Abstract group 384.4705: $(C_6\times D_4).D_4$

• Label: 384.4705
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $15$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$(C_6\times D_4).D_4$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_3:(C_2^5.C_2^4.C_2^6)$, of order \(98304\)\(\medspace = 2^{15} \cdot 3 \)
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
Elements	1	47	2	16	94	192	32	384
Conjugacy classes	1	17	1	6	31	8	8	72
Divisions	1	17	1	6	17	4	6	52
Autjugacy classes	1	5	1	2	5	1	2	17

Dimension	1	2	4	8
Irr. complex chars.	16	44	12	0	72
Irr. rational chars.	8	24	16	4	52

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$48$
Rank:	$3$
Inequivalent generating triples:	$168$

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:	$\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{8}=d^{2}=e^{6}=[a,b]=[a,d]=[a,e]=[b,d]=[b,e]=[c,d]=[d,e]=1, c^{a}=bc, c^{b}=c^{5}, e^{c}=de^{5} \rangle$
Permutation group:	Degree $15$ $\langle(1,2,5,6,4,7,3,8)(9,10)(13,14), (1,3)(2,6)(4,5)(7,8)(9,11)(10,12), (1,4) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 24 \\ 24 & 25 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 31 & 0 \\ 24 & 31 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 43 & 12 \\ 24 & 19 \end{array}\right), \left(\begin{array}{rr} 25 & 29 \\ 36 & 35 \end{array}\right), \left(\begin{array}{rr} 41 & 24 \\ 30 & 23 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/48\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_2^2$ $\,\rtimes\,$ $(C_{12}.D_4)$	$(C_6:\OD_{16})$ $\,\rtimes\,$ $C_2^2$	$(C_2\times C_{12}.D_4)$ $\,\rtimes\,$ $C_2$	$((C_2\times C_6):\OD_{16})$ $\,\rtimes\,$ $C_2$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6\times D_4)$ . $D_4$	$(D_4\times C_2^3)$ . $S_3$	$C_4$ . $(C_2^4:S_3)$	$(C_2^2\times D_4)$ . $D_6$	all 37

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{48}\Z)$.

Homology

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

Subgroups

There are 2174 subgroups in 820 conjugacy classes, 133 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^3.D_6$
Commutator:	$G' \simeq$ $C_2^2\times C_6$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2\times C_4$	$G/\Phi \simeq$ $C_2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}:C_2^4$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $(C_6\times D_4).D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^3:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.C_4$
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: $(C_6\times D_4).D_4$

Order 192: $C_2\times C_{12}.D_4$ x 4, $(C_2\times C_6):\OD_{16}$ x 2, $C_{12}:C_2^4$

Order 128: $C_2^5.C_4$

Order 96: $C_{12}:C_2^3$ x 16, $C_{12}.D_4$ x 8, $C_{12}.D_4$ x 4, $C_6:\OD_{16}$ x 4, $C_2^4\times C_6$ x 2, $C_2^3\times C_{12}$

Order 64: $\OD_{16}:C_2^2$ x 4, $C_2^2:\OD_{16}$ x 2, $D_4\times C_2^3$

Order 48: $C_6\times D_4$ x 60, $C_2^3\times C_6$ x 25, $C_2^2\times C_{12}$ x 10, $C_3:\OD_{16}$ x 8, $C_6:C_8$ x 4

Order 32: $C_2^2\times D_4$ x 16, $\OD_{16}:C_2$ x 8, $C_2^2:C_8$ x 4, $C_2\times \OD_{16}$ x 4, $C_2^5$ x 2, $C_2^3\times C_4$

Order 24: $C_2^2\times C_6$ x 97, $C_3\times D_4$ x 32, $C_2\times C_{12}$ x 18, $C_3:C_8$ x 4

Order 16: $C_2\times D_4$ x 60, $C_2^4$ x 25, $C_2^2\times C_4$ x 10, $\OD_{16}$ x 8, $C_2\times C_8$ x 4

Order 12: $C_2\times C_6$ x 85, $C_{12}$ x 6

Order 8: $C_2^3$ x 97, $D_4$ x 32, $C_2\times C_4$ x 18, $C_8$ x 4

Order 6: $C_6$ x 17

Order 4: $C_2^2$ x 85, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 17

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $(C_6\times D_4).D_4$

Order 192: $C_2\times C_{12}.D_4$, $(C_2\times C_6):\OD_{16}$, $C_{12}:C_2^4$

Order 128: $C_2^5.C_4$

Order 96: $C_{12}:C_2^3$ x 4, $C_{12}.D_4$, $C_{12}.D_4$, $C_2^4\times C_6$, $C_2^3\times C_{12}$, $C_6:\OD_{16}$

Order 64: $\OD_{16}:C_2^2$, $C_2^2:\OD_{16}$, $D_4\times C_2^3$

Order 48: $C_6\times D_4$ x 6, $C_2^3\times C_6$ x 5, $C_2^2\times C_{12}$ x 4, $C_6:C_8$, $C_3:\OD_{16}$

Order 32: $C_2^2\times D_4$ x 4, $\OD_{16}:C_2$, $C_2^5$, $C_2^2:C_8$, $C_2^3\times C_4$, $C_2\times \OD_{16}$

Order 24: $C_2^2\times C_6$ x 12, $C_2\times C_{12}$ x 5, $C_3\times D_4$ x 2, $C_3:C_8$

Order 16: $C_2\times D_4$ x 6, $C_2^4$ x 5, $C_2^2\times C_4$ x 4, $\OD_{16}$, $C_2\times C_8$

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

Order 8: $C_2^3$ x 12, $C_2\times C_4$ x 5, $D_4$ x 2, $C_8$

Order 6: $C_6$ x 5

Order 4: $C_2^2$ x 12, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $(C_6\times D_4).D_4$ ($C_1$)

Order 192: $C_2\times C_{12}.D_4$ ($C_2$) x 4, $(C_2\times C_6):\OD_{16}$ ($C_2$) x 2, $C_{12}:C_2^4$ ($C_2$)

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

Order 64: $D_4\times C_2^3$ ($S_3$)

Order 48: $C_6\times D_4$ ($D_4$) x 8, $C_2^3\times C_6$ ($C_2\times C_4$) x 5, $C_2^2\times C_{12}$ ($D_4$) x 4, $C_2^2\times C_{12}$ ($C_2^3$), $C_2^2\times C_{12}$ ($C_2\times C_4$)

Order 32: $C_2^5$ ($C_3:C_4$) x 2, $C_2^2\times D_4$ ($D_6$) x 2, $C_2^2\times D_4$ ($C_3:C_4$) x 2, $C_2^3\times C_4$ ($D_6$)

Order 24: $C_2^2\times C_6$ ($C_2^2:C_4$) x 10, $C_2\times C_{12}$ ($C_2\times D_4$) x 6, $C_2\times C_{12}$ ($C_2^2:C_4$) x 2, $C_2^2\times C_6$ ($C_2^2\times C_4$)

Order 16: $C_2\times D_4$ ($C_3:D_4$) x 8, $C_2^4$ ($C_6:C_4$) x 5, $C_2^2\times C_4$ ($C_3:D_4$) x 4, $C_2^2\times C_4$ ($C_6:C_4$), $C_2^2\times C_4$ ($C_2\times D_6$)

Order 12: $C_2\times C_6$ ($\OD_{16}:C_2$) x 4, $C_{12}$ ($C_2^2\wr C_2$) x 4, $C_2\times C_6$ ($C_2^3:C_4$) x 3

Order 8: $C_2^3$ ($C_6.D_4$) x 10, $C_2\times C_4$ ($C_6:D_4$) x 6, $C_2\times C_4$ ($C_6.D_4$) x 2, $C_2^3$ ($C_6.C_2^3$)

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

Order 4: $C_2^2$ ($C_{12}.D_4$) x 4, $C_4$ ($C_2^4:S_3$) x 4, $C_2^2$ ($C_2^3.D_6$) x 3

Order 3: $C_3$ ($C_2^5.C_4$)

Order 2: $C_2$ ($C_2\times C_{12}.D_4$) x 2, $C_2$ ($C_2^4.D_6$)

Order 1: $C_1$ ($(C_6\times D_4).D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $(C_6\times D_4).D_4$ ($C_1$)

Order 192: $C_2\times C_{12}.D_4$ ($C_2$), $(C_2\times C_6):\OD_{16}$ ($C_2$), $C_{12}:C_2^4$ ($C_2$)

Order 96: $C_2^4\times C_6$ ($C_4$), $C_{12}:C_2^3$ ($C_2^2$), $C_{12}:C_2^3$ ($C_4$), $C_2^3\times C_{12}$ ($C_2^2$), $C_6:\OD_{16}$ ($C_2^2$)

Order 64: $D_4\times C_2^3$ ($S_3$)

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

Order 32: $C_2^5$ ($C_3:C_4$), $C_2^2\times D_4$ ($D_6$), $C_2^2\times D_4$ ($C_3:C_4$), $C_2^3\times C_4$ ($D_6$)

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

Order 16: $C_2^4$ ($C_6:C_4$) x 2, $C_2\times D_4$ ($C_3:D_4$), $C_2^2\times C_4$ ($C_3:D_4$), $C_2^2\times C_4$ ($C_6:C_4$), $C_2^2\times C_4$ ($C_2\times D_6$)

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

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

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

Order 4: $C_2^2$ ($C_2^3.D_6$) x 2, $C_2^2$ ($C_{12}.D_4$), $C_4$ ($C_2^4:S_3$)

Order 3: $C_3$ ($C_2^5.C_4$)

Order 2: $C_2$ ($C_2^4.D_6$), $C_2$ ($C_2\times C_{12}.D_4$)

Order 1: $C_1$ ($(C_6\times D_4).D_4$)

Series

Derived series

$(C_6\times D_4).D_4$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_1$

Chief series

$(C_6\times D_4).D_4$

$\rhd$

$C_2\times C_{12}.D_4$

$\rhd$

$C_{12}:C_2^3$

$\rhd$

$C_2^2\times C_{12}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_6\times D_4).D_4$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3\times C_4$

$\lhd$

$D_4\times C_2^3$

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

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 $52 \times 52$ rational character table.