Abstract group 384.19728: $C_{24}:C_2^4$

• Label: 384.19728
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $96$
• Rank: $5$

Group information

Description:	$C_{24}:C_2^4$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^5.C_2^6.D_6^2$, of order \(294912\)\(\medspace = 2^{15} \cdot 3^{2} \)
Composition factors:	$C_2$ x 7, $C_3$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	63	2	64	30	128	32	64	384
Conjugacy classes	1	23	1	24	11	32	12	16	120
Divisions	1	23	1	16	11	16	8	8	84
Autjugacy classes	1	5	1	5	3	2	3	1	21

Dimension	1	2	4	8
Irr. complex chars.	64	48	8	0	120
Irr. rational chars.	32	32	16	4	84

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$96$
Rank:	$5$
Inequivalent generating 5-tuples:	$8332800$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{24}=[a,b]=[a,c]=[a,d]=[a,e]=[b,c]=[b,d]=[b,e]=[c,d]=1, e^{c}=e^{17}, e^{d}=e^{5} \rangle$
Permutation group:	Degree $15$ $\langle(4,5)(6,7)(8,9,10,13,11,12,14,15), (4,6)(5,7)(9,12)(13,15), (2,3)(4,7)(5,6) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 3 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/24\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $S_3$ $\, \times\, $ $\OD_{16}$
Semidirect product:	$C_{24}$ $\,\rtimes\,$ $C_2^4$	$(S_3\times C_8)$ $\,\rtimes\,$ $C_2^3$	$(C_8\times D_6)$ $\,\rtimes\,$ $C_2^2$	$(C_8:D_6)$ $\,\rtimes\,$ $C_2^2$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{12}$ . $C_2^5$	$(C_4\times S_3)$ . $C_2^4$ (2)	$C_2^4$ . $(C_4\times S_3)$	$(C_2^3\times D_6)$ . $C_4$	all 37

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{24}\Z)$.

Homology

Abelianization:	$C_{2}^{4} \times C_{4} $
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 3614 subgroups in 1720 conjugacy classes, 861 normal (31 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_4$	$G/Z \simeq$ $C_2\times D_6$
Commutator:	$G' \simeq$ $C_6$	$G/G' \simeq$ $C_2^4\times C_4$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2^3\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{24}:C_2^3$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{24}:C_2^4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\times \OD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

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 384: $C_{24}:C_2^4$

Order 192: $D_6\times \OD_{16}$ x 24, $C_{24}:C_2^3$ x 2, $C_{24}:C_2^3$ x 2, $C_{12}:C_2^4$, $C_{24}:C_2^3$, $C_{12}.C_2^4$

Order 128: $C_2^3\times \OD_{16}$

Order 96: $S_3\times \OD_{16}$ x 64, $C_{12}:C_2^3$ x 28, $C_8:D_6$ x 24, $C_8\times D_6$ x 24, $C_6\times \OD_{16}$ x 12, $C_6:\OD_{16}$ x 12, $C_2^2\times C_{24}$ x 2, $C_{12}.C_2^3$ x 2, $C_2^3\times D_6$, $C_2^3\times C_{12}$, $C_6.C_2^4$

Order 64: $C_2^2\times \OD_{16}$ x 28, $C_2^3\times C_8$ x 2, $C_2^4\times C_4$

Order 48: $C_4\times D_6$ x 112, $C_{24}:C_2$ x 32, $S_3\times C_8$ x 32, $C_2^2\times D_6$ x 22, $C_3\times \OD_{16}$ x 16, $C_3:\OD_{16}$ x 16, $C_2^2\times C_{12}$ x 14, $C_6.C_2^3$ x 14, $C_6:C_8$ x 12, $C_2\times C_{24}$ x 12, $C_2^3\times C_6$

Order 32: $C_2\times \OD_{16}$ x 112, $C_2^3\times C_4$ x 30, $C_2^2\times C_8$ x 28, $C_2^5$

Order 24: $C_2\times D_6$ x 88, $C_4\times S_3$ x 64, $C_2\times C_{12}$ x 28, $C_6:C_4$ x 28, $C_2^2\times C_6$ x 11, $C_{24}$ x 8, $C_3:C_8$ x 8

Order 16: $C_2^2\times C_4$ x 140, $\OD_{16}$ x 64, $C_2\times C_8$ x 56, $C_2^4$ x 23

Order 12: $D_6$ x 76, $C_2\times C_6$ x 23, $C_{12}$ x 8, $C_3:C_4$ x 8

Order 8: $C_2\times C_4$ x 120, $C_2^3$ x 99, $C_8$ x 16

Order 6: $S_3$ x 12, $C_6$ x 11

Order 4: $C_2^2$ x 99, $C_4$ x 16

Order 3: $C_3$

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $C_{24}:C_2^4$

Order 192: $C_{12}:C_2^4$, $C_{24}:C_2^3$, $C_{12}.C_2^4$, $D_6\times \OD_{16}$, $C_{24}:C_2^3$, $C_{24}:C_2^3$

Order 128: $C_2^3\times \OD_{16}$

Order 96: $C_{12}:C_2^3$ x 6, $C_2^3\times D_6$, $C_2^3\times C_{12}$, $C_6.C_2^4$, $C_6\times \OD_{16}$, $C_2^2\times C_{24}$, $C_6:\OD_{16}$, $C_{12}.C_2^3$, $S_3\times \OD_{16}$, $C_8:D_6$, $C_8\times D_6$

Order 64: $C_2^2\times \OD_{16}$ x 4, $C_2^4\times C_4$, $C_2^3\times C_8$

Order 48: $C_4\times D_6$ x 8, $C_2^2\times D_6$ x 4, $C_2^2\times C_{12}$ x 4, $C_6.C_2^3$ x 3, $C_6:C_8$, $C_2^3\times C_6$, $C_{24}:C_2$, $S_3\times C_8$, $C_3\times \OD_{16}$, $C_2\times C_{24}$, $C_3:\OD_{16}$

Order 32: $C_2^3\times C_4$ x 8, $C_2\times \OD_{16}$ x 4, $C_2^2\times C_8$ x 3, $C_2^5$

Order 24: $C_4\times S_3$ x 6, $C_2\times D_6$ x 6, $C_2\times C_{12}$ x 5, $C_6:C_4$ x 3, $C_2^2\times C_6$ x 3, $C_{24}$, $C_3:C_8$

Order 16: $C_2^2\times C_4$ x 15, $C_2^4$ x 5, $\OD_{16}$ x 3, $C_2\times C_8$ x 3

Order 12: $D_6$ x 5, $C_2\times C_6$ x 4, $C_{12}$ x 3, $C_3:C_4$ x 2

Order 8: $C_2\times C_4$ x 14, $C_2^3$ x 9, $C_8$ x 2

Order 6: $C_6$ x 3, $S_3$ x 2

Order 4: $C_2^2$ x 9, $C_4$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $C_{24}:C_2^4$ ($C_1$)

Order 192: $D_6\times \OD_{16}$ ($C_2$) x 24, $C_{24}:C_2^3$ ($C_2$) x 2, $C_{24}:C_2^3$ ($C_2$) x 2, $C_{12}:C_2^4$ ($C_2$), $C_{24}:C_2^3$ ($C_2$), $C_{12}.C_2^4$ ($C_2$)

Order 96: $S_3\times \OD_{16}$ ($C_2^2$) x 64, $C_8:D_6$ ($C_2^2$) x 24, $C_8\times D_6$ ($C_2^2$) x 24, $C_{12}:C_2^3$ ($C_2^2$) x 14, $C_{12}:C_2^3$ ($C_4$) x 14, $C_6\times \OD_{16}$ ($C_2^2$) x 12, $C_6:\OD_{16}$ ($C_2^2$) x 12, $C_2^2\times C_{24}$ ($C_2^2$) x 2, $C_{12}.C_2^3$ ($C_2^2$) x 2, $C_2^3\times D_6$ ($C_4$), $C_2^3\times C_{12}$ ($C_2^2$), $C_6.C_2^4$ ($C_4$)

Order 64: $C_2^2\times \OD_{16}$ ($S_3$)

Order 48: $C_4\times D_6$ ($C_2\times C_4$) x 84, $C_{24}:C_2$ ($C_2^3$) x 32, $S_3\times C_8$ ($C_2^3$) x 32, $C_4\times D_6$ ($C_2^3$) x 28, $C_3\times \OD_{16}$ ($C_2^3$) x 16, $C_3:\OD_{16}$ ($C_2^3$) x 16, $C_2^2\times D_6$ ($C_2\times C_4$) x 14, $C_6.C_2^3$ ($C_2\times C_4$) x 14, $C_6:C_8$ ($C_2^3$) x 12, $C_2\times C_{24}$ ($C_2^3$) x 12, $C_2^2\times C_{12}$ ($C_2^3$) x 7, $C_2^2\times C_{12}$ ($C_2\times C_4$) x 7, $C_2^3\times C_6$ ($C_2\times C_4$)

Order 32: $C_2\times \OD_{16}$ ($D_6$) x 12, $C_2^2\times C_8$ ($D_6$) x 2, $C_2^3\times C_4$ ($D_6$)

Order 24: $C_4\times S_3$ ($C_2^2\times C_4$) x 56, $C_6:C_4$ ($C_2^2\times C_4$) x 28, $C_2\times D_6$ ($C_2^2\times C_4$) x 28, $C_2\times C_{12}$ ($C_2^2\times C_4$) x 21, $C_4\times S_3$ ($C_2^4$) x 8, $C_{24}$ ($C_2^4$) x 8, $C_2\times D_6$ ($\OD_{16}$) x 8, $C_3:C_8$ ($C_2^4$) x 8, $C_2\times C_{12}$ ($C_2^4$) x 7, $C_2^2\times C_6$ ($C_2^2\times C_4$) x 7

Order 16: $\OD_{16}$ ($C_2\times D_6$) x 16, $C_2\times C_8$ ($C_2\times D_6$) x 12, $C_2^2\times C_4$ ($C_4\times S_3$) x 7, $C_2^2\times C_4$ ($C_2\times D_6$) x 7, $C_2^4$ ($C_4\times S_3$)

Order 12: $D_6$ ($C_2\times \OD_{16}$) x 24, $D_6$ ($C_2^3\times C_4$) x 8, $C_3:C_4$ ($C_2^3\times C_4$) x 8, $C_2\times C_6$ ($C_2^3\times C_4$) x 7, $C_{12}$ ($C_2^3\times C_4$) x 7, $C_2\times C_6$ ($C_2\times \OD_{16}$) x 4, $C_{12}$ ($C_2^5$)

Order 8: $C_2\times C_4$ ($C_4\times D_6$) x 21, $C_8$ ($C_2^2\times D_6$) x 8, $C_2^3$ ($C_4\times D_6$) x 7, $C_2\times C_4$ ($C_2^2\times D_6$) x 7

Order 6: $S_3$ ($C_2^2\times \OD_{16}$) x 8, $C_6$ ($C_2^2\times \OD_{16}$) x 6, $C_6$ ($C_2^4\times C_4$)

Order 4: $C_2^2$ ($C_{12}:C_2^3$) x 7, $C_4$ ($C_{12}:C_2^3$) x 7, $C_2^2$ ($S_3\times \OD_{16}$) x 4, $C_4$ ($C_2^3\times D_6$)

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

Order 2: $C_2$ ($D_6\times \OD_{16}$) x 6, $C_2$ ($C_{12}:C_2^4$)

Order 1: $C_1$ ($C_{24}:C_2^4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $C_{24}:C_2^4$ ($C_1$)

Order 192: $C_{12}:C_2^4$ ($C_2$), $C_{24}:C_2^3$ ($C_2$), $C_{12}.C_2^4$ ($C_2$), $D_6\times \OD_{16}$ ($C_2$), $C_{24}:C_2^3$ ($C_2$), $C_{24}:C_2^3$ ($C_2$)

Order 96: $C_{12}:C_2^3$ ($C_2^2$) x 3, $C_{12}:C_2^3$ ($C_4$) x 3, $C_2^3\times D_6$ ($C_4$), $C_2^3\times C_{12}$ ($C_2^2$), $C_6.C_2^4$ ($C_4$), $C_6\times \OD_{16}$ ($C_2^2$), $C_2^2\times C_{24}$ ($C_2^2$), $C_6:\OD_{16}$ ($C_2^2$), $C_{12}.C_2^3$ ($C_2^2$), $S_3\times \OD_{16}$ ($C_2^2$), $C_8:D_6$ ($C_2^2$), $C_8\times D_6$ ($C_2^2$)

Order 64: $C_2^2\times \OD_{16}$ ($S_3$)

Order 48: $C_4\times D_6$ ($C_2\times C_4$) x 5, $C_2^2\times D_6$ ($C_2\times C_4$) x 3, $C_6.C_2^3$ ($C_2\times C_4$) x 3, $C_4\times D_6$ ($C_2^3$) x 3, $C_2^2\times C_{12}$ ($C_2^3$) x 2, $C_2^2\times C_{12}$ ($C_2\times C_4$) x 2, $C_6:C_8$ ($C_2^3$), $C_2^3\times C_6$ ($C_2\times C_4$), $C_{24}:C_2$ ($C_2^3$), $S_3\times C_8$ ($C_2^3$), $C_3\times \OD_{16}$ ($C_2^3$), $C_2\times C_{24}$ ($C_2^3$), $C_3:\OD_{16}$ ($C_2^3$)

Order 32: $C_2^3\times C_4$ ($D_6$), $C_2\times \OD_{16}$ ($D_6$), $C_2^2\times C_8$ ($D_6$)

Order 24: $C_4\times S_3$ ($C_2^2\times C_4$) x 4, $C_2\times C_{12}$ ($C_2^2\times C_4$) x 3, $C_6:C_4$ ($C_2^2\times C_4$) x 3, $C_2\times D_6$ ($C_2^2\times C_4$) x 3, $C_2\times C_{12}$ ($C_2^4$) x 2, $C_4\times S_3$ ($C_2^4$) x 2, $C_2^2\times C_6$ ($C_2^2\times C_4$) x 2, $C_{24}$ ($C_2^4$), $C_2\times D_6$ ($\OD_{16}$), $C_3:C_8$ ($C_2^4$)

Order 16: $C_2^2\times C_4$ ($C_4\times S_3$) x 2, $C_2^2\times C_4$ ($C_2\times D_6$) x 2, $\OD_{16}$ ($C_2\times D_6$), $C_2\times C_8$ ($C_2\times D_6$), $C_2^4$ ($C_4\times S_3$)

Order 12: $C_2\times C_6$ ($C_2^3\times C_4$) x 2, $D_6$ ($C_2^3\times C_4$) x 2, $C_{12}$ ($C_2^3\times C_4$) x 2, $C_3:C_4$ ($C_2^3\times C_4$) x 2, $C_2\times C_6$ ($C_2\times \OD_{16}$), $D_6$ ($C_2\times \OD_{16}$), $C_{12}$ ($C_2^5$)

Order 8: $C_2\times C_4$ ($C_4\times D_6$) x 3, $C_2^3$ ($C_4\times D_6$) x 2, $C_2\times C_4$ ($C_2^2\times D_6$) x 2, $C_8$ ($C_2^2\times D_6$)

Order 6: $C_6$ ($C_2^4\times C_4$), $C_6$ ($C_2^2\times \OD_{16}$), $S_3$ ($C_2^2\times \OD_{16}$)

Order 4: $C_2^2$ ($C_{12}:C_2^3$) x 2, $C_4$ ($C_{12}:C_2^3$) x 2, $C_2^2$ ($S_3\times \OD_{16}$), $C_4$ ($C_2^3\times D_6$)

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

Order 2: $C_2$ ($C_{12}:C_2^4$), $C_2$ ($D_6\times \OD_{16}$)

Order 1: $C_1$ ($C_{24}:C_2^4$)

Series

Derived series

$C_{24}:C_2^4$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$C_{24}:C_2^4$

$\rhd$

$C_{12}:C_2^4$

$\rhd$

$C_{12}:C_2^3$

$\rhd$

$C_2^2\times C_{12}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{24}:C_2^4$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_2^2\times \OD_{16}$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $120 \times 120$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $84 \times 84$ rational character table.