Abstract group 41472.ek: $C_2^3.D_6\wr C_3$

• Label: 41472.ek
• Order: \( 2^{9} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $21$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_2^3.D_6\wr C_3$
Order:	\(41472\)\(\medspace = 2^{9} \cdot 3^{4} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_{794}:C_{198}$, of order \(3981312\)\(\medspace = 2^{14} \cdot 3^{5} \)
Composition factors:	$C_2$ x 9, $C_3$ x 4
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	9	12	18
Elements	1	2271	1178	1824	15846	2304	11136	6912	41472
Conjugacy classes	1	51	5	10	124	2	25	6	224
Divisions	1	51	4	10	120	1	23	3	213
Autjugacy classes	1	16	4	6	30	1	9	2	69

Minimal presentations

Permutation degree:	$21$
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	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid b^{4}=c^{2}=d^{6}=e^{6}=f^{6}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $21$ $\langle(1,2,4)(3,6,9,8,5,7)(10,11,13,16,17,20,21,12,15,14,18,19), (1,4,3,7,5,8,9,6,2) \!\cdots\! \rangle$
Transitive group:	36T15400	36T15547			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:	$C_2^4$ . $(S_3^3:A_4)$	$(C_2^5\times S_3^3)$ . $C_6$ (3)	$C_2^5$ . $(S_3^3:C_6)$	$(C_6^3.C_2^4)$ . $A_4$ (5)	all 41

Elements of the group are displayed as permutations of degree 21.

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 71 normal subgroups (41 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 $C_2^3$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

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 18 (order at least 2304) are shown, as well as all normal subgroups of any index.

Order 41472: $C_2^3.D_6\wr C_3$

Order 20736: $(D_6\times D_6^2).A_4$, $C_2^5.S_3\wr C_3$, $C_6^3:D_4.A_4$

Order 13824: $C_6^3.C_2^6$

Order 10368: $(D_6\times D_6^2):C_6$ x 3, $(S_3\times D_6^2).A_4$, $(C_2^2\times C_6^3).A_4$, $(D_6\times D_6^2):C_6$, $C_6:D_6^2:A_4$, $C_6^3.(C_2^2\times A_4)$

Order 6912: $C_6^3.C_2^5$ x 22, $C_2^5\times S_3^3$

Order 5184: $D_6\wr C_3$ x 12, $C_6^3.(C_2\times A_4)$ x 3, $C_2\times C_6^3:A_4$ x 3, $(C_2\times C_6^3).A_4$, $C_6^3.(C_2\times A_4)$, $C_6^2.(A_4\times D_6)$, $C_2\times C_6^3:A_4$, $(C_2\times C_6^3):C_{12}$, $C_2\times C_6^3:A_4$

Order 4608: $C_6^2.C_2^6.C_2$

Order 3456: $C_6^3.C_2^4$ x 186, $C_3^3.C_2^6.C_2$ x 46, $C_2^4\times S_3^3$ x 44, $C_3^3.C_2^5.C_2^2$ x 14, $C_2^3\times C_2\times S_3^3$ x 2, $(C_6^2\times S_3).C_2^4$ x 2, $C_2^5\times S_3\times C_3:S_3$, $C_2^5\times C_3^3:C_2^2$, $C_2^4\times C_6\times S_3^2$, $C_3^2:C_2\wr C_6$

Order 2592: $S_3^3:A_4$ x 20, $C_6^3:A_4$ x 13, $(C_2\times S_3^3):C_6$ x 4, $C_6^3:A_4$ x 3, $(C_2\times C_6^3):C_6$, $C_6^3:A_4$, $(C_2\times C_6^3):C_6$, $C_6^3:A_4$

Order 2304: $C_6^2.C_2^5.C_2$ x 54, $C_6^2.C_2^6$ x 7, $(C_2\times C_6^2).C_2^5$ x 2, $C_2^6\times S_3^2$

Order 1728: $D_6\times D_6^2$ x 4, $C_6^3:C_2^3$ x 3, $C_2^3.S_3^3$ x 3, $C_6^3:C_2^3$ x 2, $C_6^3:C_2^3$, $C_2^3.S_3^3$, $C_6^3:D_4$, $C_6^3.D_4$

Order 864: $C_6:D_6^2$ x 4, $C_2^2\times C_6^3$, $C_6^3:C_2^2$

Order 432: $C_3:D_6^2$ x 4, $C_2\times S_3^3$ x 4, $C_2\times C_6^3$ x 2, $C_6^3:C_2$, $C_6^3.C_2$

Order 216: $C_6:S_3^2$ x 4, $C_6^3$

Order 108: $C_3:S_3^2$ x 4, $C_3\times C_6^2$, $C_3^2:D_6$

Order 64: $C_2^3:D_4$

Order 54: $C_3^2\times C_6$

Order 32: $C_2^5$

Order 27: $C_3^3$

Order 16: $C_2^4$ x 2

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 41472: $C_2^3.D_6\wr C_3$ ($C_1$)

Order 20736: $(D_6\times D_6^2).A_4$ ($C_2$), $C_2^5.S_3\wr C_3$ ($C_2$), $C_6^3:D_4.A_4$ ($C_2$)

Order 13824: $C_6^3.C_2^6$ ($C_3$)

Order 10368: $(C_2^2\times C_6^3).A_4$ ($C_2^2$)

Order 6912: $C_6^3.C_2^5$ ($C_6$) x 2, $C_2^5\times S_3^3$ ($C_6$)

Order 5184: $(C_2\times C_6^3).A_4$ ($D_4$)

Order 3456: $C_6^3.C_2^4$ ($A_4$) x 4, $C_3^3.C_2^6.C_2$ ($A_4$), $C_2^5\times C_3^3:C_2^2$ ($C_2\times C_6$)

Order 1728: $D_6\times D_6^2$ ($C_2\times A_4$) x 4, $C_6^3:C_2^3$ ($C_2\times A_4$) x 3, $C_2^3.S_3^3$ ($C_2\times A_4$) x 3, $C_6^3:C_2^3$ ($C_2\times A_4$), $C_6^3:C_2^3$ ($C_2\times A_4$), $C_6^3:C_2^3$ ($C_3\times D_4$), $C_2^3.S_3^3$ ($C_2\times A_4$), $C_6^3:D_4$ ($C_2\times A_4$), $C_6^3.D_4$ ($C_2\times A_4$)

Order 864: $C_6:D_6^2$ ($C_2^2\times A_4$) x 4, $C_2^2\times C_6^3$ ($C_2^2\times A_4$), $C_6^3:C_2^2$ ($C_2^2:A_4$)

Order 432: $C_3:D_6^2$ ($D_4\times A_4$) x 4, $C_2\times S_3^3$ ($C_2^3:A_4$) x 4, $C_2\times C_6^3$ ($C_2^3:A_4$), $C_2\times C_6^3$ ($D_4\times A_4$), $C_6^3:C_2$ ($C_2^3:A_4$), $C_6^3.C_2$ ($C_2^3:A_4$)

Order 216: $C_6:S_3^2$ ($C_2^4:A_4$) x 4, $C_6^3$ ($C_2^2\wr C_3$)

Order 108: $C_3:S_3^2$ ($C_2\wr C_6$) x 4, $C_3\times C_6^2$ ($C_2^6:C_6$), $C_3^2:D_6$ ($C_2^5:A_4$)

Order 64: $C_2^3:D_4$ ($S_3\wr C_3$)

Order 54: $C_3^2\times C_6$ ($C_2^6:A_4$)

Order 32: $C_2^5$ ($S_3^3:C_6$)

Order 27: $C_3^3$ ($C_2^8:C_6$)

Order 16: $C_2^4$ ($(C_2\times S_3^3):C_6$), $C_2^4$ ($S_3^3:A_4$)

Order 8: $C_2^3$ ($D_6\wr C_3$)

Order 4: $C_2^2$ ($(D_6\times D_6^2):C_6$)

Order 2: $C_2$ ($(D_6\times D_6^2).A_4$)

Order 1: $C_1$ ($C_2^3.D_6\wr C_3$)

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

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

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

Character theory

Complex character table

The $224 \times 224$ character table is not available for this group.

Rational character table

The $213 \times 213$ rational character table is not available for this group.