Abstract group 1358954496.bf: $C_2^{16}.C_3^4.\OD_{16}.C_2^4$

• Label: 1358954496.bf
• Order: \( 2^{24} \cdot 3^{4} \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{29} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: not computed
• Trans deg.: $36$
• Rank: $5$

Group information

Description:	$C_2^{16}.C_3^4.\OD_{16}.C_2^4$
Order:	\(1358954496\)\(\medspace = 2^{24} \cdot 3^{4} \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	Group of order \(43486543872\)\(\medspace = 2^{29} \cdot 3^{4} \)
Composition factors:	$C_2$ x 24, $C_3$ x 4
Derived length:	$4$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12	16	24
Elements	1	688639	263168	85851648	72743936	521109504	267780096	382205952	28311552	1358954496
Conjugacy classes	1	141	3	474	85	196	124	44	12	1080
Divisions	1	141	3	474	85	176	124	32	12	1048
Autjugacy classes	1	93	2	197	44	56	43	9	4	449

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$36$
Rank:	$5$
Inequivalent generating 5-tuples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u \mid e^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,12,17,8,2,11,18,7)(5,15,14,10,6,16,13,9)(19,24,25,22)(20,23,26,21)(27,29,35,34) \!\cdots\! \rangle$
Transitive group:	36T94719				more information
Direct product:	$C_2$ $\, \times\, $ $(C_2^{16}.C_3^4.C_2.D_4^2)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^{17}$ . $(C_3^4:D_8:D_4)$	$(C_2^{16}.C_3^4.C_4)$ . $D_4^2$ (4)	$C_2^{16}$ . $(C_3^2:S_3^2.D_4^2)$	$(C_2^{16}.C_3^4.D_8:C_2)$ . $D_4$ (8)	all 64

Elements of the group are displayed as permutations of degree 36.

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{10}$
Commutator length:	$1$

Subgroups

There are 491 normal subgroups (21 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 $C_2$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^{11}.C_2.C_2^6.C_2^6$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

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 12 (order at least 113246208) are shown, as well as all normal subgroups of any index.

Order 1358954496: $C_2^{16}.C_3^4.\OD_{16}.C_2^4$

Order 679477248: $C_2^{16}.C_3^4.C_2.D_4^2$ x 16, $C_2^{16}.C_3^4.\OD_{16}.C_2^3$ x 4, $C_2^{12}.A_4^2.F_9.C_2^4$ x 4, $C_2^{16}.C_3^4.C_4^2.C_2^3$ x 2, $C_2^{16}.C_3^4.C_4^2.C_2^3$ x 2, $C_2^{16}.C_3^4.C_4^2.C_2^3$ x 2, $C_2^{16}.C_3^4.C_2^2.C_2^4.C_2$

Order 339738624: $C_2^{12}.A_4^2.F_9.C_2^3$ x 48, $C_2^{16}.C_3^4.D_4.D_4$ x 32, $C_2^{16}.C_3^4.D_4:D_4$ x 16, $C_2^{16}.C_3^4.D_4.D_4$ x 16, $C_2^{16}.C_3^4.D_8.C_4$ x 16, $C_2^{16}.C_3^4.C_4:\SD_{16}$ x 8, $C_2^{16}.C_3^4.C_4.C_2^4$ x 4, $C_2^{16}.C_3^4.C_4^2.C_2^2$ x 4, $C_2^{12}.A_4^2.F_9.C_2^3$ x 4, $C_2^9.A_4^2\wr C_2.\SD_{16}$ x 4, $C_2^{12}.A_4^2.F_9.C_2^3$ x 4, $C_2^{12}.A_4^2.F_9.C_2^3$ x 4, $C_2^{12}.A_4^2.F_9.C_2^3$ x 4, $C_2^{12}.A_4^2.F_9.C_2^3$ x 4, $C_2^{12}.A_4^2.F_9.C_2^3$ x 4, $C_2^{16}.C_3^4.\OD_{16}.C_2^2$ x 2, $C_2^{16}.C_3^4.\OD_{16}.C_2^2$ x 2, $C_2^{16}.C_3^4.\OD_{16}.C_2^2$ x 2, $C_2^{16}.C_3^4.C_4.C_2^4$ x 2, $C_2^{12}.A_4^2:S_3^2.C_2^4$ x 2, $C_2^{12}.A_4^2:D_6^2.C_2^2$ x 2, $C_2^{16}.C_3^4.C_2^4.C_2^2$, $C_2^{16}.C_3^4.C_2^3.C_2^3$, $C_2^{16}.C_3^4.(C_2\times C_4\times C_8)$

Order 169869312: $C_2^{12}.A_4^2:S_3^2.D_4$ x 32, $C_2^8.A_4^2\wr C_2.\SD_{16}$ x 32, $C_2^{12}.A_4^2:S_3^2.D_4$ x 32, $C_2^{12}.A_4^2:S_3^2.D_4$ x 32, $C_2^{12}.A_4^2.F_9.C_2^2$ x 24, $C_2^{12}.A_4^2.F_9.C_2^2$ x 24, $C_2^{12}.A_4^2.F_9.C_2^2$ x 24, $C_2^{12}.A_4^2:S_3^2.C_2^3$ x 18, $C_2^{12}.A_4^2:S_3^2.C_2^3$ x 18, $C_2^{16}.C_3^4.D_4.C_2^2$ x 17, $C_2^{16}.C_3^4.C_4.C_2^3$ x 17, $C_2^{16}.C_3^4.D_8:C_2$ x 16, $C_2^{16}.C_3^4.C_4\wr C_2$ x 16, $C_2^{16}.C_3^4.\OD_{16}.C_2$ x 8, $C_2^{16}.C_3^4.(C_8.C_4)$ x 8, $C_2^{16}.C_3^4.\OD_{16}.C_2$ x 8, $C_2^{12}.A_4^2:S_3^2.C_2^3$ x 6, $C_2^{16}.C_3^4.C_4:Q_8$ x 5, $C_2^{16}.C_3^4.C_4:D_4$ x 5, $C_2^{16}.C_3^4.C_4^2.C_2$ x 4, $C_2\times C_2^{16}.C_3^4.\SD_{16}$ x 4, $C_2\times C_2^{16}.C_3^4.\SD_{16}$ x 4, $C_2\times C_2^{16}.C_3^4.Q_{16}$ x 4, $C_2\times C_2^{16}.C_3^4.D_8$ x 4, $C_2^{12}.A_4^2.F_9:C_2^2$ x 4, $C_2^{12}.A_4^2.F_9.C_2^2$ x 4, $C_2^{12}.A_4^2:S_3^2.C_2^3$ x 4, $C_2^9.A_4^2\wr C_2.Q_8$ x 4, $C_2^9.A_4^2\wr C_2.D_4$ x 4, $C_2^{12}.A_4^2:S_3^2.C_2^3$ x 4, $C_2^{12}.A_4^2.F_9.C_2^2$ x 4, $C_2^9.A_4^2\wr C_2.C_8$ x 4, $C_2^{16}.C_3^4.C_2^4.C_2$ x 2, $C_2^{16}.C_3^4.C_2.C_2^4$ x 2, $C_2^{16}.C_3^4.Q_{16}.C_2$ x 2, $C_2\times C_2^{16}.C_3^4.D_8$ x 2, $C_2^9.(A_4^2\wr C_2.D_4)$ x 2, $C_2^{16}.C_3^4.C_2^3.C_2^2$, $C_2^8.C_3^2.C_2^4.A_4^2.C_2^3.C_2^2$, $C_2^8.C_3^2.C_2^4.A_4^2.C_2.C_2^4$

Order 84934656: $C_2^{16}.C_3^4.\SD_{16}$ x 16, $C_2^{12}.A_4^2:S_3^2.C_2^2$ x 16, $C_2^{16}.C_3^4.C_4.C_2^2$ x 12, $C_2^{16}.C_3^4.D_8$ x 8, $C_2^{16}.C_3^4.Q_{16}$ x 8, $C_2^{16}.C_3^4.\OD_{16}$ x 8, $C_2^{16}.C_3^4.C_2^3.C_2$ x 5, $C_2^{16}.C_3^4.C_2.C_2^3$ x 4, $C_2\times C_2^{16}.C_3^4.D_4$ x 4, $C_2\times C_2^{16}.C_3^4.Q_8$ x 4, $C_2^{16}.C_3^4.C_4^2$ x 2

Order 42467328: $C_2^{16}.C_3^4.C_8$ x 8, $C_2^{16}.C_3^4.D_4$ x 8, $C_2^{16}.C_3^4.Q_8$ x 8, $C_2^{16}.C_3^4.C_2^2.C_2$ x 6, $C_2^{16}.C_3^4.C_2^3$

Order 21233664: $C_2^{16}.C_3^4.C_4$ x 4, $C_2^{16}.C_3^3.D_6$ x 3

Order 16777216: $C_2^{11}.C_2.C_2^6.C_2^6$

Order 10616832: $C_2^{16}.C_3^4.C_2$ x 3

Order 5308416: $C_2^{16}.C_3^4$

Order 131072: $C_2^{17}$

Order 65536: $C_2^{16}$

Order 20736: $C_3^2:S_3^2.D_4^2$

Order 81: $C_3^4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 1358954496: $C_2^{16}.C_3^4.\OD_{16}.C_2^4$ ($C_1$)

Order 679477248: $C_2^{16}.C_3^4.C_2.D_4^2$ ($C_2$) x 16, $C_2^{16}.C_3^4.\OD_{16}.C_2^3$ ($C_2$) x 4, $C_2^{12}.A_4^2.F_9.C_2^4$ ($C_2$) x 4, $C_2^{16}.C_3^4.C_4^2.C_2^3$ ($C_2$) x 2, $C_2^{16}.C_3^4.C_4^2.C_2^3$ ($C_2$) x 2, $C_2^{16}.C_3^4.C_4^2.C_2^3$ ($C_2$) x 2, $C_2^{16}.C_3^4.C_2^2.C_2^4.C_2$ ($C_2$)

Order 339738624: $C_2^{12}.A_4^2.F_9.C_2^3$ ($C_2^2$) x 32, $C_2^{16}.C_3^4.D_4.D_4$ ($C_2^2$) x 32, $C_2^{16}.C_3^4.D_4:D_4$ ($C_2^2$) x 16, $C_2^{16}.C_3^4.D_4.D_4$ ($C_2^2$) x 16, $C_2^{16}.C_3^4.D_8.C_4$ ($C_2^2$) x 16, $C_2^{16}.C_3^4.C_4:\SD_{16}$ ($C_2^2$) x 8, $C_2^{16}.C_3^4.C_4.C_2^4$ ($C_2^2$) x 4, $C_2^{16}.C_3^4.C_4^2.C_2^2$ ($C_2^2$) x 4, $C_2^{12}.A_4^2.F_9.C_2^3$ ($C_2^2$) x 4, $C_2^{12}.A_4^2.F_9.C_2^3$ ($C_2^2$) x 4, $C_2^{12}.A_4^2.F_9.C_2^3$ ($C_2^2$) x 4, $C_2^{16}.C_3^4.\OD_{16}.C_2^2$ ($C_2^2$) x 2, $C_2^{16}.C_3^4.\OD_{16}.C_2^2$ ($C_2^2$) x 2, $C_2^{16}.C_3^4.\OD_{16}.C_2^2$ ($C_2^2$) x 2, $C_2^{16}.C_3^4.C_4.C_2^4$ ($C_2^2$) x 2, $C_2^{12}.A_4^2:S_3^2.C_2^4$ ($C_2^2$) x 2, $C_2^{12}.A_4^2:D_6^2.C_2^2$ ($C_2^2$) x 2, $C_2^{16}.C_3^4.C_2^4.C_2^2$ ($C_2^2$), $C_2^{16}.C_3^4.C_2^3.C_2^3$ ($C_2^2$), $C_2^{16}.C_3^4.(C_2\times C_4\times C_8)$ ($C_2^2$)

Order 169869312: $C_2^{12}.A_4^2.F_9.C_2^2$ ($C_2^3$) x 16, $C_2^{12}.A_4^2.F_9.C_2^2$ ($C_2^3$) x 16, $C_2^{16}.C_3^4.C_4\wr C_2$ ($C_2^3$) x 16, $C_2^{12}.A_4^2.F_9.C_2^2$ ($C_2^3$) x 16, $C_2^{16}.C_3^4.D_4.C_2^2$ ($C_2^3$) x 12, $C_2^{12}.A_4^2:S_3^2.C_2^3$ ($C_2^3$) x 8, $C_2^{16}.C_3^4.\OD_{16}.C_2$ ($C_2^3$) x 8, $C_2^{16}.C_3^4.(C_8.C_4)$ ($C_2^3$) x 8, $C_2^{16}.C_3^4.D_8:C_2$ ($C_2^3$) x 8, $C_2^{16}.C_3^4.D_8:C_2$ ($D_4$) x 8, $C_2^{16}.C_3^4.\OD_{16}.C_2$ ($C_2^3$) x 8, $C_2^{12}.A_4^2:S_3^2.C_2^3$ ($C_2^3$) x 8, $C_2^{16}.C_3^4.C_4.C_2^3$ ($C_2^3$) x 6, $C_2^{16}.C_3^4.C_4^2.C_2$ ($C_2^3$) x 4, $C_2^{16}.C_3^4.C_4:Q_8$ ($C_2^3$) x 4, $C_2^{16}.C_3^4.C_4:D_4$ ($C_2^3$) x 4, $C_2^{12}.A_4^2.F_9:C_2^2$ ($D_4$) x 4, $C_2^{12}.A_4^2:S_3^2.C_2^3$ ($C_2^3$) x 4, $C_2^{12}.A_4^2.F_9.C_2^2$ ($C_2^3$) x 4, $C_2^{16}.C_3^4.C_2^4.C_2$ ($C_2^3$) x 2, $C_2^{16}.C_3^4.C_2.C_2^4$ ($C_2^3$) x 2, $C_2^{16}.C_3^4.Q_{16}.C_2$ ($D_4$) x 2, $C_2\times C_2^{16}.C_3^4.D_8$ ($D_4$) x 2, $C_2^{16}.C_3^4.C_2^3.C_2^2$ ($C_2^3$)

Order 84934656: $C_2^{16}.C_3^4.\SD_{16}$ ($C_2\times D_4$) x 16, $C_2^{16}.C_3^4.C_4.C_2^2$ ($C_2\times D_4$) x 8, $C_2^{16}.C_3^4.D_8$ ($C_2\times D_4$) x 8, $C_2^{16}.C_3^4.Q_{16}$ ($C_2\times D_4$) x 8, $C_2^{12}.A_4^2:S_3^2.C_2^2$ ($C_2^4$) x 8, $C_2^{12}.A_4^2:S_3^2.C_2^2$ ($C_2\times D_4$) x 8, $C_2^{16}.C_3^4.\OD_{16}$ ($C_2^4$) x 8, $C_2^{16}.C_3^4.C_2^3.C_2$ ($C_2^4$) x 5, $C_2^{16}.C_3^4.C_4.C_2^2$ ($C_2^4$) x 4, $C_2^{16}.C_3^4.C_2.C_2^3$ ($C_2^4$) x 4, $C_2\times C_2^{16}.C_3^4.D_4$ ($C_2\times D_4$) x 4, $C_2\times C_2^{16}.C_3^4.Q_8$ ($C_2\times D_4$) x 4, $C_2^{16}.C_3^4.C_4^2$ ($C_2^4$) x 2

Order 42467328: $C_2^{16}.C_3^4.C_8$ ($C_2^2\times D_4$) x 8, $C_2^{16}.C_3^4.D_4$ ($C_2^2\times D_4$) x 8, $C_2^{16}.C_3^4.Q_8$ ($C_2^2\times D_4$) x 8, $C_2^{16}.C_3^4.C_2^2.C_2$ ($C_2^2\times D_4$) x 4, $C_2^{16}.C_3^4.C_2^3$ ($D_4:C_2^2$), $C_2^{16}.C_3^4.C_2^2.C_2$ ($C_2^5$), $C_2^{16}.C_3^4.C_2^2.C_2$ ($D_4:C_2^2$)

Order 21233664: $C_2^{16}.C_3^4.C_4$ ($D_4\times C_2^3$) x 2, $C_2^{16}.C_3^4.C_4$ ($D_4^2$) x 2, $C_2^{16}.C_3^3.D_6$ ($D_4^2$) x 2, $C_2^{16}.C_3^3.D_6$ ($D_4:C_2^3$)

Order 10616832: $C_2^{16}.C_3^4.C_2$ ($D_8:D_4$) x 2, $C_2^{16}.C_3^4.C_2$ ($C_2\times D_4^2$)

Order 5308416: $C_2^{16}.C_3^4$ ($C_4.D_4^2$)

Order 131072: $C_2^{17}$ ($C_3^4:D_8:D_4$)

Order 65536: $C_2^{16}$ ($C_3^2:S_3^2.D_4^2$)

Order 2: $C_2$ ($C_2^{16}.C_3^4.C_2.D_4^2$)

Order 1: $C_1$ ($C_2^{16}.C_3^4.\OD_{16}.C_2^4$)

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

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $1080 \times 1080$ character table is not available for this group.

Rational character table

The $1048 \times 1048$ rational character table is not available for this group.