Abstract group 384.20153: $C_2^5\times C_{12}$

• Label: 384.20153
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{22} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \)
• Perm deg.: $17$
• Trans deg.: $384$
• Rank: $6$

Group information

Description:	$C_2^5\times C_{12}$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^7.C_2^5.\GL(5,2)$, of order \(40957378560\)\(\medspace = 2^{22} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \)
Composition factors:	$C_2$ x 7, $C_3$
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).

Group statistics

Order	1	2	3	4	6	12
Elements	1	63	2	64	126	128	384
Conjugacy classes	1	63	2	64	126	128	384
Divisions	1	63	1	32	63	32	192
Autjugacy classes	1	2	1	1	2	1	8

Dimension	1	2	4
Irr. complex chars.	384	0	0	384
Irr. rational chars.	64	96	32	192

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$384$
Rank:	$6$
Inequivalent generating 6-tuples:	$22932$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	6	not computed	not computed

Constructions

Presentation:	Abelian group $\langle a, b, c, d, e, f \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{2}=f^{12}=1 \rangle$
Permutation group:	Degree $17$ $\langle(11,14,12,13), (1,2), (3,4), (5,6), (7,8), (9,10), (15,17,16), (11,12)(13,14)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 12 & 13 \end{array}\right), \left(\begin{array}{rr} 17 & 12 \\ 12 & 5 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 11 & 6 \\ 6 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 6 \\ 6 & 11 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/24\Z)$
Direct product:	$C_2$ ${}^5$ $\, \times\, $ $C_4$ $\, \times\, $ $C_3$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^6$ . $C_6$	$C_6$ . $C_2^6$	$(C_2^5\times C_6)$ . $C_2$	$C_2^5$ . $(C_2\times C_6)$	all 12

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

Homology

Primary decomposition:	$C_{2}^{5} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{15}$
Commutator length:	$0$

Subgroups

There are 10552 subgroups, all normal (8 characteristic).

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

Special subgroups

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

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

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 384: $C_2^5\times C_{12}$

Order 192: $C_2^4\times C_{12}$ x 62, $C_2^5\times C_6$

Order 128: $C_2^5\times C_4$

Order 96: $C_2^3\times C_{12}$ x 620, $C_2^4\times C_6$ x 63

Order 64: $C_2^4\times C_4$ x 62, $C_2^6$

Order 48: $C_2^2\times C_{12}$ x 1240, $C_2^3\times C_6$ x 651

Order 32: $C_2^3\times C_4$ x 620, $C_2^5$ x 63

Order 24: $C_2^2\times C_6$ x 1395, $C_2\times C_{12}$ x 496

Order 16: $C_2^2\times C_4$ x 1240, $C_2^4$ x 651

Order 12: $C_2\times C_6$ x 651, $C_{12}$ x 32

Order 8: $C_2^3$ x 1395, $C_2\times C_4$ x 496

Order 6: $C_6$ x 63

Order 4: $C_2^2$ x 651, $C_4$ x 32

Order 3: $C_3$

Order 2: $C_2$ x 63

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $C_2^5\times C_{12}$

Order 192: $C_2^5\times C_6$, $C_2^4\times C_{12}$

Order 128: $C_2^5\times C_4$

Order 96: $C_2^4\times C_6$ x 2, $C_2^3\times C_{12}$

Order 64: $C_2^6$, $C_2^4\times C_4$

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

Order 32: $C_2^5$ x 2, $C_2^3\times C_4$

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

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

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

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

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $C_2^5\times C_{12}$ ($C_1$)

Order 192: $C_2^4\times C_{12}$ ($C_2$) x 62, $C_2^5\times C_6$ ($C_2$)

Order 128: $C_2^5\times C_4$ ($C_3$)

Order 96: $C_2^3\times C_{12}$ ($C_2^2$) x 620, $C_2^4\times C_6$ ($C_4$) x 32, $C_2^4\times C_6$ ($C_2^2$) x 31

Order 64: $C_2^4\times C_4$ ($C_6$) x 62, $C_2^6$ ($C_6$)

Order 48: $C_2^2\times C_{12}$ ($C_2^3$) x 1240, $C_2^3\times C_6$ ($C_2\times C_4$) x 496, $C_2^3\times C_6$ ($C_2^3$) x 155

Order 32: $C_2^3\times C_4$ ($C_2\times C_6$) x 620, $C_2^5$ ($C_{12}$) x 32, $C_2^5$ ($C_2\times C_6$) x 31

Order 24: $C_2^2\times C_6$ ($C_2^2\times C_4$) x 1240, $C_2\times C_{12}$ ($C_2^4$) x 496, $C_2^2\times C_6$ ($C_2^4$) x 155

Order 16: $C_2^2\times C_4$ ($C_2^2\times C_6$) x 1240, $C_2^4$ ($C_2\times C_{12}$) x 496, $C_2^4$ ($C_2^2\times C_6$) x 155

Order 12: $C_2\times C_6$ ($C_2^3\times C_4$) x 620, $C_{12}$ ($C_2^5$) x 32, $C_2\times C_6$ ($C_2^5$) x 31

Order 8: $C_2^3$ ($C_2^2\times C_{12}$) x 1240, $C_2\times C_4$ ($C_2^3\times C_6$) x 496, $C_2^3$ ($C_2^3\times C_6$) x 155

Order 6: $C_6$ ($C_2^4\times C_4$) x 62, $C_6$ ($C_2^6$)

Order 4: $C_2^2$ ($C_2^3\times C_{12}$) x 620, $C_4$ ($C_2^4\times C_6$) x 32, $C_2^2$ ($C_2^4\times C_6$) x 31

Order 3: $C_3$ ($C_2^5\times C_4$)

Order 2: $C_2$ ($C_2^4\times C_{12}$) x 62, $C_2$ ($C_2^5\times C_6$)

Order 1: $C_1$ ($C_2^5\times C_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $C_2^5\times C_{12}$ ($C_1$)

Order 192: $C_2^5\times C_6$ ($C_2$), $C_2^4\times C_{12}$ ($C_2$)

Order 128: $C_2^5\times C_4$ ($C_3$)

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

Order 64: $C_2^6$ ($C_6$), $C_2^4\times C_4$ ($C_6$)

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

Order 32: $C_2^5$ ($C_2\times C_6$), $C_2^5$ ($C_{12}$), $C_2^3\times C_4$ ($C_2\times C_6$)

Order 24: $C_2\times C_{12}$ ($C_2^4$), $C_2^2\times C_6$ ($C_2^4$), $C_2^2\times C_6$ ($C_2^2\times C_4$)

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

Order 12: $C_2\times C_6$ ($C_2^5$), $C_2\times C_6$ ($C_2^3\times C_4$), $C_{12}$ ($C_2^5$)

Order 8: $C_2^3$ ($C_2^3\times C_6$), $C_2^3$ ($C_2^2\times C_{12}$), $C_2\times C_4$ ($C_2^3\times C_6$)

Order 6: $C_6$ ($C_2^6$), $C_6$ ($C_2^4\times C_4$)

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

Order 3: $C_3$ ($C_2^5\times C_4$)

Order 2: $C_2$ ($C_2^5\times C_6$), $C_2$ ($C_2^4\times C_{12}$)

Order 1: $C_1$ ($C_2^5\times C_{12}$)

Series

Derived series

$C_2^5\times C_{12}$

$\rhd$

$C_1$

Chief series

$C_2^5\times C_{12}$

$\rhd$

$C_2^5\times C_6$

$\rhd$

$C_2^4\times C_6$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2^5\times C_{12}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^5\times C_{12}$

Character theory

Complex character table

See the $384 \times 384$ 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 $192 \times 192$ rational character table (warning: may be slow to load).