\times C_4$ x 3, $Q_8$ x 2

Order 6: $C_6$, $S_3$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $C_{16}.S_4$ ($C_1$)

Order 192: $\GL(2,3):C_4$ ($C_2$), $C_{16}.A_4$ ($C_2$), $C_8.S_4$ ($C_2$)

Order 96: $C_8.A_4$ ($C_2^2$), $\Unitary(2,3)$ ($C_4$), $\GL(2,3):C_2$ ($C_4$)

Order 64: $\OD_{32}:C_2$ ($S_3$)

Order 48: $\SL(2,3):C_2$ ($C_2\times C_4$), $\GL(2,3)$ ($C_8$), $C_2.S_4$ ($C_8$)

Order 32: $\OD_{16}:C_2$ ($D_6$)

Order 24: $\SL(2,3)$ ($C_2\times C_8$)

Order 16: $D_4:C_2$ ($C_4\times S_3$), $C_{16}$ ($S_4$)

Order 8: $Q_8$ ($S_3\times C_8$), $C_8$ ($C_2\times S_4$)

Order 4: $C_4$ ($C_4\times S_4$)

Order 2: $C_2$ ($C_8\times S_4$)

Order 1: $C_1$ ($C_{16}.S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $C_{16}.S_4$ ($C_1$)

Order 192: $\GL(2,3):C_4$ ($C_2$), $C_{16}.A_4$ ($C_2$), $C_8.S_4$ ($C_2$)

Order 96: $C_8.A_4$ ($C_2^2$), $\Unitary(2,3)$ ($C_4$), $\GL(2,3):C_2$ ($C_4$)

Order 64: $\OD_{32}:C_2$ ($S_3$)

Order 48: $\SL(2,3):C_2$ ($C_2\times C_4$), $\GL(2,3)$ ($C_8$), $C_2.S_4$ ($C_8$)

Order 32: $\OD_{16}:C_2$ ($D_6$)

Order 24: $\SL(2,3)$ ($C_2\times C_8$)

Order 16: $D_4:C_2$ ($C_4\times S_3$), $C_{16}$ ($S_4$)

Order 8: $Q_8$ ($S_3\times C_8$), $C_8$ ($C_2\times S_4$)

Order 4: $C_4$ ($C_4\times S_4$)

Order 2: $C_2$ ($C_8\times S_4$)

Order 1: $C_1$ ($C_{16}.S_4$)

Series

Derived series

$C_{16}.S_4$

$\rhd$

$\SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_{16}.S_4$

$\rhd$

$\GL(2,3):C_4$

$\rhd$

$C_8.A_4$

$\rhd$

$\SL(2,3):C_2$

$\rhd$

$\SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_{16}.S_4$

$\rhd$

$\SL(2,3)$

Upper central series

$C_1$

$\lhd$

$C_8$

$\lhd$

$C_{16}$

Character theory

Complex character table

See the $52 \times 52$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $22 \times 22$ rational character table.