Abstract group 384.3993: $C_2\times C_4.D_{24}$

• Label: 384.3993
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $37$
• Trans deg.: $192$
• Rank: $3$

Group information

Description:	$C_2\times C_4.D_{24}$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_3:(C_2^2.C_2^6.C_2^3)$, of order \(6144\)\(\medspace = 2^{11} \cdot 3 \)
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	48
Elements	1	55	2	56	14	112	16	32	32	64	384
Conjugacy classes	1	7	1	8	7	12	8	16	16	32	108
Divisions	1	7	1	6	5	6	4	2	4	2	38
Autjugacy classes	1	5	1	5	4	5	4	1	4	1	31

Dimension	1	2	4	8	16	32
Irr. complex chars.	16	92	0	0	0	0	108
Irr. rational chars.	8	12	10	4	2	2	38

Minimal presentations

Permutation degree:	$37$
Transitive degree:	$192$
Rank:	$3$
Inequivalent generating triples:	$2688$

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:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{2}=d^{48}=[a,b]=[a,c]=[a,d]=[c,d]=1, c^{b}=cd^{24}, d^{b}=cd^{11} \rangle$
Permutation group:	Degree $37$ $\langle(1,2,6,11,4,9,7,21)(3,12,15,29,13,28,22,27)(5,17,19,26,16,31,20,10)(8,18,24,30,23,32,25,14) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 28 & 27 \\ 3 & 4 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 16 & 17 \end{array}\right), \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 28 & 11 \\ 25 & 3 \end{array}\right), \left(\begin{array}{rr} 13 & 10 \\ 14 & 31 \end{array}\right), \left(\begin{array}{rr} 13 & 8 \\ 24 & 21 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
	$\left\langle \left(\begin{array}{rr} 23 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 17 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 16 \end{array}\right), \left(\begin{array}{rr} 34 & 39 \\ 6 & 17 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 38 & 0 \\ 0 & 38 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/51\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_4.D_{24})$
Semidirect product:	$C_6$ $\,\rtimes\,$ $(C_4.D_8)$ (2)	$(C_2\times C_{16})$ $\,\rtimes\,$ $D_6$ (2)	$(C_2^2\times C_{16})$ $\,\rtimes\,$ $S_3$	$(C_2^2\times C_{48})$ $\,\rtimes\,$ $C_2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_8$ . $(C_4\times D_6)$	$C_8$ . $(D_6:C_4)$ (2)	$C_8$ . $(C_6:D_4)$	$(C_6:Q_{16})$ . $C_4$	all 46

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{32}\Z)$.

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 878 subgroups in 224 conjugacy classes, 85 normal (63 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $D_{24}$
Commutator:	$G' \simeq$ $C_{24}$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_8$	$G/\Phi \simeq$ $C_2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{48}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_2\times C_4.D_{24}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $D_4:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 384: $C_2\times C_4.D_{24}$

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

Order 128: $C_2^3.\SD_{16}$

Order 96: $D_{24}:C_2$ x 6, $C_2\times C_{48}$ x 4, $C_{24}.C_4$ x 3, $D_{12}:C_2^2$, $C_2^2\times C_{24}$, $C_6:\OD_{16}$, $C_6:Q_{16}$, $C_2\times D_{24}$, $C_8:D_6$

Order 64: $C_4.D_8$ x 4, $D_8:C_2^2$, $C_2^2\times C_{16}$, $C_2^3.Q_8$

Order 48: $D_{12}:C_2$ x 6, $C_2\times C_{24}$ x 6, $C_{24}:C_2$ x 4, $C_3:Q_{16}$ x 3, $D_{24}$ x 3, $C_3:\OD_{16}$ x 3, $C_{48}$ x 2, $C_6:C_8$, $C_2^2\times C_{12}$, $C_6:D_4$, $C_2\times D_{12}$, $C_4\times D_6$, $C_6:Q_8$

Order 32: $D_8:C_2$ x 6, $C_2\times C_{16}$ x 4, $C_8.C_4$ x 3, $D_4:C_2^2$, $C_2\times Q_{16}$, $C_2\times \SD_{16}$, $C_2\times D_8$, $C_2\times \OD_{16}$, $C_2^2\times C_8$

Order 24: $C_2\times C_{12}$ x 6, $C_3:D_4$ x 4, $C_4\times S_3$ x 4, $C_{24}$ x 4, $D_{12}$ x 3, $C_3:Q_8$ x 3, $C_3:C_8$ x 2, $C_6:C_4$, $C_2^2\times C_6$, $C_2\times D_6$

Order 16: $C_2\times C_8$ x 7, $D_4:C_2$ x 6, $\SD_{16}$ x 4, $Q_{16}$ x 3, $D_8$ x 3, $\OD_{16}$ x 3, $C_2\times D_4$ x 2, $C_2^2\times C_4$ x 2, $C_{16}$ x 2, $C_2\times Q_8$

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

Order 8: $C_2\times C_4$ x 11, $D_4$ x 7, $C_8$ x 6, $Q_8$ x 3, $C_2^3$ x 2

Order 6: $C_6$ x 5, $S_3$ x 2

Order 4: $C_2^2$ x 9, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $C_2\times C_4.D_{24}$

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

Order 128: $C_2^3.\SD_{16}$

Order 96: $D_{24}:C_2$ x 4, $C_2\times C_{48}$ x 3, $C_{24}.C_4$ x 2, $D_{12}:C_2^2$, $C_2^2\times C_{24}$, $C_6:\OD_{16}$, $C_6:Q_{16}$, $C_2\times D_{24}$, $C_8:D_6$

Order 64: $C_4.D_8$, $D_8:C_2^2$, $C_2^2\times C_{16}$, $C_2^3.Q_8$

Order 48: $C_2\times C_{24}$ x 6, $D_{12}:C_2$ x 4, $C_3:Q_{16}$ x 2, $D_{24}$ x 2, $C_{24}:C_2$ x 2, $C_3:\OD_{16}$ x 2, $C_6:C_8$, $C_2^2\times C_{12}$, $C_6:D_4$, $C_2\times D_{12}$, $C_4\times D_6$, $C_6:Q_8$, $C_{48}$

Order 32: $D_8:C_2$ x 4, $C_2\times C_{16}$ x 3, $C_8.C_4$ x 2, $D_4:C_2^2$, $C_2\times Q_{16}$, $C_2\times \SD_{16}$, $C_2\times D_8$, $C_2\times \OD_{16}$, $C_2^2\times C_8$

Order 24: $C_2\times C_{12}$ x 6, $C_{24}$ x 4, $C_3:D_4$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3:Q_8$ x 2, $C_6:C_4$, $C_2^2\times C_6$, $C_2\times D_6$, $C_3:C_8$

Order 16: $C_2\times C_8$ x 7, $D_4:C_2$ x 4, $Q_{16}$ x 2, $\SD_{16}$ x 2, $D_8$ x 2, $\OD_{16}$ x 2, $C_2\times D_4$ x 2, $C_2^2\times C_4$ x 2, $C_2\times Q_8$, $C_{16}$

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

Order 8: $C_2\times C_4$ x 9, $C_8$ x 5, $D_4$ x 4, $C_2^3$ x 2, $Q_8$ x 2

Order 6: $C_6$ x 4, $S_3$

Order 4: $C_2^2$ x 6, $C_4$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $C_2\times C_4.D_{24}$ ($C_1$)

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

Order 96: $C_2\times C_{48}$ ($C_2^2$) x 2, $C_{24}.C_4$ ($C_2^2$) x 2, $D_{24}:C_2$ ($C_2^2$) x 2, $D_{24}:C_2$ ($C_4$) x 2, $C_2^2\times C_{24}$ ($C_2^2$), $C_6:Q_{16}$ ($C_4$), $C_2\times D_{24}$ ($C_4$)

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

Order 48: $C_2\times C_{24}$ ($D_4$) x 3, $C_3:Q_{16}$ ($C_2\times C_4$) x 2, $D_{24}$ ($C_2\times C_4$) x 2, $C_2\times C_{24}$ ($C_2\times C_4$) x 2, $C_2^2\times C_{12}$ ($D_4$), $C_2\times C_{24}$ ($C_2^3$)

Order 32: $C_2\times C_{16}$ ($D_6$) x 2, $C_2^2\times C_8$ ($D_6$)

Order 24: $C_2\times C_{12}$ ($D_8$) x 2, $C_2\times C_{12}$ ($C_2^2:C_4$) x 2, $C_{24}$ ($C_2^2:C_4$) x 2, $C_2\times C_{12}$ ($\SD_{16}$), $C_2\times C_{12}$ ($C_2\times D_4$), $C_{24}$ ($C_2\times D_4$), $C_{24}$ ($C_2^2\times C_4$), $C_2^2\times C_6$ ($\SD_{16}$)

Order 16: $C_2\times C_8$ ($C_3:D_4$) x 2, $C_2\times C_8$ ($C_4\times S_3$) x 2, $C_2\times C_8$ ($D_{12}$), $C_2\times C_8$ ($C_2\times D_6$), $C_2^2\times C_4$ ($D_{12}$)

Order 12: $C_2\times C_6$ ($D_4:C_4$) x 2, $C_{12}$ ($D_4:C_4$) x 2, $C_2\times C_6$ ($C_2\times \SD_{16}$), $C_{12}$ ($C_2\times D_8$), $C_{12}$ ($C_2^3:C_4$)

Order 8: $C_2\times C_4$ ($D_{24}$) x 2, $C_2\times C_4$ ($D_6:C_4$) x 2, $C_8$ ($D_6:C_4$) x 2, $C_2^3$ ($C_{24}:C_2$), $C_2\times C_4$ ($C_{24}:C_2$), $C_2\times C_4$ ($C_2\times D_{12}$), $C_8$ ($C_6:D_4$), $C_8$ ($C_4\times D_6$)

Order 6: $C_6$ ($C_4.D_8$) x 2, $C_6$ ($C_2^2.D_8$)

Order 4: $C_2^2$ ($D_{12}:C_4$) x 2, $C_4$ ($D_{12}:C_4$) x 2, $C_2^2$ ($C_8:D_6$), $C_4$ ($C_2^2.D_{12}$), $C_4$ ($C_2\times D_{24}$)

Order 3: $C_3$ ($C_2^3.\SD_{16}$)

Order 2: $C_2$ ($C_4.D_{24}$) x 2, $C_2$ ($C_2^2.D_{24}$)

Order 1: $C_1$ ($C_2\times C_4.D_{24}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $C_2\times C_4.D_{24}$ ($C_1$)

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

Order 96: $C_2\times C_{48}$ ($C_2^2$), $C_{24}.C_4$ ($C_2^2$), $C_2^2\times C_{24}$ ($C_2^2$), $C_6:Q_{16}$ ($C_4$), $D_{24}:C_2$ ($C_2^2$), $D_{24}:C_2$ ($C_4$), $C_2\times D_{24}$ ($C_4$)

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

Order 48: $C_2\times C_{24}$ ($D_4$) x 3, $C_2\times C_{24}$ ($C_2\times C_4$) x 2, $C_3:Q_{16}$ ($C_2\times C_4$), $D_{24}$ ($C_2\times C_4$), $C_2^2\times C_{12}$ ($D_4$), $C_2\times C_{24}$ ($C_2^3$)

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

Order 24: $C_2\times C_{12}$ ($D_8$) x 2, $C_2\times C_{12}$ ($C_2^2:C_4$) x 2, $C_{24}$ ($C_2^2:C_4$) x 2, $C_2\times C_{12}$ ($\SD_{16}$), $C_2\times C_{12}$ ($C_2\times D_4$), $C_{24}$ ($C_2\times D_4$), $C_{24}$ ($C_2^2\times C_4$), $C_2^2\times C_6$ ($\SD_{16}$)

Order 16: $C_2\times C_8$ ($C_3:D_4$) x 2, $C_2\times C_8$ ($C_4\times S_3$) x 2, $C_2\times C_8$ ($D_{12}$), $C_2\times C_8$ ($C_2\times D_6$), $C_2^2\times C_4$ ($D_{12}$)

Order 12: $C_2\times C_6$ ($D_4:C_4$) x 2, $C_{12}$ ($D_4:C_4$) x 2, $C_2\times C_6$ ($C_2\times \SD_{16}$), $C_{12}$ ($C_2\times D_8$), $C_{12}$ ($C_2^3:C_4$)

Order 8: $C_2\times C_4$ ($D_{24}$) x 2, $C_2\times C_4$ ($D_6:C_4$) x 2, $C_8$ ($D_6:C_4$) x 2, $C_2^3$ ($C_{24}:C_2$), $C_2\times C_4$ ($C_{24}:C_2$), $C_2\times C_4$ ($C_2\times D_{12}$), $C_8$ ($C_6:D_4$), $C_8$ ($C_4\times D_6$)

Order 6: $C_6$ ($C_2^2.D_8$), $C_6$ ($C_4.D_8$)

Order 4: $C_2^2$ ($D_{12}:C_4$) x 2, $C_4$ ($D_{12}:C_4$) x 2, $C_2^2$ ($C_8:D_6$), $C_4$ ($C_2^2.D_{12}$), $C_4$ ($C_2\times D_{24}$)

Order 3: $C_3$ ($C_2^3.\SD_{16}$)

Order 2: $C_2$ ($C_4.D_{24}$), $C_2$ ($C_2^2.D_{24}$)

Order 1: $C_1$ ($C_2\times C_4.D_{24}$)

Series

Derived series

$C_2\times C_4.D_{24}$

$\rhd$

$C_{24}$

$\rhd$

$C_1$

Chief series

$C_2\times C_4.D_{24}$

$\rhd$

$D_{24}:C_2^2$

$\rhd$

$C_2^2\times C_{24}$

$\rhd$

$C_2\times C_{24}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2\times C_4.D_{24}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_2^2\times C_8$

$\lhd$

$C_2^2\times C_{16}$

Supergroups

This group is a maximal subgroup of 2 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 $108 \times 108$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.