Abstract group 384.2088: $(C_2^2\times C_6).D_8$

• Label: 384.2088
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $19$
• Trans deg.: $192$
• Rank: $3$

Group information

Description:	$(C_2^2\times C_6).D_8$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_3:(C_2^7.C_2^6)$, of order \(24576\)\(\medspace = 2^{13} \cdot 3 \)
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	15	2	144	30	96	96	384
Conjugacy classes	1	11	1	16	11	16	16	72
Divisions	1	11	1	10	11	6	8	48
Autjugacy classes	1	6	1	6	6	2	5	27

Dimension	1	2	4	8
Irr. complex chars.	16	44	12	0	72
Irr. rational chars.	8	16	20	4	48

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$192$
Rank:	$3$
Inequivalent generating triples:	$672$

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:	$\langle a, b, c, d \mid a^{2}=b^{4}=c^{8}=d^{6}=[a,c]=[a,d]=[b,d]=1, b^{a}=bd^{3}, c^{b}=c^{7}d^{3}, d^{c}=d^{5} \rangle$
Permutation group:	Degree $19$ $\langle(1,2)(3,5)(4,6)(7,8)(12,13,14,16)(15,17,18,19), (1,3,8,6,4,7,5,2)(10,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 39 \\ 42 & 11 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 13 & 33 \\ 30 & 19 \end{array}\right), \left(\begin{array}{rr} 25 & 24 \\ 24 & 25 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/48\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_2^2$ $\,\rtimes\,$ $(C_6.D_8)$	$(C_{12}.C_4^2)$ $\,\rtimes\,$ $C_2$	$(C_{12}.C_2^3)$ $\,\rtimes\,$ $C_4$	$(C_2\times C_6.D_8)$ $\,\rtimes\,$ $C_2$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $(D_6.D_4)$	$(C_6:C_8)$ . $D_4$	$C_6$ . $(C_8:D_4)$	$(C_2^3.D_4)$ . $S_3$	all 67

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

Homology

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

Subgroups

There are 894 subgroups in 336 conjugacy classes, 125 normal (39 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_6:D_4$
Commutator:	$G' \simeq$ $C_2\times C_{12}$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2\times C_4$	$G/\Phi \simeq$ $C_2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $(C_2^2\times C_{12}):C_4$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $(C_2^2\times C_6).D_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.D_8$
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: $(C_2^2\times C_6).D_8$

Order 192: $C_{12}.C_4^2$ x 2, $C_2\times C_6.D_8$ x 2, $(C_2^2\times C_{12}):C_4$, $C_2^3.D_{12}$, $C_{12}.C_2^4$

Order 128: $C_2^3.D_8$

Order 96: $C_{12}.C_2^3$ x 10, $C_6.D_8$ x 4, $C_2\times C_4:C_{12}$ x 2, $C_2^2.D_{12}$ x 2, $C_6.C_4^2$, $C_6.C_4^2$, $C_2^3\times C_{12}$, $C_2^3:C_{12}$, $C_2^3.D_6$

Order 64: $C_2^3.D_4$ x 2, $C_4.C_4^2$ x 2, $C_2^2.D_8$ x 2, $C_2^3\times C_8$

Order 48: $C_6:C_8$ x 18, $C_2^2\times C_{12}$ x 12, $C_4:C_{12}$ x 4, $C_{12}:C_4$ x 4, $C_6.C_2^3$ x 2, $C_2^2:C_{12}$ x 2, $C_6.D_4$ x 2, $C_2^3\times C_6$

Order 32: $C_2^2\times C_8$ x 10, $C_2^2.D_4$ x 4, $C_8:C_4$ x 4, $C_2^3:C_4$ x 2, $C_2.C_4^2$ x 2, $C_2^3\times C_4$

Order 24: $C_2\times C_{12}$ x 24, $C_2^2\times C_6$ x 11, $C_6:C_4$ x 6, $C_3:C_8$ x 6

Order 16: $C_2\times C_8$ x 18, $C_2^2\times C_4$ x 14, $C_4:C_4$ x 8, $C_2^2:C_4$ x 4, $C_2^4$

Order 12: $C_2\times C_6$ x 23, $C_{12}$ x 8, $C_3:C_4$ x 2

Order 8: $C_2\times C_4$ x 30, $C_2^3$ x 11, $C_8$ x 6

Order 6: $C_6$ x 11

Order 4: $C_2^2$ x 23, $C_4$ x 10

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $(C_2^2\times C_6).D_8$

Order 192: $C_{12}.C_4^2$, $(C_2^2\times C_{12}):C_4$, $C_2^3.D_{12}$, $C_2\times C_6.D_8$, $C_{12}.C_2^4$

Order 128: $C_2^3.D_8$

Order 96: $C_{12}.C_2^3$ x 4, $C_6.C_4^2$, $C_6.C_4^2$, $C_2^3\times C_{12}$, $C_2\times C_4:C_{12}$, $C_2^3:C_{12}$, $C_2^3.D_6$, $C_6.D_8$, $C_2^2.D_{12}$

Order 64: $C_2^3.D_4$ x 2, $C_2^3\times C_8$, $C_4.C_4^2$, $C_2^2.D_8$

Order 48: $C_2^2\times C_{12}$ x 7, $C_6:C_8$ x 5, $C_4:C_{12}$ x 2, $C_{12}:C_4$ x 2, $C_2^3\times C_6$, $C_6.C_2^3$, $C_2^2:C_{12}$, $C_6.D_4$

Order 32: $C_2^2\times C_8$ x 4, $C_2^2.D_4$ x 2, $C_2^3:C_4$ x 2, $C_2.C_4^2$ x 2, $C_2^3\times C_4$, $C_8:C_4$

Order 24: $C_2\times C_{12}$ x 12, $C_2^2\times C_6$ x 6, $C_6:C_4$ x 3, $C_3:C_8$ x 2

Order 16: $C_2^2\times C_4$ x 8, $C_2\times C_8$ x 5, $C_4:C_4$ x 4, $C_2^2:C_4$ x 2, $C_2^4$

Order 12: $C_2\times C_6$ x 10, $C_{12}$ x 5, $C_3:C_4$

Order 8: $C_2\times C_4$ x 15, $C_2^3$ x 6, $C_8$ x 2

Order 6: $C_6$ x 6

Order 4: $C_2^2$ x 10, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $(C_2^2\times C_6).D_8$ ($C_1$)

Order 192: $C_{12}.C_4^2$ ($C_2$) x 2, $C_2\times C_6.D_8$ ($C_2$) x 2, $(C_2^2\times C_{12}):C_4$ ($C_2$), $C_2^3.D_{12}$ ($C_2$), $C_{12}.C_2^4$ ($C_2$)

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

Order 64: $C_2^3.D_4$ ($S_3$)

Order 48: $C_6:C_8$ ($D_4$) x 4, $C_6:C_8$ ($C_2\times C_4$) x 4, $C_2^2\times C_{12}$ ($Q_8$) x 2, $C_2^2\times C_{12}$ ($C_2\times C_4$) x 2, $C_2^3\times C_6$ ($D_4$), $C_2^2\times C_{12}$ ($C_2^3$), $C_2^2\times C_{12}$ ($D_4$)

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

Order 24: $C_3:C_8$ ($C_2^2:C_4$) x 4, $C_2\times C_{12}$ ($C_4:C_4$) x 2, $C_2\times C_{12}$ ($D_4:C_2$) x 2, $C_2\times C_{12}$ ($C_2\times D_4$) x 2, $C_2^2\times C_6$ ($Q_{16}$) x 2, $C_2^2\times C_6$ ($D_8$) x 2, $C_2^2\times C_6$ ($C_4:C_4$) x 2, $C_2\times C_{12}$ ($C_2\times Q_8$), $C_2\times C_{12}$ ($C_2^2\times C_4$), $C_2^2\times C_6$ ($C_2\times D_4$)

Order 16: $C_2^2\times C_4$ ($C_4\times S_3$) x 2, $C_2^2\times C_4$ ($C_3:Q_8$) x 2, $C_2^4$ ($C_3:D_4$), $C_2^2\times C_4$ ($C_3:D_4$), $C_2^2\times C_4$ ($C_2\times D_6$)

Order 12: $C_2\times C_6$ ($C_8:C_4$) x 4, $C_2\times C_6$ ($D_8:C_2$) x 2, $C_2\times C_6$ ($C_4:D_4$) x 2, $C_{12}$ ($C_2^2:Q_8$) x 2, $C_2\times C_6$ ($C_2\times Q_{16}$), $C_2\times C_6$ ($C_2\times D_8$), $C_2\times C_6$ ($C_2^2.D_4$), $C_{12}$ ($C_4^2:C_2$), $C_{12}$ ($C_2^3:C_4$)

Order 8: $C_2^3$ ($C_3:Q_{16}$) x 2, $C_2^3$ ($C_3:D_8$) x 2, $C_2^3$ ($C_6.D_4$) x 2, $C_2\times C_4$ ($D_4:S_3$) x 2, $C_2\times C_4$ ($S_3\times D_4$) x 2, $C_2\times C_4$ ($C_6.D_4$) x 2, $C_2^3$ ($C_6:D_4$), $C_2\times C_4$ ($C_4\times D_6$), $C_2\times C_4$ ($C_6:Q_8$)

Order 6: $C_6$ ($C_2^2:Q_{16}$) x 2, $C_6$ ($C_8:D_4$) x 2, $C_6$ ($C_2^3.D_4$), $C_6$ ($C_2^3.D_4$), $C_6$ ($C_2^2.D_8$)

Order 4: $C_2^2$ ($C_6.D_8$) x 4, $C_2^2$ ($C_{12}.D_4$) x 2, $C_2^2$ ($C_2^3.D_6$) x 2, $C_4$ ($C_2^3.D_6$) x 2, $C_2^2$ ($C_6:Q_{16}$), $C_2^2$ ($C_6:D_8$), $C_2^2$ ($C_2^3.D_6$), $C_4$ ($D_6.D_4$), $C_4$ ($C_2^3.D_6$)

Order 3: $C_3$ ($C_2^3.D_8$)

Order 2: $C_2$ ($(C_2\times C_6):Q_{16}$) x 2, $C_2$ ($(C_2\times C_6):D_8$) x 2, $C_2$ ($C_4.(C_4\times D_6)$), $C_2$ ($C_2\times C_6.D_8$), $C_2$ ($C_2^4.D_6$)

Order 1: $C_1$ ($(C_2^2\times C_6).D_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $(C_2^2\times C_6).D_8$ ($C_1$)

Order 192: $C_{12}.C_4^2$ ($C_2$), $(C_2^2\times C_{12}):C_4$ ($C_2$), $C_2^3.D_{12}$ ($C_2$), $C_2\times C_6.D_8$ ($C_2$), $C_{12}.C_2^4$ ($C_2$)

Order 96: $C_{12}.C_2^3$ ($C_2^2$) x 2, $C_2^3\times C_{12}$ ($C_2^2$), $C_2\times C_4:C_{12}$ ($C_2^2$), $C_2^2.D_{12}$ ($C_2^2$), $C_{12}.C_2^3$ ($C_4$)

Order 64: $C_2^3.D_4$ ($S_3$)

Order 48: $C_6:C_8$ ($C_2\times C_4$) x 2, $C_6:C_8$ ($D_4$), $C_2^3\times C_6$ ($D_4$), $C_2^2\times C_{12}$ ($C_2^3$), $C_2^2\times C_{12}$ ($Q_8$), $C_2^2\times C_{12}$ ($D_4$), $C_2^2\times C_{12}$ ($C_2\times C_4$)

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

Order 24: $C_2\times C_{12}$ ($C_4:C_4$), $C_2\times C_{12}$ ($D_4:C_2$), $C_2\times C_{12}$ ($C_2\times Q_8$), $C_2\times C_{12}$ ($C_2\times D_4$), $C_2\times C_{12}$ ($C_2^2\times C_4$), $C_2^2\times C_6$ ($Q_{16}$), $C_2^2\times C_6$ ($D_8$), $C_2^2\times C_6$ ($C_4:C_4$), $C_2^2\times C_6$ ($C_2\times D_4$), $C_3:C_8$ ($C_2^2:C_4$)

Order 16: $C_2^4$ ($C_3:D_4$), $C_2^2\times C_4$ ($C_3:D_4$), $C_2^2\times C_4$ ($C_4\times S_3$), $C_2^2\times C_4$ ($C_3:Q_8$), $C_2^2\times C_4$ ($C_2\times D_6$)

Order 12: $C_2\times C_6$ ($D_8:C_2$), $C_2\times C_6$ ($C_2\times Q_{16}$), $C_2\times C_6$ ($C_2\times D_8$), $C_2\times C_6$ ($C_4:D_4$), $C_2\times C_6$ ($C_2^2.D_4$), $C_2\times C_6$ ($C_8:C_4$), $C_{12}$ ($C_2^2:Q_8$), $C_{12}$ ($C_4^2:C_2$), $C_{12}$ ($C_2^3:C_4$)

Order 8: $C_2^3$ ($C_6:D_4$), $C_2^3$ ($C_3:Q_{16}$), $C_2^3$ ($C_3:D_8$), $C_2^3$ ($C_6.D_4$), $C_2\times C_4$ ($D_4:S_3$), $C_2\times C_4$ ($S_3\times D_4$), $C_2\times C_4$ ($C_4\times D_6$), $C_2\times C_4$ ($C_6:Q_8$), $C_2\times C_4$ ($C_6.D_4$)

Order 6: $C_6$ ($C_2^3.D_4$), $C_6$ ($C_2^2:Q_{16}$), $C_6$ ($C_8:D_4$), $C_6$ ($C_2^3.D_4$), $C_6$ ($C_2^2.D_8$)

Order 4: $C_2^2$ ($C_{12}.D_4$), $C_2^2$ ($C_6:Q_{16}$), $C_2^2$ ($C_2^3.D_6$), $C_2^2$ ($C_6.D_8$), $C_2^2$ ($C_6:D_8$), $C_2^2$ ($C_2^3.D_6$), $C_4$ ($D_6.D_4$), $C_4$ ($C_2^3.D_6$), $C_4$ ($C_2^3.D_6$)

Order 3: $C_3$ ($C_2^3.D_8$)

Order 2: $C_2$ ($(C_2\times C_6):Q_{16}$), $C_2$ ($(C_2\times C_6):D_8$), $C_2$ ($C_4.(C_4\times D_6)$), $C_2$ ($C_2\times C_6.D_8$), $C_2$ ($C_2^4.D_6$)

Order 1: $C_1$ ($(C_2^2\times C_6).D_8$)

Series

Derived series

$(C_2^2\times C_6).D_8$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_1$

Chief series

$(C_2^2\times C_6).D_8$

$\rhd$

$(C_2^2\times C_{12}):C_4$

$\rhd$

$C_2^3\times C_{12}$

$\rhd$

$C_2^2\times C_{12}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_2^2\times C_6).D_8$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^3\times C_4$

$\lhd$

$C_2^3.D_4$

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.