Abstract group 384.4485: $D_4^2:S_3$

• Label: 384.4485
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $11$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$D_4^2:S_3$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$S_3\times C_2^3.D_4^2$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \)
Composition factors:	$C_2$ x 7, $C_3$
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	59	2	148	70	48	56	384
Conjugacy classes	1	10	1	8	15	1	9	45
Divisions	1	10	1	8	9	1	5	35
Autjugacy classes	1	6	1	5	5	1	3	22

Dimension	1	2	4	8
Irr. complex chars.	8	22	14	1	45
Irr. rational chars.	8	10	12	5	35

Minimal presentations

Permutation degree:	$11$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$5376$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{4}=c^{4}=d^{12}=[a,c]=[c,d]=1, b^{a}=b^{3}, d^{a}=c^{2}d^{7}, c^{b}=c^{3}d^{9}, d^{b}=c^{2}d^{5} \rangle$
Permutation group:	Degree $11$ $\langle(1,2)(3,5)(4,6)(7,8)(10,11), (1,3)(2,5)(6,8), (2,6)(5,8), (4,7)(6,8), (1,4)(2,6)(3,7)(5,8), (2,5)(6,8), (1,3)(2,5)(4,7)(6,8), (9,10,11)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & -1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
Transitive group:	24T714				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_4^2$ $\,\rtimes\,$ $S_3$	$(C_6\times D_4)$ $\,\rtimes\,$ $D_4$	$(D_{12}:D_4)$ $\,\rtimes\,$ $C_2$	$(C_4\times C_{12})$ $\,\rtimes\,$ $D_4$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4:D_4)$ . $D_6$	$(C_6\times D_4)$ . $C_2^3$	$C_2^3$ . $(C_6:D_4)$ (2)	$(C_{12}:D_4)$ . $C_2^2$	all 14

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

Homology

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

Subgroups

There are 1526 subgroups in 354 conjugacy classes, 45 normal (23 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $(C_2\times C_{12}):D_4$
Commutator:	$G' \simeq$ $C_6\times D_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2\times D_4$	$G/\Phi \simeq$ $C_2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times D_4^2$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_4^2:S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_2\wr C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\wr D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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

Normal subgroups up to automorphism

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

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Classes of subgroups up to conjugation

Order 384: $D_4^2:S_3$

Order 192: $(C_2^3\times C_6):C_4$ x 2, $C_2^4:D_6$ x 2, $D_{12}:D_4$, $C_3\times D_4^2$, $(C_4\times C_{12}):C_4$

Order 128: $C_2\wr D_4$

Order 96: $C_2^4:C_6$ x 3, $C_2^3.D_6$ x 2, $C_{12}:C_2^3$ x 2, $C_{12}:D_4$ x 2, $C_2^4:S_3$ x 2, $C_2^2.D_{12}$ x 2, $C_{12}.D_4$, $D_4:D_6$, $C_{12}:D_4$, $D_4\times C_{12}$, $D_{12}:C_2^2$, $D_{12}:C_4$

Order 64: $C_2\wr C_4$ x 2, $C_2\wr C_2^2$ x 2, $C_4^2:C_4$, $D_4^2$, $D_4:D_4$

Order 48: $C_6\times D_4$ x 17, $C_2^2:C_{12}$ x 5, $C_6:D_4$ x 4, $C_6.D_4$ x 4, $C_2^3\times C_6$ x 3, $C_2^2\times C_{12}$ x 2, $D_4:S_3$, $S_3\times D_4$, $D_{12}:C_2$, $C_4:C_{12}$, $C_4\times C_{12}$, $C_3:\SD_{16}$, $C_3:D_8$, $C_3:\OD_{16}$

Order 32: $C_2^2\wr C_2$ x 5, $C_2^3:C_4$ x 4, $C_2^2\times D_4$ x 2, $C_4:D_4$ x 2, $\OD_{16}:C_2$, $D_4:C_2^2$, $D_8:C_2$, $C_4:D_4$, $C_4\times D_4$, $C_4\wr C_2$

Order 24: $C_3\times D_4$ x 18, $C_2^2\times C_6$ x 16, $C_2\times C_{12}$ x 8, $C_3:D_4$ x 7, $C_6:C_4$ x 4, $C_2\times D_6$ x 2, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $C_3:C_8$

Order 16: $C_2\times D_4$ x 22, $C_2^2:C_4$ x 9, $C_2^4$ x 3, $D_4:C_2$ x 2, $C_2^2\times C_4$ x 2, $\SD_{16}$, $D_8$, $\OD_{16}$, $C_4:C_4$, $C_4^2$

Order 12: $C_2\times C_6$ x 26, $C_{12}$ x 5, $D_6$ x 3, $C_3:C_4$ x 3

Order 8: $D_4$ x 26, $C_2^3$ x 18, $C_2\times C_4$ x 13, $Q_8$, $C_8$

Order 6: $C_6$ x 9, $S_3$

Order 4: $C_2^2$ x 29, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 10

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $D_4^2:S_3$

Order 192: $(C_2^3\times C_6):C_4$, $D_{12}:D_4$, $C_2^4:D_6$, $C_3\times D_4^2$, $(C_4\times C_{12}):C_4$

Order 128: $C_2\wr D_4$

Order 96: $C_2^4:C_6$ x 2, $C_2^3.D_6$, $C_{12}.D_4$, $C_{12}:C_2^3$, $D_4:D_6$, $C_{12}:D_4$, $C_{12}:D_4$, $D_4\times C_{12}$, $C_2^4:S_3$, $D_{12}:C_2^2$, $C_2^2.D_{12}$, $D_{12}:C_4$

Order 64: $C_4^2:C_4$, $C_2\wr C_4$, $D_4^2$, $C_2\wr C_2^2$, $D_4:D_4$

Order 48: $C_6\times D_4$ x 7, $C_2^2:C_{12}$ x 3, $C_2^3\times C_6$ x 2, $C_6:D_4$ x 2, $C_6.D_4$ x 2, $C_2^2\times C_{12}$, $D_4:S_3$, $S_3\times D_4$, $D_{12}:C_2$, $C_4:C_{12}$, $C_4\times C_{12}$, $C_3:\SD_{16}$, $C_3:D_8$, $C_3:\OD_{16}$

Order 32: $C_2^2\wr C_2$ x 3, $C_2^3:C_4$ x 2, $\OD_{16}:C_2$, $D_4:C_2^2$, $C_2^2\times D_4$, $D_8:C_2$, $C_4:D_4$, $C_4:D_4$, $C_4\times D_4$, $C_4\wr C_2$

Order 24: $C_2^2\times C_6$ x 7, $C_3\times D_4$ x 6, $C_2\times C_{12}$ x 4, $C_3:D_4$ x 3, $C_6:C_4$ x 2, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $C_2\times D_6$, $C_3:C_8$

Order 16: $C_2\times D_4$ x 10, $C_2^2:C_4$ x 5, $C_2^4$ x 2, $D_4:C_2$ x 2, $\SD_{16}$, $D_8$, $\OD_{16}$, $C_4:C_4$, $C_4^2$, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 11, $C_{12}$ x 3, $D_6$ x 2, $C_3:C_4$ x 2

Order 8: $D_4$ x 10, $C_2^3$ x 8, $C_2\times C_4$ x 7, $Q_8$, $C_8$

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

Order 4: $C_2^2$ x 13, $C_4$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $D_4^2:S_3$ ($C_1$)

Order 192: $(C_2^3\times C_6):C_4$ ($C_2$) x 2, $C_2^4:D_6$ ($C_2$) x 2, $D_{12}:D_4$ ($C_2$), $C_3\times D_4^2$ ($C_2$), $(C_4\times C_{12}):C_4$ ($C_2$)

Order 96: $C_2^3.D_6$ ($C_2^2$) x 2, $C_2^4:C_6$ ($C_2^2$) x 2, $C_{12}.D_4$ ($C_2^2$), $D_4:D_6$ ($C_2^2$), $C_{12}:D_4$ ($C_2^2$)

Order 64: $D_4^2$ ($S_3$)

Order 48: $C_2^3\times C_6$ ($D_4$) x 2, $C_2^2:C_{12}$ ($D_4$) x 2, $C_6\times D_4$ ($C_2^3$), $C_6\times D_4$ ($D_4$), $C_4\times C_{12}$ ($D_4$)

Order 32: $C_2^2\wr C_2$ ($D_6$) x 2, $C_4:D_4$ ($D_6$)

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

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

Order 12: $C_2\times C_6$ ($C_2^2\wr C_2$)

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

Order 6: $C_6$ ($C_2\wr C_2^2$)

Order 4: $C_2^2$ ($C_2^4:S_3$)

Order 3: $C_3$ ($C_2\wr D_4$)

Order 2: $C_2$ ($(C_2\times C_{12}):D_4$)

Order 1: $C_1$ ($D_4^2:S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $D_4^2:S_3$ ($C_1$)

Order 192: $(C_2^3\times C_6):C_4$ ($C_2$), $D_{12}:D_4$ ($C_2$), $C_2^4:D_6$ ($C_2$), $C_3\times D_4^2$ ($C_2$), $(C_4\times C_{12}):C_4$ ($C_2$)

Order 96: $C_2^3.D_6$ ($C_2^2$), $C_{12}.D_4$ ($C_2^2$), $D_4:D_6$ ($C_2^2$), $C_{12}:D_4$ ($C_2^2$), $C_2^4:C_6$ ($C_2^2$)

Order 64: $D_4^2$ ($S_3$)

Order 48: $C_2^3\times C_6$ ($D_4$), $C_6\times D_4$ ($C_2^3$), $C_6\times D_4$ ($D_4$), $C_2^2:C_{12}$ ($D_4$), $C_4\times C_{12}$ ($D_4$)

Order 32: $C_4:D_4$ ($D_6$), $C_2^2\wr C_2$ ($D_6$)

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

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

Order 12: $C_2\times C_6$ ($C_2^2\wr C_2$)

Order 8: $C_2^3$ ($C_6:D_4$), $C_2\times C_4$ ($C_6:D_4$)

Order 6: $C_6$ ($C_2\wr C_2^2$)

Order 4: $C_2^2$ ($C_2^4:S_3$)

Order 3: $C_3$ ($C_2\wr D_4$)

Order 2: $C_2$ ($(C_2\times C_{12}):D_4$)

Order 1: $C_1$ ($D_4^2:S_3$)

Series

Derived series

$D_4^2:S_3$

$\rhd$

$C_6\times D_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$D_4^2:S_3$

$\rhd$

$C_2^4:D_6$

$\rhd$

$C_2^4:C_6$

$\rhd$

$C_6\times D_4$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_4^2:S_3$

$\rhd$

$C_6\times D_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times D_4$

$\lhd$

$D_4^2$

Supergroups

This group is a maximal subgroup of 12 larger groups in the database.

This group is a maximal quotient of 20 larger groups in the database.

Character theory

Complex character table

See the $45 \times 45$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $35 \times 35$ rational character table.