Abstract group 576.3886: $D_{12}.C_{24}$

• Label: 576.3886
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $38$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$D_{12}.C_{24}$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_4\times C_{12}:C_2^4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6	8	12	16	24	48
Elements	1	15	8	16	48	32	56	64	112	224	576
Conjugacy classes	1	4	5	5	14	10	19	20	38	64	180
Divisions	1	4	3	4	8	4	8	4	8	8	52
Autjugacy classes	1	3	3	3	7	3	7	3	7	5	42

Dimension	1	2	4	8	16	32
Irr. complex chars.	96	72	12	0	0	0	180
Irr. rational chars.	8	16	14	8	4	2	52

Minimal presentations

Permutation degree:	$38$
Transitive degree:	$96$
Rank:	$3$
Inequivalent generating triples:	$69888$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	32
Arbitrary	4	6	20

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{6}=c^{48}=1, b^{a}=bc^{24}, c^{a}=c^{25}, c^{b}=c^{17} \rangle$
Permutation group:	Degree $38$ $\langle(2,7)(8,11)(9,18)(10,19)(20,23)(21,24)(22,29)(30,31)(33,34)(35,37)(36,38) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 67 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 17 & 39 \\ 6 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 51 \\ 51 & 52 \end{array}\right), \left(\begin{array}{rr} 67 & 0 \\ 0 & 67 \end{array}\right), \left(\begin{array}{rr} 1 & 34 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 97 & 0 \\ 51 & 97 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/102\Z)$
Direct product:	$C_3$ $\, \times\, $ $(\OD_{32}:S_3)$
Semidirect product:	$(S_3\times C_{48})$ $\,\rtimes\,$ $C_2$ (2)	$(S_3\times C_{16})$ $\,\rtimes\,$ $C_6$ (2)	$(C_6:C_{48})$ $\,\rtimes\,$ $C_2$	$(C_{48}:C_6)$ $\,\rtimes\,$ $C_2$ (2)	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{48}$ . $D_6$ (2)	$D_{12}$ . $C_{24}$	$(C_8.D_6)$ . $C_6$	$C_8$ . $(C_6\times D_6)$	all 62

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

Homology

Abelianization:	$C_{2}^{2} \times C_{24} \simeq C_{2}^{2} \times C_{8} \times C_{3}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 332 subgroups in 181 conjugacy classes, 102 normal (58 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{24}$	$G/Z \simeq$ $C_2\times D_6$
Commutator:	$G' \simeq$ $C_6$	$G/G' \simeq$ $C_2^2\times C_{24}$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $C_6\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2\times \OD_{32}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{12}.C_{24}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{32}:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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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 576: $D_{12}.C_{24}$

Order 288: $C_{48}:C_6$ x 2, $S_3\times C_{48}$ x 2, $C_{24}.D_6$, $C_3^2\times \OD_{32}$, $C_6:C_{48}$

Order 192: $\OD_{32}:C_6$, $\OD_{32}:S_3$

Order 144: $C_{24}:C_6$ x 2, $S_3\times C_{24}$ x 2, $C_3\times C_{48}$ x 2, $C_3:C_{48}$ x 2, $C_{12}.C_{12}$, $D_{12}:C_6$, $C_6\times C_{24}$

Order 96: $C_3\times \OD_{32}$ x 5, $C_2\times C_{48}$ x 3, $C_{48}:C_2$ x 2, $S_3\times C_{16}$ x 2, $C_6:C_{16}$, $\OD_{16}:C_6$, $C_8.D_6$

Order 72: $C_6\wr C_2$ x 2, $S_3\times C_{12}$ x 2, $C_3\times C_{24}$ x 2, $C_3:C_{24}$ x 2, $C_6\times C_{12}$, $C_3\times D_{12}$, $C_3^2:Q_8$

Order 64: $\OD_{32}:C_2$

Order 48: $C_{48}$ x 8, $C_2\times C_{24}$ x 5, $C_3\times \OD_{16}$ x 3, $C_{24}:C_2$ x 2, $S_3\times C_8$ x 2, $C_3:C_{16}$ x 2, $D_4:C_6$, $D_{12}:C_2$, $C_3:\OD_{16}$

Order 36: $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_6^2$

Order 32: $\OD_{32}$ x 3, $C_2\times C_{16}$ x 3, $\OD_{16}:C_2$

Order 24: $C_{24}$ x 8, $C_2\times C_{12}$ x 5, $C_3\times D_4$ x 3, $C_3:D_4$ x 2, $C_4\times S_3$ x 2, $C_3:C_8$ x 2, $D_{12}$, $C_3:Q_8$, $C_3\times Q_8$

Order 18: $C_3\times C_6$ x 2, $C_3\times S_3$ x 2

Order 16: $C_{16}$ x 4, $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $D_4:C_2$

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 5, $D_6$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$

Order 8: $C_8$ x 4, $D_4$ x 3, $C_2\times C_4$ x 3, $Q_8$

Order 6: $C_6$ x 8, $S_3$ x 2

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $D_{12}.C_{24}$

Order 288: $C_{24}.D_6$, $C_3^2\times \OD_{32}$, $C_6:C_{48}$, $C_{48}:C_6$, $S_3\times C_{48}$

Order 192: $\OD_{32}:C_6$, $\OD_{32}:S_3$

Order 144: $C_3:C_{48}$ x 2, $C_{12}.C_{12}$, $C_{24}:C_6$, $S_3\times C_{24}$, $C_3\times C_{48}$, $D_{12}:C_6$, $C_6\times C_{24}$

Order 96: $C_3\times \OD_{32}$ x 4, $C_2\times C_{48}$ x 2, $C_{48}:C_2$, $S_3\times C_{16}$, $C_6:C_{16}$, $\OD_{16}:C_6$, $C_8.D_6$

Order 72: $C_3\times C_{24}$ x 2, $C_6\times C_{12}$, $C_6\wr C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3:C_{24}$

Order 64: $\OD_{32}:C_2$

Order 48: $C_{48}$ x 5, $C_2\times C_{24}$ x 4, $C_3\times \OD_{16}$ x 2, $C_3:C_{16}$ x 2, $C_{24}:C_2$, $D_4:C_6$, $S_3\times C_8$, $D_{12}:C_2$, $C_3:\OD_{16}$

Order 36: $C_3\times C_{12}$ x 2, $C_3:C_{12}$, $C_6^2$, $C_6\times S_3$

Order 32: $\OD_{32}$ x 2, $C_2\times C_{16}$ x 2, $\OD_{16}:C_2$

Order 24: $C_{24}$ x 7, $C_2\times C_{12}$ x 4, $C_3\times D_4$ x 2, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $C_3\times Q_8$, $C_3:C_8$

Order 18: $C_3\times C_6$ x 2, $C_3\times S_3$

Order 16: $C_{16}$ x 3, $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $D_4:C_2$

Order 12: $C_{12}$ x 7, $C_2\times C_6$ x 4, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

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

Order 6: $C_6$ x 7, $S_3$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $D_{12}.C_{24}$ ($C_1$)

Order 288: $C_{48}:C_6$ ($C_2$) x 2, $S_3\times C_{48}$ ($C_2$) x 2, $C_{24}.D_6$ ($C_2$), $C_3^2\times \OD_{32}$ ($C_2$), $C_6:C_{48}$ ($C_2$)

Order 192: $\OD_{32}:S_3$ ($C_3$)

Order 144: $C_{24}:C_6$ ($C_4$) x 2, $S_3\times C_{24}$ ($C_2^2$) x 2, $C_3\times C_{48}$ ($C_2^2$) x 2, $C_3:C_{48}$ ($C_2^2$) x 2, $C_{12}.C_{12}$ ($C_4$), $D_{12}:C_6$ ($C_4$), $C_6\times C_{24}$ ($C_2^2$)

Order 96: $C_{48}:C_2$ ($C_6$) x 2, $S_3\times C_{16}$ ($C_6$) x 2, $C_3\times \OD_{32}$ ($C_6$), $C_3\times \OD_{32}$ ($S_3$), $C_6:C_{16}$ ($C_6$), $C_8.D_6$ ($C_6$)

Order 72: $C_6\wr C_2$ ($C_8$) x 2, $S_3\times C_{12}$ ($C_2\times C_4$) x 2, $C_3:C_{24}$ ($C_2\times C_4$) x 2, $C_6\times C_{12}$ ($C_2\times C_4$), $C_3\times D_{12}$ ($C_8$), $C_3^2:Q_8$ ($C_8$), $C_3\times C_{24}$ ($C_2^3$), $C_3\times C_{24}$ ($C_2\times C_4$)

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

Order 36: $C_3:C_{12}$ ($C_2\times C_8$) x 2, $C_6\times S_3$ ($C_2\times C_8$) x 2, $C_3\times C_{12}$ ($C_2\times C_8$), $C_3\times C_{12}$ ($C_2^2\times C_4$), $C_6^2$ ($C_2\times C_8$)

Order 32: $\OD_{32}$ ($C_3\times S_3$)

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

Order 18: $C_3\times C_6$ ($C_2^2\times C_8$)

Order 16: $C_{16}$ ($C_6\times S_3$) x 2, $C_2\times C_8$ ($C_6\times S_3$)

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

Order 9: $C_3^2$ ($\OD_{32}:C_2$)

Order 8: $C_2\times C_4$ ($S_3\times C_{12}$), $C_8$ ($C_6\times D_6$), $C_8$ ($S_3\times C_{12}$)

Order 6: $C_6$ ($C_2^2\times C_{24}$), $C_6$ ($C_8\times D_6$)

Order 4: $C_2^2$ ($S_3\times C_{24}$), $C_4$ ($S_3\times C_{24}$), $C_4$ ($C_{12}\times D_6$)

Order 3: $C_3$ ($\OD_{32}:C_6$), $C_3$ ($\OD_{32}:S_3$)

Order 2: $C_2$ ($D_6\times C_{24}$)

Order 1: $C_1$ ($D_{12}.C_{24}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $D_{12}.C_{24}$ ($C_1$)

Order 288: $C_{24}.D_6$ ($C_2$), $C_3^2\times \OD_{32}$ ($C_2$), $C_6:C_{48}$ ($C_2$), $C_{48}:C_6$ ($C_2$), $S_3\times C_{48}$ ($C_2$)

Order 192: $\OD_{32}:S_3$ ($C_3$)

Order 144: $C_3:C_{48}$ ($C_2^2$) x 2, $C_{12}.C_{12}$ ($C_4$), $C_{24}:C_6$ ($C_4$), $S_3\times C_{24}$ ($C_2^2$), $C_3\times C_{48}$ ($C_2^2$), $D_{12}:C_6$ ($C_4$), $C_6\times C_{24}$ ($C_2^2$)

Order 96: $C_3\times \OD_{32}$ ($C_6$), $C_3\times \OD_{32}$ ($S_3$), $C_{48}:C_2$ ($C_6$), $S_3\times C_{16}$ ($C_6$), $C_6:C_{16}$ ($C_6$), $C_8.D_6$ ($C_6$)

Order 72: $C_6\times C_{12}$ ($C_2\times C_4$), $C_6\wr C_2$ ($C_8$), $C_3\times D_{12}$ ($C_8$), $S_3\times C_{12}$ ($C_2\times C_4$), $C_3^2:Q_8$ ($C_8$), $C_3\times C_{24}$ ($C_2^3$), $C_3\times C_{24}$ ($C_2\times C_4$), $C_3:C_{24}$ ($C_2\times C_4$)

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

Order 36: $C_3\times C_{12}$ ($C_2\times C_8$), $C_3\times C_{12}$ ($C_2^2\times C_4$), $C_3:C_{12}$ ($C_2\times C_8$), $C_6^2$ ($C_2\times C_8$), $C_6\times S_3$ ($C_2\times C_8$)

Order 32: $\OD_{32}$ ($C_3\times S_3$)

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

Order 18: $C_3\times C_6$ ($C_2^2\times C_8$)

Order 16: $C_2\times C_8$ ($C_6\times S_3$), $C_{16}$ ($C_6\times S_3$)

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

Order 9: $C_3^2$ ($\OD_{32}:C_2$)

Order 8: $C_2\times C_4$ ($S_3\times C_{12}$), $C_8$ ($C_6\times D_6$), $C_8$ ($S_3\times C_{12}$)

Order 6: $C_6$ ($C_2^2\times C_{24}$), $C_6$ ($C_8\times D_6$)

Order 4: $C_2^2$ ($S_3\times C_{24}$), $C_4$ ($S_3\times C_{24}$), $C_4$ ($C_{12}\times D_6$)

Order 3: $C_3$ ($\OD_{32}:C_6$), $C_3$ ($\OD_{32}:S_3$)

Order 2: $C_2$ ($D_6\times C_{24}$)

Order 1: $C_1$ ($D_{12}.C_{24}$)

Series

Derived series

$D_{12}.C_{24}$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$D_{12}.C_{24}$

$\rhd$

$C_3^2\times \OD_{32}$

$\rhd$

$C_3\times C_{48}$

$\rhd$

$C_{48}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_{12}.C_{24}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_{24}$

$\lhd$

$C_3\times \OD_{32}$

Supergroups

This group is a maximal subgroup of 11 larger groups in the database.

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

Character theory

Complex character table

See the $180 \times 180$ 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 $52 \times 52$ rational character table.