Abstract group 1358954496.by: $C_2^{16}.C_3^4.C_4^2.C_2^3.C_2$

• Label: 1358954496.by
• Order: \( 2^{24} \cdot 3^{4} \)
• Exponent: \( 2^{5} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{27} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: not computed
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_2^{16}.C_3^4.C_4^2.C_2^3.C_2$
Order:	\(1358954496\)\(\medspace = 2^{24} \cdot 3^{4} \)
Exponent:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Automorphism group:	Group of order \(10871635968\)\(\medspace = 2^{27} \cdot 3^{4} \)
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	32
Elements	1	614911	263168	56434176	34995200	276922368	154533888	467140608	28311552	339738624	1358954496
Conjugacy classes	1	117	2	384	72	138	114	44	12	16	900
Divisions	1	117	2	384	72	118	114	20	12	4	844
Autjugacy classes	1	92	2	203	43	81	44	16	8	2	492

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	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:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u \mid c^{4}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,23,10,36,18,33,4,20,15,21,8,28,11,29,13,25)(2,24,9,35,17,34,3,19,16,22,7,27,12,30,14,26) \!\cdots\! \rangle$
Transitive group:	36T94746	36T94747	36T94748		more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^{17}$ . $(C_3^2:F_9.D_8)$	$C_2^{16}$ . $(C_2\times C_3^2:F_9.D_8)$	$(C_2^{16}.C_3^4.C_4.C_2^3)$ . $D_4$ (4)	$(C_2^{16}.C_3^4.C_4.C_2^2)$ . $D_8$ (4)	all 28

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

Homology

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

Subgroups

There are 75 normal subgroups (25 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:	$P_{ 2 } \simeq$ $C_2^{12}.C_2^4.C_2.C_2^6.C_2$
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 81 (order at least 16777216) are shown, as well as all normal subgroups of any index.

Order 1358954496: $C_2^{16}.C_3^4.C_4^2.C_2^3.C_2$

Order 679477248: $C_2^{16}.C_3^4.C_4:\SD_{16}.C_2$ x 4, $C_2^{16}.C_3^4.\OD_{32}.C_2^2$ x 2, $C_2^{16}.C_3^4.C_2^2.C_2^4.C_2$

Order 339738624: $C_2^{16}.C_3^4.C_4:\SD_{16}$ x 6, $C_2^{16}.C_3^4.(C_8.C_8)$ x 6, $C_2^{16}.C_3^4.C_4.C_2^4$ x 2, $C_2^{16}.C_3^4.C_8.C_2^3$ 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: $C_2^{16}.C_3^4.D_4.C_2^2$ x 11, $C_2^{16}.C_3^4.C_4.C_2^3$ x 7, $C_2^{16}.C_3^4.\OD_{32}$ x 6, $C_2^{16}.C_3^4.C_4^2.C_2$ x 4, $C_2^{16}.C_3^4.C_4:Q_8$ x 4, $C_2^{16}.C_3^4.C_4:D_4$ x 4, $C_2^{16}.C_3^4.C_{16}.C_2$ x 2, $C_2^{16}.C_3^4.C_2^4.C_2$, $C_2^{16}.C_3^4.C_2^3.C_2^2$, $C_2^{16}.C_3^4.C_2.C_2^4$, $C_2^8.C_3^2.C_2^6.A_4.C_6.C_2^4$, $C_2^8.C_3^2.C_2^4.A_4^2.C_2^3.C_2^2$

Order 84934656: $C_2^{16}.C_3^4.\SD_{16}$ x 16, $C_2^{16}.C_3^4.C_4.C_2^2$ x 12, $C_2^{16}.C_3^4.C_2^3.C_2$ x 9, $C_2^{16}.C_3^4.C_2.C_2^3$ x 8, $C_2^8.C_3^2.C_2^4.A_4^2.C_4:C_4$ x 4, $C_2^{16}.C_3^4.C_{16}$ x 4, $C_2^{16}.C_3^4.C_4^2$ x 3, $C_2^8.C_6^2.C_2^6.C_4:S_3^2$ x 3, $C_2^{16}.C_3^4.C_2^4$, $C_2^8.C_6^2.C_2^6.C_6^2.C_2^2$, $C_2^8.C_3^2.C_2^6.A_4.D_{12}.C_2$, $C_2^8.C_3^2.C_2^6.A_4.C_6.C_2^3$, $C_2^8.C_3^2.C_2^6.(C_6\times A_4).C_2^3$, $C_2^8.C_3^2.C_2^4.A_4^2.D_4.C_2$

Order 75497472: $C_2^8.C_6^2.C_2^5.C_2^5.C_2^3$

Order 56623104: $C_2^8.C_6^2.C_2^4.C_6.C_2.C_2^5$

Order 42467328: $C_2^{16}.C_3^4.C_2^2.C_2$ x 11, $C_2^{16}.C_3^4.C_8$ x 8, $C_2^{16}.C_3^4.Q_8$ x 6, $C_2^{16}.C_3^4.D_4$ x 6, $C_2^{16}.C_3.S_3^3$ x 5, $C_2^{16}.C_3^4.C_4.C_2$ x 4, $C_2^8.C_6^2.C_2^6.D_6:S_3$ x 4, $C_2^{16}.C_3^4.C_2^3$ x 2, $C_2^{14}.C_3^3.A_4.C_2^3$ x 2, $C_2^8.C_6^2.C_2^4.(A_4\times D_{12})$ x 2, $C_2^8.C_3^2.C_2^6.(A_4\times D_{12})$ x 2, $C_2^8.C_6^2.C_2^6.C_6^2.C_2$, $C_2^8.C_6^2.A_4^2.C_2.C_2^4$

Order 37748736: $C_2^8.C_6^2.C_2^5.C_2^4.C_2^3$ x 14, $C_2^8.C_6^2.C_2^5.C_2^5.C_2^2$ x 10, $C_2^8.C_6^2.C_2^6.C_2.C_2^5$ x 3, $C_2^8.C_6^2.C_2^6.C_2^4.C_2^2$ x 2, $C_2^8.C_6^2.C_2^6.C_2^6$, $C_2^8.C_6^2.C_2.C_2^6.C_2.C_2^4$

Order 28311552: $C_2^8.C_6^2.C_2^4.C_3.D_4^2$ x 16, $C_2^8.C_6^2.C_2^4.C_6.C_2.C_2^4$ x 7, $C_2^{14}.C_6^3.C_2^3$ x 2, $C_2^8.C_6^2.C_2^4.C_6.C_2^5$ x 2, $C_2^8.C_6^2.C_2^4.C_6.C_2^4.C_2$ x 2, $C_2^{14}.C_3^3.C_2^4.C_2^2$, $C_2^8.C_3^2.C_2^6.A_4.C_2^4$

Order 21233664: $C_2^{16}.C_3^4.C_4$ x 8, $C_2^{16}.C_3^3.D_6$ x 7, $C_2^{14}.C_3^3.A_4.C_2^2$ x 6, $C_2^8.C_3^2.C_2^6.C_4:S_3^2$ x 5, $C_2^{14}.C_3^4.C_2^4$ x 2, $C_2^8.C_6^2.C_2^6.C_3.C_{12}$ x 2, $C_2^8.C_6^2.A_4^2.D_4.C_2$ x 2, $C_2^{14}.C_3^4.C_2.C_2^3$, $C_2^{12}.C_6^2.A_4.C_6.C_2$, $C_2^{12}.C_3^4.C_2^4.C_2^2$, $C_2^8.C_6^2.C_2^6.C_6^2$, $C_2^8.C_6^2.C_2^4.(C_6\times D_{12})$, $C_2^8.C_6^2.A_4^2.C_4.C_2^2$, $C_2^8.C_6^2.A_4^2.C_2.C_2^3$, $C_2^8.C_3^2.C_2^6.C_6^2.C_2^2$, $C_2^8.C_3^2.C_2^4.C_6.A_4.C_2^3$

Order 18874368: $C_2^8.C_6^2.C_2^5.C_2^4.C_2^2$ x 64, $C_2^8.C_6^2.C_2^6.C_2^5$ x 34, $C_2^8.C_6^2.C_2^6.C_2.C_2^4$ x 34, $C_2^8.C_6^2.C_2^5.C_2.C_4:D_4$ x 16, $C_2^8.C_6^2.C_2.C_2^6.D_4.C_2$ x 16, $C_2^8.C_6^2.C_2^5.C_2.C_2^5$ x 14, $C_2^8.C_6^2.C_2^6.C_2^4.C_2$ x 10, $C_2^8.C_6^2.C_2^5.C_2^5.C_2$ x 8, $C_2^8.C_6^2.C_2.C_2^6.C_2^4$ x 7, $C_2^8.C_6^2.C_2^6.C_2^3.C_2^2$ x 6, $C_2^8.C_6^2.C_2^5.C_4^2.C_2^2$ x 6, $C_2^8.C_6^2.C_2^4.C_2\wr C_4.C_2$ x 6, $C_2^8.C_6^2.C_2^5.C_2^6$ x 4, $C_2^8.C_6^2.C_2^5.C_4.C_2^4$ x 2, $C_2^8.C_6^2.C_2^5.C_2^2.C_2^4$ x 2, $C_2^8.C_6^2.C_2.C_2^6.C_2^3.C_2$ x 2, $C_2^8.C_6^2.C_2.C_2^6.C_2.C_2^3$ x 2, $C_2^{16}.F_9.C_2^2$ x 2, $C_2^9.C_6^2.C_2^6.C_2^4$, $C_2^9.C_6^2.C_2^5.C_2^5$

Order 16777216: $C_2^{12}.C_2^4.C_2.C_2^6.C_2$

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: $C_2\times C_3^2:F_9.D_8$

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: $C_2^{16}.C_3^4.C_4^2.C_2^3.C_2$ ($C_1$)

Order 679477248: $C_2^{16}.C_3^4.C_4:\SD_{16}.C_2$ ($C_2$) x 4, $C_2^{16}.C_3^4.\OD_{32}.C_2^2$ ($C_2$) x 2, $C_2^{16}.C_3^4.C_2^2.C_2^4.C_2$ ($C_2$)

Order 339738624: $C_2^{16}.C_3^4.(C_8.C_8)$ ($C_2^2$) x 4, $C_2^{16}.C_3^4.C_4:\SD_{16}$ ($C_2^2$) x 2, $C_2^{16}.C_3^4.C_4:\SD_{16}$ ($C_4$) x 2, $C_2^{16}.C_3^4.C_2^4.C_2^2$ ($C_4$), $C_2^{16}.C_3^4.C_2^3.C_2^3$ ($C_4$), $C_2^{16}.C_3^4.(C_2\times C_4\times C_8)$ ($C_2^2$)

Order 169869312: $C_2^{16}.C_3^4.C_4^2.C_2$ ($D_4$) x 2, $C_2^{16}.C_3^4.C_4:Q_8$ ($C_2\times C_4$) x 2, $C_2^{16}.C_3^4.C_4:D_4$ ($C_2\times C_4$) x 2, $C_2^{16}.C_3^4.C_4.C_2^3$ ($D_4$) x 2, $C_2^{16}.C_3^4.C_4^2.C_2$ ($C_2^3$), $C_2^{16}.C_3^4.C_4^2.C_2$ ($C_2\times C_4$), $C_2^{16}.C_3^4.C_2^3.C_2^2$ ($C_2\times C_4$)

Order 84934656: $C_2^{16}.C_3^4.C_4.C_2^2$ ($\SD_{16}$) x 4, $C_2^{16}.C_3^4.C_4.C_2^2$ ($D_8$) x 4, $C_2^{16}.C_3^4.C_4.C_2^2$ ($C_2^2:C_4$) x 2, $C_2^{16}.C_3^4.C_4.C_2^2$ ($C_2\times D_4$) x 2, $C_2^{16}.C_3^4.C_4^2$ ($C_2^2:C_4$), $C_2^{16}.C_3^4.C_4^2$ ($C_2^2\times C_4$), $C_2^{16}.C_3^4.C_2^3.C_2$ ($C_2^2:C_4$)

Order 42467328: $C_2^{16}.C_3^4.C_8$ ($D_4:C_4$) x 4, $C_2^{16}.C_3^4.C_2^2.C_2$ ($D_4:C_4$) x 4, $C_2^{16}.C_3^4.C_8$ ($C_2\times \SD_{16}$) x 2, $C_2^{16}.C_3^4.C_8$ ($C_2\times D_8$) x 2, $C_2^{16}.C_3^4.C_2^3$ ($\OD_{16}:C_2$), $C_2^{16}.C_3^4.C_2^2.C_2$ ($\OD_{16}:C_2$), $C_2^{16}.C_3^4.C_2^2.C_2$ ($C_2^3:C_4$)

Order 21233664: $C_2^{16}.C_3^4.C_4$ ($C_2^2.D_8$) x 2, $C_2^{16}.C_3^4.C_4$ ($C_4.D_8$) x 2, $C_2^{16}.C_3^3.D_6$ ($C_4.D_8$) x 2, $C_2^{16}.C_3^3.D_6$ ($\OD_{16}:C_2^2$)

Order 10616832: $C_2^{16}.C_3^4.C_2$ ($C_8.D_8$) x 2, $C_2^{16}.C_3^4.C_2$ ($C_4^2.D_4$)

Order 5308416: $C_2^{16}.C_3^4$ ($C_2\times C_8.D_8$)

Order 131072: $C_2^{17}$ ($C_3^2:F_9.D_8$)

Order 65536: $C_2^{16}$ ($C_2\times C_3^2:F_9.D_8$)

Order 2: $C_2$ ($C_2^{16}.C_3^4.C_4:\SD_{16}.C_2$)

Order 1: $C_1$ ($C_2^{16}.C_3^4.C_4^2.C_2^3.C_2$)

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 6 larger groups in the database.

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

Character theory

Complex character table

The $900 \times 900$ character table is not available for this group.

Rational character table

The $844 \times 844$ rational character table is not available for this group.