Abstract group 131072.m: $C_2^{12}.C_4.C_2^3$

• Label: 131072.m
• Order: \( 2^{17} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: 8
• $\card{\Aut(G)}$: \( 2^{34} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{20} \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $6$

Group information

Description:	$C_2^{12}.C_4.C_2^3$
Order:	\(131072\)\(\medspace = 2^{17} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	Group of order \(17179869184\)\(\medspace = 2^{34} \)
Composition factors:	$C_2$ x 17
Nilpotency class:	$4$
Derived length:	$3$

This group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).

Group statistics

Order	1	2	4	8
Elements	1	9599	105088	16384	131072
Conjugacy classes	1	496	681	24	1202
Divisions	1	496	681	16	1194
Autjugacy classes	1	84	125	3	213

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
Rank:	$6$
Inequivalent generating 6-tuples:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l \mid a^{2}=b^{4}=c^{2}=d^{4}= \!\cdots\! \rangle}$
Permutation group:	Degree $64$ $\langle(1,33)(4,36)(5,37)(8,40)(9,41)(12,44)(13,45)(16,48)(17,49)(20,52)(21,53) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^{12}.D_4)$ . $C_2^2$ (4)	$(C_2^{10}.C_2^6)$ . $C_2$	$(C_2^{12}.C_4)$ . $C_2^3$ (8)	$(C_2^9.C_2^6)$ . $C_2^2$	all 15
Aut. group:	$\Aut(C_2^3.C_2^4)$	$\Aut(C_2^4.D_4)$

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

Homology

Abelianization:	$C_{2}^{6} $
Schur multiplier:	$C_{2}^{21}$
Commutator length:	$2$

Subgroups

There are 24018 normal subgroups (89 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2^3$
Commutator:	not computed
Frattini:	not computed
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^{12}.C_4.C_2^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 2 (order at least 65536) are shown, as well as normal subgroups of index up to 8 or of order up to 16.

Order 131072: $C_2^{12}.C_4.C_2^3$

Order 65536: $C_2^6.C_2^6.C_2^4$ x 53, $C_2^5.C_2^6.C_2^5$ x 5, $C_2^{12}.C_4.C_2^2$ x 2, $C_2^9.C_2^6.C_2$, $C_2^7.C_2^5.C_2^4$, $C_2^{10}.C_2^6$

Order 32768: $C_2^6.C_2^6.C_2^3$ x 221, $C_2^6.C_2^5.C_2^4$ x 170, $C_2^5.C_2^6.C_2^4$ x 148, $C_2^9.C_2^6$ x 32, $C_2^{10}.C_2^5$ x 21, $C_2^7.C_2^4.C_2^4$ x 16, $C_2^7.C_2^6.C_2^2$ x 11, $C_2^7.C_2^5.C_2^3$ x 9, $C_2^8.C_2^3.C_2^4$ x 4, $C_2^{12}.D_4$ x 4, $C_2^8.C_2^6.C_2$ x 3, $C_2\times C_2^8.C_2^6$ x 3, $C_2^8.C_2^4.C_2^3$ x 2, $C_2^9.C_2.C_2^5$, $C_2^8.C_2.C_2^6$, $C_2^3\times C_2^7.C_2^5$, $C_2^2\times C_2^7.C_2^6$, $C_2\times C_2^6.C_2^5.C_2^3$, $C_2\times C_2^5.C_2^6.C_2^3$, $C_2^9.C_2^6$

Order 16384: $C_2^5.C_2^6.C_2^3$ x 591, $C_2^6.C_2^6.C_2^2$ x 198, $C_2^6.C_2^5.C_2^3$ x 172, $C_2^9.C_2^5$ x 167, $C_2^8.C_2^6$ x 86, $C_2^7.C_2^6.C_2$ x 76, $C_2^7.C_2^4.C_2^3$ x 48, $C_2^7.C_2.C_2^6$ x 43, $C_2\times C_2^8.C_2^5$ x 29, $C_2\times C_2^9.C_2^4$ x 27, $C_2^{10}.C_2^4$ x 24, $C_2^7.C_2^5.C_2^2$ x 24, $C_2^8.C_2.C_2^5$ x 18, $C_2^3\times C_2^6.C_2^5$ x 14, $C_2^8.C_2^3.C_2^3$ x 12, $C_2\times C_2^7.C_2^6$ x 9, $C_2^8.C_2^4.C_2^2$ x 8, $C_2^{12}.C_4$ x 8, $C_2^3\times C_2^7.C_2^4$ x 7, $C_2^2\times C_2^7.C_2^5$ x 7, $C_2^2\times C_2^6.C_2^6$ x 5, $C_2\times C_2^6.C_2^6.C_2$ x 5, $C_2\times C_2^6.C_2^4.C_2^3$ x 5, $C_2\times C_2^5.C_2^6.C_2^2$ x 5, $C_2\times C_2^5.C_2^5.C_2^3$ x 5, $C_2\times C_2^6.C_2^5.C_2^2$ x 4, $C_2^2\times C_2^8.C_2^4$ x 2, $C_2\times C_2^7.C_2.C_2^5$ x 2, $C_2^8.C_2^5.C_2$, $C_2^3\times C_2^4.C_2^6.C_2$

Order 16: $C_2^4$ x 399

Order 8: $C_2^3$ x 163

Order 4: $C_2^2$ x 59

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

All normal subgroups of index up to 8 or order up to 16 are shown.

Order 131072: $C_2^{12}.C_4.C_2^3$ ($C_1$)

Order 65536: $C_2^6.C_2^6.C_2^4$ ($C_2$) x 53, $C_2^5.C_2^6.C_2^5$ ($C_2$) x 5, $C_2^{12}.C_4.C_2^2$ ($C_2$) x 2, $C_2^9.C_2^6.C_2$ ($C_2$), $C_2^7.C_2^5.C_2^4$ ($C_2$), $C_2^{10}.C_2^6$ ($C_2$)

Order 32768: $C_2^6.C_2^6.C_2^3$ ($C_2^2$) x 221, $C_2^6.C_2^5.C_2^4$ ($C_2^2$) x 170, $C_2^5.C_2^6.C_2^4$ ($C_2^2$) x 148, $C_2^9.C_2^6$ ($C_2^2$) x 32, $C_2^{10}.C_2^5$ ($C_2^2$) x 21, $C_2^7.C_2^4.C_2^4$ ($C_2^2$) x 16, $C_2^7.C_2^6.C_2^2$ ($C_2^2$) x 11, $C_2^7.C_2^5.C_2^3$ ($C_2^2$) x 9, $C_2^8.C_2^3.C_2^4$ ($C_2^2$) x 4, $C_2^{12}.D_4$ ($C_2^2$) x 4, $C_2^8.C_2^6.C_2$ ($C_2^2$) x 3, $C_2\times C_2^8.C_2^6$ ($C_2^2$) x 3, $C_2^8.C_2^4.C_2^3$ ($C_2^2$) x 2, $C_2^9.C_2.C_2^5$ ($C_2^2$), $C_2^8.C_2.C_2^6$ ($C_2^2$), $C_2^3\times C_2^7.C_2^5$ ($C_2^2$), $C_2^2\times C_2^7.C_2^6$ ($C_2^2$), $C_2\times C_2^6.C_2^5.C_2^3$ ($C_2^2$), $C_2\times C_2^5.C_2^6.C_2^3$ ($C_2^2$), $C_2^9.C_2^6$ ($C_2^2$)

Order 16384: $C_2^5.C_2^6.C_2^3$ ($C_2^3$) x 511, $C_2^6.C_2^6.C_2^2$ ($C_2^3$) x 182, $C_2^6.C_2^5.C_2^3$ ($C_2^3$) x 172, $C_2^9.C_2^5$ ($C_2^3$) x 151, $C_2^5.C_2^6.C_2^3$ ($D_4$) x 80, $C_2^7.C_2^4.C_2^3$ ($C_2^3$) x 48, $C_2^7.C_2^6.C_2$ ($C_2^3$) x 46, $C_2^8.C_2^6$ ($D_4$) x 44, $C_2^7.C_2.C_2^6$ ($C_2^3$) x 43, $C_2^8.C_2^6$ ($C_2^3$) x 42, $C_2^7.C_2^6.C_2$ ($D_4$) x 30, $C_2\times C_2^8.C_2^5$ ($C_2^3$) x 29, $C_2^{10}.C_2^4$ ($C_2^3$) x 24, $C_2^7.C_2^5.C_2^2$ ($C_2^3$) x 24, $C_2^8.C_2.C_2^5$ ($C_2^3$) x 18, $C_2^9.C_2^5$ ($D_4$) x 16, $C_2^6.C_2^6.C_2^2$ ($D_4$) x 16, $C_2^3\times C_2^6.C_2^5$ ($C_2^3$) x 14, $C_2\times C_2^9.C_2^4$ ($D_4$) x 14, $C_2\times C_2^9.C_2^4$ ($C_2^3$) x 13, $C_2^8.C_2^3.C_2^3$ ($C_2^3$) x 12, $C_2^8.C_2^4.C_2^2$ ($C_2^3$) x 8, $C_2^{12}.C_4$ ($C_2^3$) x 8, $C_2^3\times C_2^7.C_2^4$ ($C_2^3$) x 7, $C_2^2\times C_2^7.C_2^5$ ($C_2^3$) x 7, $C_2^2\times C_2^6.C_2^6$ ($C_2^3$) x 5, $C_2\times C_2^7.C_2^6$ ($C_2^3$) x 5, $C_2\times C_2^6.C_2^4.C_2^3$ ($C_2^3$) x 5, $C_2\times C_2^5.C_2^6.C_2^2$ ($C_2^3$) x 5, $C_2\times C_2^5.C_2^5.C_2^3$ ($C_2^3$) x 5, $C_2\times C_2^7.C_2^6$ ($D_4$) x 4, $C_2\times C_2^6.C_2^5.C_2^2$ ($C_2^3$) x 4, $C_2\times C_2^6.C_2^6.C_2$ ($C_2^3$) x 3, $C_2^2\times C_2^8.C_2^4$ ($D_4$) x 2, $C_2\times C_2^7.C_2.C_2^5$ ($C_2^3$) x 2, $C_2\times C_2^6.C_2^6.C_2$ ($D_4$) x 2, $C_2^8.C_2^5.C_2$ ($C_2^3$), $C_2^3\times C_2^4.C_2^6.C_2$ ($C_2^3$)

Order 8192: ? (unidentified group of order 16) x 2211

Order 4096: ? (unidentified group of order 32) x 2803

Order 2048: ? (unidentified group of order 64) x 2979

Order 1024: ? (unidentified group of order 128) x 3587

Order 512: ? (unidentified group of order 256) x 3219

Order 256: ? (unidentified group of order 512) x 2831

Order 128: ? (unidentified group of order 1024) x 1811

Order 64: ? (unidentified group of order 2048) x 1043

Order 32: ? (unidentified group of order 4096) x 587

Order 16: $C_2^4$ ($C_2^7.C_2^6$) x 248, $C_2^4$ ($C_2.C_2^6.C_2^6$) x 112, $C_2^4$ ($C_2^8.C_2^5$) x 31, $C_2^4$ ($C_2^9.(C_2\times D_4)$) x 8

Order 8: $C_2^3$ ($C_2^8.C_2^6$) x 74, $C_2^3$ ($C_2^4.C_2^6.C_2^4$) x 48, $C_2^3$ ($C_2^5.C_2^5.C_2^4$) x 30, $C_2^3$ ($C_2^3.C_2^6.C_2^5$) x 8, $C_2^3$ ($C_2^{10}.(C_2\times D_4)$) x 2, $C_2^3$ ($C_2^9.C_2^5$)

Order 4: $C_2^2$ ($C_2^5.C_2^6.C_2^4$) x 32, $C_2^2$ ($C_2^9.C_2^6$) x 15, $C_2^2$ ($C_2^4.C_2^6.C_2^5$) x 12

Order 2: $C_2$ ($C_2^5.C_2^6.C_2^5$) x 6, $C_2$ ($C_2^{10}.C_2^6$)

Order 1: $C_1$ ($C_2^{12}.C_4.C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Character theory

Complex character table

The $1202 \times 1202$ character table is not available for this group.

Rational character table

The $1194 \times 1194$ rational character table is not available for this group.