Abstract group 384.16396: $C_{24}.C_2^4$

• Label: 384.16396
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 3 \)
• Perm deg.: $23$
• Trans deg.: $192$
• Rank: $4$

Group information

Description:	$C_{24}.C_2^4$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_2^6.A_4.C_6.C_2^5$, of order \(147456\)\(\medspace = 2^{14} \cdot 3^{2} \)
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	16	24
Elements	1	39	2	104	78	16	16	96	32	384
Conjugacy classes	1	11	1	8	15	8	4	16	8	72
Divisions	1	11	1	8	11	4	4	4	4	48
Autjugacy classes	1	3	1	3	3	2	2	1	2	18

Dimension	1	2	4	8
Irr. complex chars.	16	44	12	0	72
Irr. rational chars.	16	12	12	8	48

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$192$
Rank:	$4$
Inequivalent generating quadruples:	$43680$

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:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{16}=d^{6}=[a,b]=[a,c]=[a,d]=[b,d]=1, c^{b}=c^{7}, d^{c}=d^{5} \rangle$
Permutation group:	Degree $23$ $\langle(1,2,3,7)(4,6,10,14)(5,13,11,9)(8,16,15,12)(18,19)(20,21)(22,23), (2,6) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 31 & 36 \\ 6 & 25 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 21 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right), \left(\begin{array}{rr} 31 & 24 \\ 6 & 25 \end{array}\right), \left(\begin{array}{rr} 25 & 22 \\ 18 & 17 \end{array}\right), \left(\begin{array}{rr} 13 & 36 \\ 30 & 25 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/42\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_3:\SD_{32})$
Semidirect product:	$(C_2\times C_6)$ $\,\rtimes\,$ $\SD_{32}$	$C_6$ $\,\rtimes\,$ $(C_2\times \SD_{32})$	$(C_6:Q_{16})$ $\,\rtimes\,$ $C_2^2$	$(C_3:Q_{16})$ $\,\rtimes\,$ $C_2^3$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{24}$ . $C_2^4$	$(C_2\times D_8)$ . $D_6$	$D_8$ . $(C_2\times D_6)$	$C_8$ . $(C_6:D_4)$	all 33

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 1342 subgroups in 400 conjugacy classes, 149 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_3:D_8$
Commutator:	$G' \simeq$ $C_{24}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $C_2^2\times D_6$
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:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $D_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times \SD_{32}$
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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

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

Order 192: $C_6:\SD_{32}$ x 12, $C_{24}.C_2^3$, $C_{24}:C_2^3$, $C_{12}.C_2^4$

Order 128: $C_2^2\times \SD_{32}$

Order 96: $C_3:\SD_{32}$ x 16, $C_6\times D_8$ x 9, $C_6:Q_{16}$ x 9, $C_6:C_{16}$ x 6, $C_{12}:C_2^3$, $C_6.C_2^4$, $C_2^2\times C_{24}$

Order 64: $C_2\times \SD_{32}$ x 12, $C_2^2\times Q_{16}$, $C_2^2\times D_8$, $C_2^2\times C_{16}$

Order 48: $C_3:Q_{16}$ x 10, $C_3\times D_8$ x 10, $C_6\times D_4$ x 9, $C_6:Q_8$ x 9, $C_2\times C_{24}$ x 6, $C_3:C_{16}$ x 4, $C_2^3\times C_6$, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 32: $\SD_{32}$ x 16, $C_2\times Q_{16}$ x 9, $C_2\times D_8$ x 9, $C_2\times C_{16}$ x 6, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_2^2\times C_8$

Order 24: $C_2^2\times C_6$ x 11, $C_3:Q_8$ x 10, $C_3\times D_4$ x 10, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6, $C_{24}$ x 4

Order 16: $Q_{16}$ x 10, $D_8$ x 10, $C_2\times Q_8$ x 9, $C_2\times D_4$ x 9, $C_2\times C_8$ x 6, $C_{16}$ x 4, $C_2^2\times C_4$ x 2, $C_2^4$

Order 12: $C_2\times C_6$ x 23, $C_{12}$ x 4, $C_3:C_4$ x 4

Order 8: $C_2\times C_4$ x 12, $C_2^3$ x 11, $Q_8$ x 10, $D_4$ x 10, $C_8$ x 4

Order 6: $C_6$ x 11

Order 4: $C_2^2$ x 23, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 192: $C_6:\SD_{32}$, $C_{24}.C_2^3$, $C_{24}:C_2^3$, $C_{12}.C_2^4$

Order 128: $C_2^2\times \SD_{32}$

Order 96: $C_6\times D_8$ x 2, $C_6:Q_{16}$ x 2, $C_3:\SD_{32}$, $C_{12}:C_2^3$, $C_6.C_2^4$, $C_6:C_{16}$, $C_2^2\times C_{24}$

Order 64: $C_2^2\times Q_{16}$, $C_2^2\times D_8$, $C_2\times \SD_{32}$, $C_2^2\times C_{16}$

Order 48: $C_3:Q_{16}$ x 2, $C_6\times D_4$ x 2, $C_6:Q_8$ x 2, $C_3\times D_8$ x 2, $C_2\times C_{24}$ x 2, $C_2^3\times C_6$, $C_2^2\times C_{12}$, $C_6.C_2^3$, $C_3:C_{16}$

Order 32: $C_2\times Q_{16}$ x 2, $C_2\times D_8$ x 2, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_2^2\times C_8$, $\SD_{32}$, $C_2\times C_{16}$

Order 24: $C_2^2\times C_6$ x 3, $C_2\times C_{12}$ x 2, $C_3:Q_8$ x 2, $C_{24}$ x 2, $C_3\times D_4$ x 2, $C_6:C_4$

Order 16: $Q_{16}$ x 2, $D_8$ x 2, $C_2\times C_8$ x 2, $C_2\times Q_8$ x 2, $C_2\times D_4$ x 2, $C_2^2\times C_4$ x 2, $C_2^4$, $C_{16}$

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

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

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 192: $C_6:\SD_{32}$ ($C_2$) x 12, $C_{24}.C_2^3$ ($C_2$), $C_{24}:C_2^3$ ($C_2$), $C_{12}.C_2^4$ ($C_2$)

Order 96: $C_3:\SD_{32}$ ($C_2^2$) x 16, $C_6:C_{16}$ ($C_2^2$) x 6, $C_6\times D_8$ ($C_2^2$) x 6, $C_6:Q_{16}$ ($C_2^2$) x 6, $C_2^2\times C_{24}$ ($C_2^2$)

Order 64: $C_2^2\times D_8$ ($S_3$)

Order 48: $C_3:Q_{16}$ ($C_2^3$) x 4, $C_3\times D_8$ ($C_2^3$) x 4, $C_3:C_{16}$ ($C_2^3$) x 4, $C_2\times C_{24}$ ($C_2^3$) x 3, $C_2\times C_{24}$ ($D_4$) x 3, $C_2^2\times C_{12}$ ($D_4$)

Order 32: $C_2\times D_8$ ($D_6$) x 6, $C_2^2\times C_8$ ($D_6$)

Order 24: $C_2\times C_{12}$ ($D_8$) x 3, $C_2\times C_{12}$ ($C_2\times D_4$) x 3, $C_{24}$ ($C_2\times D_4$) x 3, $C_{24}$ ($C_2^4$), $C_2^2\times C_6$ ($D_8$)

Order 16: $D_8$ ($C_2\times D_6$) x 4, $C_2\times C_8$ ($C_3:D_4$) x 3, $C_2\times C_8$ ($C_2\times D_6$) x 3, $C_2^2\times C_4$ ($C_3:D_4$)

Order 12: $C_2\times C_6$ ($\SD_{32}$) x 4, $C_2\times C_6$ ($C_2\times D_8$) x 3, $C_{12}$ ($C_2\times D_8$) x 3, $C_{12}$ ($C_2^2\times D_4$)

Order 8: $C_2\times C_4$ ($C_6:D_4$) x 3, $C_2\times C_4$ ($C_3:D_8$) x 3, $C_8$ ($C_6:D_4$) x 3, $C_2^3$ ($C_3:D_8$), $C_8$ ($C_2^2\times D_6$)

Order 6: $C_6$ ($C_2\times \SD_{32}$) x 6, $C_6$ ($C_2^2\times D_8$)

Order 4: $C_2^2$ ($C_3:\SD_{32}$) x 4, $C_2^2$ ($C_6:D_8$) x 3, $C_4$ ($C_6:D_8$) x 3, $C_4$ ($C_2^3:D_6$)

Order 3: $C_3$ ($C_2^2\times \SD_{32}$)

Order 2: $C_2$ ($C_6:\SD_{32}$) x 6, $C_2$ ($D_{12}:C_2^3$)

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: $C_6:\SD_{32}$ ($C_2$), $C_{24}.C_2^3$ ($C_2$), $C_{24}:C_2^3$ ($C_2$), $C_{12}.C_2^4$ ($C_2$)

Order 96: $C_3:\SD_{32}$ ($C_2^2$), $C_6:C_{16}$ ($C_2^2$), $C_6\times D_8$ ($C_2^2$), $C_2^2\times C_{24}$ ($C_2^2$), $C_6:Q_{16}$ ($C_2^2$)

Order 64: $C_2^2\times D_8$ ($S_3$)

Order 48: $C_3:Q_{16}$ ($C_2^3$), $C_2^2\times C_{12}$ ($D_4$), $C_3\times D_8$ ($C_2^3$), $C_2\times C_{24}$ ($C_2^3$), $C_2\times C_{24}$ ($D_4$), $C_3:C_{16}$ ($C_2^3$)

Order 32: $C_2\times D_8$ ($D_6$), $C_2^2\times C_8$ ($D_6$)

Order 24: $C_2\times C_{12}$ ($D_8$), $C_2\times C_{12}$ ($C_2\times D_4$), $C_{24}$ ($C_2^4$), $C_{24}$ ($C_2\times D_4$), $C_2^2\times C_6$ ($D_8$)

Order 16: $D_8$ ($C_2\times D_6$), $C_2\times C_8$ ($C_3:D_4$), $C_2\times C_8$ ($C_2\times D_6$), $C_2^2\times C_4$ ($C_3:D_4$)

Order 12: $C_2\times C_6$ ($C_2\times D_8$), $C_2\times C_6$ ($\SD_{32}$), $C_{12}$ ($C_2^2\times D_4$), $C_{12}$ ($C_2\times D_8$)

Order 8: $C_2^3$ ($C_3:D_8$), $C_2\times C_4$ ($C_6:D_4$), $C_2\times C_4$ ($C_3:D_8$), $C_8$ ($C_2^2\times D_6$), $C_8$ ($C_6:D_4$)

Order 6: $C_6$ ($C_2^2\times D_8$), $C_6$ ($C_2\times \SD_{32}$)

Order 4: $C_2^2$ ($C_3:\SD_{32}$), $C_2^2$ ($C_6:D_8$), $C_4$ ($C_2^3:D_6$), $C_4$ ($C_6:D_8$)

Order 3: $C_3$ ($C_2^2\times \SD_{32}$)

Order 2: $C_2$ ($C_6:\SD_{32}$), $C_2$ ($D_{12}:C_2^3$)

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

Series

Derived series

$C_{24}.C_2^4$

$\rhd$

$C_{24}$

$\rhd$

$C_1$

Chief series

$C_{24}.C_2^4$

$\rhd$

$C_6:\SD_{32}$

$\rhd$

$C_6\times D_8$

$\rhd$

$C_2\times C_{24}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{24}.C_2^4$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_2^2\times C_8$

$\lhd$

$C_2^2\times D_8$

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $48 \times 48$ rational character table.