Abstract group 20480.kz: $C_2^6.C_2\wr D_5$

• Label: 20480.kz
• Order: \( 2^{12} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: 4
• $\card{\Aut(G)}$: \( 2^{17} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $22$
• Trans deg.: $40$
• Rank: $3$

Group information

Description:	$C_2^6.C_2\wr D_5$
Order:	\(20480\)\(\medspace = 2^{12} \cdot 5 \)
Exponent:	\(40\)\(\medspace = 2^{3} \cdot 5 \)
Automorphism group:	$C_4^4.C_{30}.C_2^4.C_2^4$, of order \(1966080\)\(\medspace = 2^{17} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 12, $C_5$
Derived length:	$3$

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

Group statistics

Order	1	2	4	5	8	10
Elements	1	1407	3200	1024	7680	7168	20480
Conjugacy classes	1	25	86	2	48	14	176
Divisions	1	25	86	1	36	5	154
Autjugacy classes	1	8	12	1	2	3	27

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$40$
Rank:	$3$
Inequivalent generating triples:	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:	${\langle a, b, c, d, e, f, g, h \mid b^{10}=c^{4}=d^{4}=e^{4}=f^{4}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $22$ $\langle(1,2,4,10,6,8,5,3,7,12)(9,13,17,20,16,11,15,19,18,14)(21,22), (1,3,8,4) \!\cdots\! \rangle$
Transitive group:	40T11751	40T12820	40T13578	40T13835	all 5
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_4^4:C_{10})$ . $D_4$ (2)	$C_4^4$ . $(C_{10}:D_4)$	$C_2^6$ . $(C_2\wr D_5)$ (3)	$(C_2^7.C_2^4)$ . $D_5$	all 24

Elements of the group are displayed as permutations of degree 22.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 40 normal subgroups (18 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 $C_2^2$
Commutator:	a subgroup isomorphic to $C_4^4:C_{10}$
Frattini:	a subgroup isomorphic to $C_2^5$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_2^4.C_2^5.C_2^2$

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 32 (order at least 640) are shown, as well as all normal subgroups of any index.

Order 20480: $C_2^6.C_2\wr D_5$

Order 10240: $(C_2\times C_4^4).D_{10}$ x 4, $C_2\times C_4^4.D_{10}$, $(C_2\times C_4^4).D_{10}$, $C_2^2\times C_4^4.C_{10}$

Order 5120: $C_4^4:D_{10}$ x 4, $C_2\times C_4^4:C_{10}$ x 4, $(C_2\times C_4^4).D_5$ x 2, $C_2\times C_4^4:C_{10}$

Order 4096: $C_2\times C_2^4.C_2^5.C_2^2$

Order 2560: $C_4^4:C_{10}$ x 3, $C_4^4:C_{10}$ x 2, $C_4^4:D_5$ x 2

Order 2048: $C_2^4.C_2^5.C_2^2$ x 16, $C_2\times C_4^2.C_2^5.C_2$ x 2, $C_2\times C_4^2.C_2^4.C_2^2$ x 2, $C_2\times C_2.C_2^6.C_2^3$ x 2, $C_2\times (C_2^2\times C_4^3).C_2^2$ x 2, $C_2^7.C_2^4$, $C_2\times C_4^2.C_4^3$, $C_2\times C_2^4.C_2^4.C_2^2$, $C_2\times C_2^3.C_2^6.C_2$, $C_2\times C_2^3.C_2^5.C_2^2$, $C_2\times (C_2^2\times C_4^2).C_2^4$, $C_2\times (C_2\times C_4^3).C_2^3$

Order 1280: $C_2^6:D_{10}$, $C_4^4:C_5$

Order 1024: $C_4^2.C_2^5.C_2$ x 56, $C_2.C_2^6.C_2^3$ x 56, $C_4^2.C_2^4.C_2^2$ x 32, $C_2^3.C_2^5.C_2^2$ x 22, $C_2^3.C_2^6.C_2$ x 20, $(C_2\times C_4^3).C_2^3$ x 20, $C_2^6.C_2^4$ x 18, $C_4^4:C_2^2$ x 16, $C_2\times C_2^3.C_2^5.C_2$ x 14, $C_2^4.C_2^4.C_2^2$ x 12, $(C_2^2\times C_4^3).C_2^2$ x 12, $C_2\times C_4^2.C_2^4.C_2$ x 11, $C_2\times (C_2\times C_4^3).C_2^2$ x 9, $C_2\times C_4^3.C_2^3$ x 8, $C_4^4:C_4$ x 8, $C_2\times C_2.C_2^6.C_2^2$ x 7, $C_2\times C_4^2.C_2^5$ x 6, $(C_2^2\times C_4^2).C_2^4$ x 6, $C_2\times (C_2^4\times C_4).C_2^3$ x 5, $(C_2^3\times C_4^2).C_2^3$ x 4, $C_2^7.C_2^3$ x 2, $C_2^5.C_2^5$ x 2, $C_2\times C_2^4.C_2^3.C_2^2$ x 2, $C_2\times C_2^3.C_2^4.C_2^2$ x 2, $C_2\times C_2^2.C_2^6.C_2$ x 2, $C_2\times (C_2\times C_4^2).C_2^4$ x 2, $C_2\times (C_2\times C_4).C_2^6$ x 2, $C_2^2\times C_4^4$, $C_2\times C_2^6.C_2^3$, $C_2\times C_2^4.C_2^4.C_2$, $C_2\times (C_2^2\times C_4^3).C_2$, $C_2\times (C_2\times C_4).D_4^2$, $(C_2^5\times C_4).C_2^3$, $(C_2\times C_4^2).C_2^5$

Order 640: $C_2^6:D_5$ x 4, $C_2^6:C_{10}$, $C_2^5:D_{10}$, $C_2^5.D_{10}$

Order 512: $C_2\times C_4^4$ x 3

Order 256: $C_4^4$

Order 128: $C_2^7$

Order 64: $C_2^6$ x 3

Order 32: $C_2^5$ x 3

Order 16: $C_2^4$

Order 4: $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 20480: $C_2^6.C_2\wr D_5$ ($C_1$)

Order 10240: $(C_2\times C_4^4).D_{10}$ ($C_2$) x 4, $C_2\times C_4^4.D_{10}$ ($C_2$), $(C_2\times C_4^4).D_{10}$ ($C_2$), $C_2^2\times C_4^4.C_{10}$ ($C_2$)

Order 5120: $(C_2\times C_4^4).D_5$ ($C_2^2$) x 2, $C_4^4:D_{10}$ ($C_2^2$) x 2, $C_2\times C_4^4:C_{10}$ ($C_2^2$) x 2, $C_2\times C_4^4:C_{10}$ ($C_2^2$)

Order 2560: $C_4^4:C_{10}$ ($D_4$) x 2, $C_4^4:C_{10}$ ($C_2^3$)

Order 2048: $C_2^7.C_2^4$ ($D_5$)

Order 1280: $C_4^4:C_5$ ($C_2\times D_4$)

Order 1024: $C_2^6.C_2^4$ ($D_{10}$) x 2, $C_2^2\times C_4^4$ ($D_{10}$)

Order 512: $C_2\times C_4^4$ ($C_5:D_4$) x 2, $C_2\times C_4^4$ ($C_2\times D_{10}$)

Order 256: $C_4^4$ ($C_{10}:D_4$)

Order 128: $C_2^7$ ($C_2^4:D_5$)

Order 64: $C_2^6$ ($C_2\wr D_5$) x 3

Order 32: $C_2^5$ ($C_2^6:D_5$) x 2, $C_2^5$ ($C_2^5:D_{10}$)

Order 16: $C_2^4$ ($C_2^6:D_{10}$)

Order 4: $C_2^2$ ($C_4^4:D_{10}$)

Order 2: $C_2$ ($(C_2\times C_4^4).D_{10}$) x 2, $C_2$ ($C_2\times C_4^4.D_{10}$)

Order 1: $C_1$ ($C_2^6.C_2\wr D_5$)

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 66 larger groups in the database.

This group is a maximal quotient of 7 larger groups in the database.

Character theory

Complex character table

The $176 \times 176$ character table is not available for this group.

Rational character table

The $154 \times 154$ rational character table is not available for this group.