Abstract group 3840.gz: $A_5.D_4^2$

• Label: 3840.gz
• Order: \( 2^{8} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: 4
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $13$
• Trans deg.: not computed
• Rank: $4$

Group information

Description:	$A_5.D_4^2$
Order:	\(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^{12}$, of order \(61440\)\(\medspace = 2^{12} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 6, $A_5$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	10	12	20
Elements	1	479	20	1312	24	700	456	560	288	3840
Conjugacy classes	1	25	1	24	1	15	10	9	3	89
Divisions	1	25	1	24	1	15	9	9	3	88
Autjugacy classes	1	13	1	10	1	7	5	4	2	44

Dimension	1	2	4	5	6	8	10	12	16	20	24
Irr. complex chars.	16	8	17	16	8	8	8	6	1	1	0	89
Irr. rational chars.	16	8	17	16	8	8	8	4	1	1	1	88

Minimal presentations

Permutation degree:	$13$
Transitive degree:	not computed
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Permutation group:	Degree $13$ $\langle(6,7)(8,9)(10,11)(12,13), (8,9)(10,12)(11,13), (6,8)(7,9)(11,13), (6,7) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$A_5$ . $D_4^2$	$(C_2^2:S_5)$ . $D_4$ (8)	$C_2^2$ . $(D_4\times S_5)$ (4)	$(C_2^4:S_5)$ . $C_2$ (2)	all 43

Elements of the group are displayed as permutations of degree 13.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 115 normal subgroups (27 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 $C_2^2\times A_5$
Frattini:	a subgroup isomorphic to $C_2^2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:D_4^2$

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 64 (order at least 60) are shown, as well as all normal subgroups of any index.

Order 3840: $A_5.D_4^2$

Order 1920: $C_2\times D_4\times S_5$ x 2, $C_2^4:S_5$ x 2, $(C_2\times S_5):D_4$ x 2, $(C_2\times D_4):S_5$ x 2, $C_2.(D_4\times S_5)$ x 2, $C_2^2:(C_4\times S_5)$ x 2, $(C_2\times D_4):S_5$, $A_5\times C_2^2\wr C_2$, $(C_2\times D_4).S_5$

Order 960: $C_2^3:S_5$ x 18, $D_4\times S_5$ x 8, $(C_2\times C_4):S_5$ x 4, $C_2^2:C_4\times A_5$ x 3, $C_2^3\times S_5$ x 3, $C_2\times D_4\times A_5$ x 3, $C_2\times C_4:S_5$ x 3, $(C_2\times C_4).S_5$ x 2, $C_2^3.S_5$ x 2, $C_2\times C_4\times S_5$ x 2, $C_2^3.S_5$, $A_5:C_4^2$, $C_2^4\times A_5$

Order 768: $A_4:D_4^2$

Order 640: $D_5.D_4^2$

Order 480: $C_2^2:S_5$ x 24, $C_2^2\times S_5$ x 19, $C_2^3\times A_5$ x 9, $C_2^2.S_5$ x 8, $D_4\times A_5$ x 5, $C_4:S_5$ x 5, $C_4\times S_5$ x 4, $C_2\times C_4\times A_5$ x 3

Order 384: $C_2^2\times \GL(2,\mathbb{Z}/4)$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.D_6$ x 2, $C_2^4:D_{12}$ x 2, $\GL(2,\mathbb{Z}/4):C_4$ x 2, $S_3\times D_4^2$, $C_2^6:C_6$, $C_4:\GL(2,\mathbb{Z}/4)$, $C_2^2:\GL(2,\mathbb{Z}/4)$

Order 320: $(C_2\times D_{20}):C_4$ x 3, $C_2^4:F_5$ x 2, $D_{10}.C_2^4$ x 2, $C_{10}.(C_4\times D_4)$ x 2, $C_{10}.(C_4\times D_4)$ x 2, $D_{10}:C_4^2$ x 2, $C_2^4:D_{10}$, $(C_2\times D_{20}):C_4$

Order 256: $C_2^2:D_4^2$

Order 240: $C_2\times S_5$ x 26, $C_2^2\times A_5$ x 19, $A_5:C_4$ x 6, $C_4\times A_5$ x 3

Order 192: $C_2\times \GL(2,\mathbb{Z}/4)$ x 18, $D_4\times S_4$ x 8, $C_2.\GL(2,\mathbb{Z}/4)$ x 4, $C_{12}:C_2^4$ x 4, $C_2^4:D_6$ x 4, $D_{12}:D_4$ x 4, $S_3\times C_4:D_4$ x 4, $C_2^4:D_6$ x 4, $C_2^4:D_6$ x 4, $C_2^4:C_{12}$ x 3, $C_2^3\times S_4$ x 3, $C_2^5:C_6$ x 3, $C_2^3:D_{12}$ x 3, $C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.S_4$ x 2, $C_2^4.D_6$ x 2, $D_4\times D_{12}$ x 2, $C_4^2:D_6$ x 2, $C_2.\GL(2,\mathbb{Z}/4)$, $A_4:C_4^2$, $A_4\times C_2^4$, $C_3\times D_4^2$, $C_3:D_4^2$, $C_4^2:D_6$

Order 160: $C_2^3:F_5$ x 19, $D_4\times F_5$ x 8, $C_{10}.C_4^2$ x 6, $C_2^3\times F_5$ x 5, $D_4\times D_{10}$ x 3, $D_{10}.D_4$ x 3, $C_{10}:C_4^2$ x 3, $D_{10}:D_4$ x 3, $C_2^2:D_{20}$ x 3, $D_{10}.D_4$ x 3, $C_2^3\times D_{10}$, $C_2^4:C_{10}$, $C_2^4:D_5$

Order 128: $C_2.D_4^2$ x 6, $C_2^4:D_4$ x 6, $C_2\times D_4^2$ x 3, $C_2^4:D_4$ x 3, $C_2.D_4^2$ x 3, $C_2.D_4^2$ x 3, $C_2^4:D_4$ x 3, $C_2.D_4^2$ x 3, $C_2.D_4^2$

Order 120: $C_2\times A_5$ x 9, $S_5$ x 6

Order 96: $D_4\times D_6$ x 64, $\GL(2,\mathbb{Z}/4)$ x 24, $C_2^2\times S_4$ x 19, $C_2^2:D_{12}$ x 12, $D_6:D_4$ x 12, $C_2^3\times A_4$ x 9, $D_6:D_4$ x 8, $D_6.D_4$ x 8, $C_2^3:D_6$ x 8, $C_2^2.S_4$ x 8, $C_2^3.D_6$ x 8, $C_{12}:D_4$ x 6, $D_4\times A_4$ x 5, $C_4:S_4$ x 5, $C_2^3.D_6$ x 4, $C_2^3\times D_6$ x 4, $C_{12}:C_2^3$ x 4, $C_2^2\times D_{12}$ x 4, $C_{12}:C_2^3$ x 4, $C_4\times S_4$ x 4, $C_{12}:D_4$ x 4, $C_2^4:C_6$ x 4, $C_2^4:S_3$ x 4, $C_{12}:D_4$ x 4, $C_{12}:D_4$ x 4, $C_{12}:D_4$ x 4, $C_4:D_{12}$ x 4, $C_2^3:C_{12}$ x 3, $D_6.D_4$ x 2, $C_4\times D_{12}$ x 2, $D_4\times C_{12}$ x 2, $C_2^3.D_6$ x 2, $D_{12}:C_4$ x 2, $C_4:D_{12}$, $S_3\times C_4^2$, $C_{12}:D_4$

Order 80: $C_2^2\times F_5$ x 33, $D_{10}:C_4$ x 25, $C_2^2\times D_{10}$ x 17, $D_4\times D_5$ x 10, $C_{10}:D_4$ x 6, $D_{10}:C_4$ x 6, $C_{20}:C_4$ x 5, $C_4\times F_5$ x 5, $D_4\times C_{10}$ x 3, $C_2\times D_{20}$ x 3, $C_4\times D_{10}$ x 3, $C_2^2:C_{20}$ x 3, $C_{10}.D_4$ x 3, $C_2^3\times C_{10}$

Order 64: $C_2^3:D_4$ x 50, $D_4^2$ x 24, $C_2^3:D_4$ x 18, $C_2^3.D_4$ x 18, $C_2^3:D_4$ x 15, $C_2^3.D_4$ x 12, $D_4\times C_2^3$ x 12, $C_2^4:C_4$ x 9, $C_2^3.C_2^3$ x 6, $C_2^3.C_2^3$ x 6, $C_2^3.D_4$ x 6, $C_4^2:C_2^2$ x 6, $C_4^2:C_2^2$ x 4, $C_2^4:C_4$ x 3, $C_2^2:C_4^2$ x 3, $C_2^3.Q_8$ x 2, $C_2^6$

Order 60: $A_5$

Order 32: $C_2^2\wr C_2$

Order 16: $C_2^2:C_4$ x 3, $C_2\times D_4$ x 3, $C_2^4$

Order 8: $C_2^3$ x 4, $C_2\times C_4$ x 3

Order 4: $C_2^2$ x 5

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 3840: $A_5.D_4^2$ ($C_1$)

Order 1920: $C_2\times D_4\times S_5$ ($C_2$) x 2, $C_2^4:S_5$ ($C_2$) x 2, $(C_2\times S_5):D_4$ ($C_2$) x 2, $(C_2\times D_4):S_5$ ($C_2$) x 2, $C_2.(D_4\times S_5)$ ($C_2$) x 2, $C_2^2:(C_4\times S_5)$ ($C_2$) x 2, $(C_2\times D_4):S_5$ ($C_2$), $A_5\times C_2^2\wr C_2$ ($C_2$), $(C_2\times D_4).S_5$ ($C_2$)

Order 960: $C_2^3:S_5$ ($C_2^2$) x 10, $(C_2\times C_4):S_5$ ($C_2^2$) x 4, $C_2^2:C_4\times A_5$ ($C_2^2$) x 3, $C_2^3\times S_5$ ($C_2^2$) x 3, $C_2\times D_4\times A_5$ ($C_2^2$) x 3, $C_2\times C_4:S_5$ ($C_2^2$) x 3, $(C_2\times C_4).S_5$ ($C_2^2$) x 2, $C_2^3.S_5$ ($C_2^2$) x 2, $C_2\times C_4\times S_5$ ($C_2^2$) x 2, $C_2^3.S_5$ ($C_2^2$), $A_5:C_4^2$ ($C_2^2$), $C_2^4\times A_5$ ($C_2^2$)

Order 480: $C_2^2:S_5$ ($D_4$) x 8, $C_2^2.S_5$ ($C_2^3$) x 4, $C_2^3\times A_5$ ($C_2^3$) x 4, $C_2^2\times S_5$ ($C_2^3$) x 4, $C_2\times C_4\times A_5$ ($C_2^3$) x 3

Order 240: $A_5:C_4$ ($C_2\times D_4$) x 4, $C_2^2\times A_5$ ($C_2\times D_4$) x 4, $C_2\times S_5$ ($C_2\times D_4$) x 4, $C_2^2\times A_5$ ($C_2^4$)

Order 120: $C_2\times A_5$ ($C_2^2\times D_4$) x 2, $C_2\times A_5$ ($D_4:C_2^2$)

Order 60: $A_5$ ($D_4^2$)

Order 32: $C_2^2\wr C_2$ ($S_5$)

Order 16: $C_2^2:C_4$ ($C_2\times S_5$) x 3, $C_2\times D_4$ ($C_2\times S_5$) x 3, $C_2^4$ ($C_2\times S_5$)

Order 8: $C_2^3$ ($C_2^2\times S_5$) x 4, $C_2\times C_4$ ($C_2^2\times S_5$) x 3

Order 4: $C_2^2$ ($D_4\times S_5$) x 4, $C_2^2$ ($C_2^3\times S_5$)

Order 2: $C_2$ ($C_2\times D_4\times S_5$) x 2, $C_2$ ($(C_2\times D_4):S_5$)

Order 1: $C_1$ ($A_5.D_4^2$)

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 7 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $89 \times 89$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $88 \times 88$ rational character table.