Abstract group 43200.be: $A_4\times A_5^2$

• Label: 43200.be
• Order: \( 2^{6} \cdot 3^{3} \cdot 5^{2} \)
• Exponent: \( 2 \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{3} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $14$
• Trans deg.: $100$
• Rank: $2$

Group information

Description:	$A_4\times A_5^2$
Order:	\(43200\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{2} \)
Exponent:	\(30\)\(\medspace = 2 \cdot 3 \cdot 5 \)
Automorphism group:	$S_4\times S_5\wr C_2$, of order \(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \)
Composition factors:	$C_2$ x 2, $C_3$, $A_5$ x 2
Derived length:	$2$

This group is nonabelian, an A-group, and nonsolvable.

Group statistics

Order	1	2	3	5	6	10	15	30
Elements	1	1023	3968	624	10560	4752	13632	8640	43200
Conjugacy classes	1	7	11	8	17	16	28	12	100
Divisions	1	7	7	4	12	8	8	4	51
Autjugacy classes	1	5	5	2	7	4	4	2	30

Dimension	1	2	3	4	5	6	8	9	10	12	15	16	18	20	24	25	27	30	32	36	40	45	48	50	54	60	72	75	90
Irr. complex chars.	3	0	13	6	6	0	0	16	0	14	14	3	0	6	0	3	4	0	0	4	0	4	1	0	0	2	0	1	0	100
Irr. rational chars.	1	1	1	2	2	2	2	0	2	4	2	1	4	2	2	1	0	2	1	2	2	0	3	1	2	4	2	1	2	51

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$100$
Rank:	$2$
Inequivalent generating pairs:	$684$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	27	27	48
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $14$ $\langle(2,4)(3,5), (2,3)(4,5), (2,5,4), (1,12,14,11,13)(3,5,4)(6,7,8,9,10), (1,12)(2,4)(3,5)(6,9)(7,10)(11,13)\rangle$
Direct product:	$A_4$ $\, \times\, $ $A_5$ ${}^2$
Semidirect product:	$(C_2^2\times A_5^2)$ $\,\rtimes\,$ $C_3$	$(C_2^2\times A_5)$ $\,\rtimes\,$ $\GL(2,4)$	$C_2^2$ $\,\rtimes\,$ $(A_5\times \GL(2,4))$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 14.

Homology

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

Subgroups

There are 285056 subgroups in 1145 conjugacy classes, 12 normal (6 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $A_4\times A_5^2$
Commutator:	$G' \simeq$ $C_2^2\times A_5^2$	$G/G' \simeq$ $C_3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $A_4\times A_5^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2$	$G/\operatorname{Fit} \simeq$ $A_5\times \GL(2,4)$
Radical:	$R \simeq$ $A_4$	$G/R \simeq$ $A_5^2$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times A_5^2$	$G/\operatorname{soc} \simeq$ $C_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 43200: $A_4\times A_5^2$

Order 14400: $C_2^2\times A_5^2$

Order 10800: $A_5\times \GL(2,4)$

Order 8640: $A_5\times A_4^2$ x 2

Order 7200: $D_5\times A_4\times A_5$ x 2, $C_2\times A_5^2$

Order 4320: $S_3\times A_4\times A_5$ x 2

Order 3600: $C_5\times A_4\times A_5$ x 2, $A_5^2$

Order 2880: $C_2^4.\GL(2,4)$ x 4, $C_2^4:\GL(2,4)$ x 4

Order 2400: $C_2\times D_{10}\times A_5$ x 2

Order 2160: $A_4\times \GL(2,4)$ x 4

Order 1800: $D_5\times \GL(2,4)$ x 2

Order 1728: $A_4^3$

Order 1440: $C_2^3:\GL(2,4)$ x 4, $D_5\times A_4^2$ x 2, $\GL(2,4):C_2^3$ x 2

Order 1200: $D_{10}\times A_5$ x 4, $C_2\times C_{10}\times A_5$ x 2, $A_4\times D_5^2$

Order 1080: $S_3\times \GL(2,4)$ x 2

Order 960: $C_2^4\times A_5$ x 2

Order 900: $C_{15}\times A_5$ x 2

Order 864: $S_3\times A_4^2$ x 2

Order 720: $A_4\times A_5$ x 16, $C_2^2\times \GL(2,4)$ x 4, $D_6\times A_5$ x 4, $C_5\times A_4^2$ x 2, $S_3\times D_5\times A_4$ x 2

Order 600: $D_5\times A_5$ x 4, $C_{10}^2:C_6$ x 2, $C_{10}\times A_5$ x 2, $C_{10}^2:C_6$

Order 576: $C_2^2:A_4^2$ x 6, $C_2^2:A_4^2$ x 4, $C_2^2\times A_4^2$ x 3

Order 540: $C_3\times \GL(2,4)$ x 2

Order 480: $C_2^3\times A_5$ x 6, $D_5\times C_2^2:A_4$ x 4, $C_2\times A_4\times D_{10}$ x 4

Order 432: $C_3\times A_4^2$ x 3, $A_4\times S_3^2$

Order 400: $D_{10}^2$

Order 360: $C_3\times D_5\times A_4$ x 4, $C_6\times A_5$ x 4, $S_3\times A_5$ x 4, $A_4\times D_{15}$ x 2, $C_5\times S_3\times A_4$ x 2

Order 300: $C_5\times A_5$ x 2, $A_4\times C_5^2$, $C_3\times D_5^2$

Order 288: $S_3\times C_2^2:A_4$ x 4, $C_2\times A_4\times D_6$ x 4, $C_2\times A_4^2$ x 3

Order 240: $C_2^2\times A_5$ x 16, $A_4\times D_{10}$ x 8, $C_2^4:C_{15}$ x 4, $C_2^3:C_{30}$ x 4, $D_6\times D_{10}$ x 2

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

Order 200: $D_5\times D_{10}$ x 4, $C_{10}\times D_{10}$ x 2, $C_{10}:D_{10}$

Order 192: $C_2^2\wr C_3$ x 6, $C_2^4:A_4$ x 4, $A_4\times C_2^4$ x 3

Order 180: $\GL(2,4)$ x 8, $A_4\times C_{15}$ x 4, $C_{15}:D_6$ x 2

Order 160: $C_2^3\times D_{10}$ x 2

Order 150: $D_5\times C_{15}$ x 2, $C_{15}:D_5$

Order 144: $A_4^2$ x 25, $A_4\times D_6$ x 8, $C_2^4:C_3^2$ x 6, $C_2^2:C_6^2$ x 6, $D_6^2$

Order 120: $D_5\times A_4$ x 20, $S_3\times D_{10}$ x 8, $C_2\times A_5$ x 8, $C_6\times D_{10}$ x 4, $C_{10}\times A_4$ x 4, $C_2\times D_{30}$ x 2, $C_{10}\times D_6$ x 2

Order 108: $C_3^2\times A_4$ x 3, $C_3\times S_3^2$

Order 100: $D_5^2$ x 6, $C_5\times D_{10}$ x 4, $C_5:D_{10}$ x 2, $C_{10}^2$

Order 96: $C_2^3\times A_4$ x 9, $C_2^3:A_4$ x 6, $C_2^3\times D_6$ x 2

Order 90: $C_3\times D_{15}$ x 2, $S_3\times C_{15}$ x 2, $C_3^2\times D_5$ x 2

Order 80: $C_2^2\times D_{10}$ x 12, $C_2^3\times C_{10}$ x 2

Order 75: $C_5\times C_{15}$

Order 72: $S_3\times A_4$ x 19, $C_6\times A_4$ x 6, $C_6\times D_6$ x 4, $S_3\times D_6$ x 4, $C_6:D_6$

Order 64: $C_2^6$

Order 60: $C_5\times A_4$ x 16, $S_3\times D_5$ x 12, $C_3\times D_{10}$ x 8, $A_5$ x 4, $C_2\times C_{30}$ x 4, $D_{30}$ x 4, $S_3\times C_{10}$ x 4

Order 54: $S_3\times C_3^2$ x 2, $C_3^2:C_6$

Order 50: $C_5\times D_5$ x 4, $C_5:D_5$ x 2, $C_5\times C_{10}$

Order 48: $C_2^2\times A_4$ x 35, $C_2^2:A_4$ x 30, $C_2^2\times D_6$ x 12, $C_2^3\times C_6$ x 3

Order 45: $C_3\times C_{15}$ x 2

Order 40: $C_2\times D_{10}$ x 42, $C_2^2\times C_{10}$ x 6

Order 36: $C_3\times A_4$ x 28, $C_6\times S_3$ x 8, $S_3^2$ x 6, $C_6^2$ x 3, $C_6:S_3$ x 2

Order 32: $C_2^5$ x 7

Order 30: $C_3\times D_5$ x 12, $C_{30}$ x 4, $D_{15}$ x 4, $C_5\times S_3$ x 4

Order 27: $C_3^3$

Order 25: $C_5^2$

Order 24: $C_2\times D_6$ x 41, $C_2\times A_4$ x 23, $C_2^2\times C_6$ x 9

Order 20: $D_{10}$ x 36, $C_2\times C_{10}$ x 16

Order 18: $C_3\times S_3$ x 11, $C_3\times C_6$ x 3, $C_3:S_3$ x 2

Order 16: $C_2^4$ x 41

Order 15: $C_{15}$ x 8

Order 12: $D_6$ x 34, $A_4$ x 34, $C_2\times C_6$ x 24

Order 10: $D_5$ x 12, $C_{10}$ x 8

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 65

Order 6: $C_6$ x 12, $S_3$ x 10

Order 5: $C_5$ x 4

Order 4: $C_2^2$ x 41

Order 3: $C_3$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 43200: $A_4\times A_5^2$

Order 14400: $C_2^2\times A_5^2$

Order 10800: $A_5\times \GL(2,4)$

Order 8640: $A_5\times A_4^2$

Order 7200: $D_5\times A_4\times A_5$, $C_2\times A_5^2$

Order 4320: $S_3\times A_4\times A_5$

Order 3600: $C_5\times A_4\times A_5$, $A_5^2$

Order 2880: $C_2^4:\GL(2,4)$ x 2, $C_2^4.\GL(2,4)$

Order 2400: $C_2\times D_{10}\times A_5$

Order 2160: $A_4\times \GL(2,4)$ x 2

Order 1800: $D_5\times \GL(2,4)$

Order 1728: $A_4^3$

Order 1440: $C_2^3:\GL(2,4)$ x 2, $D_5\times A_4^2$, $\GL(2,4):C_2^3$

Order 1200: $D_{10}\times A_5$ x 2, $A_4\times D_5^2$, $C_2\times C_{10}\times A_5$

Order 1080: $S_3\times \GL(2,4)$

Order 960: $C_2^4\times A_5$

Order 900: $C_{15}\times A_5$

Order 864: $S_3\times A_4^2$

Order 720: $A_4\times A_5$ x 6, $C_2^2\times \GL(2,4)$ x 2, $D_6\times A_5$ x 2, $C_5\times A_4^2$, $S_3\times D_5\times A_4$

Order 600: $D_5\times A_5$ x 2, $C_{10}^2:C_6$, $C_{10}^2:C_6$, $C_{10}\times A_5$

Order 576: $C_2^2:A_4^2$ x 2, $C_2^2\times A_4^2$ x 2, $C_2^2:A_4^2$

Order 540: $C_3\times \GL(2,4)$

Order 480: $C_2^3\times A_5$ x 3, $C_2\times A_4\times D_{10}$ x 2, $D_5\times C_2^2:A_4$

Order 432: $C_3\times A_4^2$ x 2, $A_4\times S_3^2$

Order 400: $D_{10}^2$

Order 360: $C_3\times D_5\times A_4$ x 2, $C_6\times A_5$ x 2, $S_3\times A_5$ x 2, $A_4\times D_{15}$, $C_5\times S_3\times A_4$

Order 300: $A_4\times C_5^2$, $C_3\times D_5^2$, $C_5\times A_5$

Order 288: $C_2\times A_4\times D_6$ x 2, $C_2\times A_4^2$ x 2, $S_3\times C_2^2:A_4$

Order 240: $C_2^2\times A_5$ x 7, $A_4\times D_{10}$ x 4, $C_2^3:C_{30}$ x 2, $C_2^4:C_{15}$, $D_6\times D_{10}$

Order 216: $C_6^2:C_6$ x 2, $C_6^2:C_6$

Order 200: $D_5\times D_{10}$ x 3, $C_{10}:D_{10}$, $C_{10}\times D_{10}$

Order 192: $C_2^2\wr C_3$ x 2, $A_4\times C_2^4$ x 2, $C_2^4:A_4$

Order 180: $\GL(2,4)$ x 4, $A_4\times C_{15}$ x 2, $C_{15}:D_6$

Order 160: $C_2^3\times D_{10}$

Order 150: $C_{15}:D_5$, $D_5\times C_{15}$

Order 144: $A_4^2$ x 9, $A_4\times D_6$ x 4, $C_2^2:C_6^2$ x 3, $C_2^4:C_3^2$ x 2, $D_6^2$

Order 120: $D_5\times A_4$ x 8, $S_3\times D_{10}$ x 4, $C_2\times A_5$ x 4, $C_6\times D_{10}$ x 2, $C_{10}\times A_4$ x 2, $C_2\times D_{30}$, $C_{10}\times D_6$

Order 108: $C_3^2\times A_4$ x 2, $C_3\times S_3^2$

Order 100: $D_5^2$ x 4, $C_5:D_{10}$ x 2, $C_5\times D_{10}$ x 2, $C_{10}^2$

Order 96: $C_2^3\times A_4$ x 5, $C_2^3:A_4$ x 2, $C_2^3\times D_6$

Order 90: $C_3\times D_{15}$, $S_3\times C_{15}$, $C_3^2\times D_5$

Order 80: $C_2^2\times D_{10}$ x 6, $C_2^3\times C_{10}$

Order 75: $C_5\times C_{15}$

Order 72: $S_3\times A_4$ x 8, $C_6\times A_4$ x 3, $S_3\times D_6$ x 3, $C_6\times D_6$ x 2, $C_6:D_6$

Order 64: $C_2^6$

Order 60: $C_5\times A_4$ x 6, $S_3\times D_5$ x 5, $C_3\times D_{10}$ x 4, $A_5$ x 2, $C_2\times C_{30}$ x 2, $D_{30}$ x 2, $S_3\times C_{10}$ x 2

Order 54: $C_3^2:C_6$, $S_3\times C_3^2$

Order 50: $C_5:D_5$ x 2, $C_5\times D_5$ x 2, $C_5\times C_{10}$

Order 48: $C_2^2\times A_4$ x 15, $C_2^2:A_4$ x 7, $C_2^2\times D_6$ x 6, $C_2^3\times C_6$ x 2

Order 45: $C_3\times C_{15}$

Order 40: $C_2\times D_{10}$ x 17, $C_2^2\times C_{10}$ x 3

Order 36: $C_3\times A_4$ x 12, $C_6\times S_3$ x 4, $S_3^2$ x 4, $C_6^2$ x 2, $C_6:S_3$ x 2

Order 32: $C_2^5$ x 5

Order 30: $C_3\times D_5$ x 6, $C_{30}$ x 2, $D_{15}$ x 2, $C_5\times S_3$ x 2

Order 27: $C_3^3$

Order 25: $C_5^2$

Order 24: $C_2\times D_6$ x 17, $C_2\times A_4$ x 10, $C_2^2\times C_6$ x 5

Order 20: $D_{10}$ x 15, $C_2\times C_{10}$ x 7

Order 18: $C_3\times S_3$ x 6, $C_3\times C_6$ x 2, $C_3:S_3$ x 2

Order 16: $C_2^4$ x 19

Order 15: $C_{15}$ x 4

Order 12: $D_6$ x 15, $C_2\times C_6$ x 12, $A_4$ x 12

Order 10: $D_5$ x 6, $C_{10}$ x 4

Order 9: $C_3^2$ x 5

Order 8: $C_2^3$ x 26

Order 6: $C_6$ x 7, $S_3$ x 6

Order 5: $C_5$ x 2

Order 4: $C_2^2$ x 19

Order 3: $C_3$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 43200: $A_4\times A_5^2$ ($C_1$)

Order 14400: $C_2^2\times A_5^2$ ($C_3$)

Order 3600: $A_5^2$ ($A_4$)

Order 720: $A_4\times A_5$ ($A_5$) x 2

Order 240: $C_2^2\times A_5$ ($\GL(2,4)$) x 2

Order 60: $A_5$ ($A_4\times A_5$) x 2

Order 12: $A_4$ ($A_5^2$)

Order 4: $C_2^2$ ($A_5\times \GL(2,4)$)

Order 1: $C_1$ ($A_4\times A_5^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 43200: $A_4\times A_5^2$ ($C_1$)

Order 14400: $C_2^2\times A_5^2$ ($C_3$)

Order 3600: $A_5^2$ ($A_4$)

Order 720: $A_4\times A_5$ ($A_5$)

Order 240: $C_2^2\times A_5$ ($\GL(2,4)$)

Order 60: $A_5$ ($A_4\times A_5$)

Order 12: $A_4$ ($A_5^2$)

Order 4: $C_2^2$ ($A_5\times \GL(2,4)$)

Order 1: $C_1$ ($A_4\times A_5^2$)

Series

Derived series

$A_4\times A_5^2$

$\rhd$

$A_4\times A_5^2$

$\rhd$

$C_2^2\times A_5^2$

$\rhd$

$C_2^2\times A_5^2$

$\rhd$

$A_5^2$

$\rhd$

$A_5^2$

Chief series

$A_4\times A_5^2$

$\rhd$

$A_4\times A_5^2$

$\rhd$

$A_4\times A_5$

$\rhd$

$A_4\times A_5$

$\rhd$

$A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$A_4\times A_5^2$

$\rhd$

$A_4\times A_5^2$

$\rhd$

$C_2^2\times A_5^2$

$\rhd$

$C_2^2\times A_5^2$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 9 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $100 \times 100$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $51 \times 51$ rational character table.