Abstract group 120.39: $D_5\times A_4$

• Label: 120.39
• Order: \( 2^{3} \cdot 3 \cdot 5 \)
• Exponent: \( 2 \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $9$
• Trans deg.: $20$
• Rank: $2$

Group information

Description:	$D_5\times A_4$
Order:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Exponent:	\(30\)\(\medspace = 2 \cdot 3 \cdot 5 \)
Automorphism group:	$F_5\times S_4$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 3, $C_3$, $C_5$
Derived length:	$2$

This group is nonabelian, monomial (hence solvable), metabelian, and an A-group.

Group statistics

Order	1	2	3	5	6	10	15
Elements	1	23	8	4	40	12	32	120
Conjugacy classes	1	3	2	2	2	2	4	16
Divisions	1	3	1	1	1	1	1	9
Autjugacy classes	1	3	1	1	1	1	1	9

Dimension	1	2	3	4	6	8	12
Irr. complex chars.	6	6	2	0	2	0	0	16
Irr. rational chars.	2	2	2	1	0	1	1	9

Minimal presentations

Permutation degree:	$9$
Transitive degree:	$20$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	6	12
Arbitrary	5	5	7

Constructions

Presentation:	$\langle a, b, c \mid a^{6}=b^{2}=c^{10}=[b,c]=1, b^{a}=c^{5}, c^{a}=bc^{9} \rangle$
Permutation group:	$\langle(6,7)(8,9), (2,4,3), (1,2)(3,4), (1,3)(2,4), (5,6,8,9,7)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ \alpha & \alpha^{2} & 0 & \alpha \\ \alpha & 0 & \alpha^{2} & 1 \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & \alpha^{2} & 1 & 1 \\ 0 & 0 & \alpha^{2} & 1 \\ \alpha^{2} & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ \alpha & 1 & 0 & \alpha^{2} \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}0 & 0 & 0 & \alpha^{2} \\ 0 & 0 & 1 & \alpha \\ \alpha & 1 & \alpha & 0 \\ \alpha & 0 & 0 & \alpha \\ \end{array}\right), \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ \alpha^{2} & 1 & 0 & 1 \\ \alpha & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{4}) = \GL_{4}(\F_{2}[\alpha]/(\alpha^{2} + \alpha + 1))$
Transitive group:	20T37	30T20	30T28	40T65	more information
Direct product:	$D_5$ $\, \times\, $ $A_4$
Semidirect product:	$(C_5\times A_4)$ $\,\rtimes\,$ $C_2$	$C_5$ $\,\rtimes\,$ $(C_2\times A_4)$	$(C_2\times D_{10})$ $\,\rtimes\,$ $C_3$	$(C_2\times C_{10})$ $\,\rtimes\,$ $C_6$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

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

Subgroups

There are 116 subgroups in 24 conjugacy classes, 9 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $D_5\times A_4$
Commutator:	$G' \simeq$ $C_2\times C_{10}$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_5\times A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{10}$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $D_5\times A_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{10}$	$G/\operatorname{soc} \simeq$ $C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 120: $D_5\times A_4$

Order 60: $C_5\times A_4$

Order 40: $C_2\times D_{10}$

Order 30: $C_3\times D_5$

Order 24: $C_2\times A_4$

Order 20: $D_{10}$ x 2, $C_2\times C_{10}$

Order 15: $C_{15}$

Order 12: $A_4$

Order 10: $D_5$ x 2, $C_{10}$

Order 8: $C_2^3$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_2^2$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 120: $D_5\times A_4$

Order 60: $C_5\times A_4$

Order 40: $C_2\times D_{10}$

Order 30: $C_3\times D_5$

Order 24: $C_2\times A_4$

Order 20: $D_{10}$ x 2, $C_2\times C_{10}$

Order 15: $C_{15}$

Order 12: $A_4$

Order 10: $D_5$ x 2, $C_{10}$

Order 8: $C_2^3$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_2^2$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 120: $D_5\times A_4$ ($C_1$)

Order 60: $C_5\times A_4$ ($C_2$)

Order 40: $C_2\times D_{10}$ ($C_3$)

Order 20: $C_2\times C_{10}$ ($C_6$)

Order 12: $A_4$ ($D_5$)

Order 10: $D_5$ ($A_4$)

Order 5: $C_5$ ($C_2\times A_4$)

Order 4: $C_2^2$ ($C_3\times D_5$)

Order 1: $C_1$ ($D_5\times A_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 120: $D_5\times A_4$ ($C_1$)

Order 60: $C_5\times A_4$ ($C_2$)

Order 40: $C_2\times D_{10}$ ($C_3$)

Order 20: $C_2\times C_{10}$ ($C_6$)

Order 12: $A_4$ ($D_5$)

Order 10: $D_5$ ($A_4$)

Order 5: $C_5$ ($C_2\times A_4$)

Order 4: $C_2^2$ ($C_3\times D_5$)

Order 1: $C_1$ ($D_5\times A_4$)

Series

Derived series	$D_5\times A_4$	$\rhd$	$C_2\times C_{10}$	$\rhd$	$C_1$
Chief series	$D_5\times A_4$	$\rhd$	$C_5\times A_4$	$\rhd$	$C_2\times C_{10}$	$\rhd$	$C_5$	$\rhd$	$C_1$
Lower central series	$D_5\times A_4$	$\rhd$	$C_2\times C_{10}$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 34 larger groups in the database.

This group is a maximal quotient of 37 larger groups in the database.

Character theory

Complex character table

		1A	2A	2B	2C	3A1	3A-1	5A1	5A2	6A1	6A-1	10A1	10A3	15A1	15A-1	15A2	15A-2
	Size	1	3	5	15	4	4	2	2	20	20	6	6	8	8	8	8
	2 P	1A	1A	1A	1A	3A-1	3A1	5A2	5A1	3A-1	3A1	5A1	5A2	15A2	15A-2	15A1	15A-1
	3 P	1A	2A	2B	2C	1A	1A	5A2	5A1	2B	2B	10A3	10A1	5A2	5A2	5A1	5A1
	5 P	1A	2A	2B	2C	3A-1	3A1	1A	1A	6A-1	6A1	2A	2A	3A1	3A-1	3A-1	3A1
	Type
120.39.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
120.39.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$
120.39.1c1	C	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
120.39.1c2	C	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
120.39.1d1	C	$1$	$1$	$-1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
120.39.1d2	C	$1$	$1$	$-1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
120.39.2a1	R	$2$	$2$	$0$	$0$	$2$	$2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	$0$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-1}+{\zeta }_5$
120.39.2a2	R	$2$	$2$	$0$	$0$	$2$	$2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	$0$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-2}+{\zeta }_5^2$
120.39.2b1	C	$2$	$2$	$0$	$0$	$2{\zeta }_{15}^{-5}$	$2{\zeta }_{15}^5$	${\zeta }_{15}^{-6}+{\zeta }_{15}^6$	${\zeta }_{15}^{-3}+{\zeta }_{15}^3$	$0$	$0$	${\zeta }_{15}^{-3}+{\zeta }_{15}^3$	${\zeta }_{15}^{-6}+{\zeta }_{15}^6$	$1-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3-{\zeta }_{15}^4-{\zeta }_{15}^7$	${\zeta }_{15}+{\zeta }_{15}^4$	$1-{\zeta }_{15}-{\zeta }_{15}^4+{\zeta }_{15}^5$	$-1+{\zeta }_{15}+{\zeta }_{15}^2-{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^5+{\zeta }_{15}^7$
120.39.2b2	C	$2$	$2$	$0$	$0$	$2{\zeta }_{15}^5$	$2{\zeta }_{15}^{-5}$	${\zeta }_{15}^{-6}+{\zeta }_{15}^6$	${\zeta }_{15}^{-3}+{\zeta }_{15}^3$	$0$	$0$	${\zeta }_{15}^{-3}+{\zeta }_{15}^3$	${\zeta }_{15}^{-6}+{\zeta }_{15}^6$	${\zeta }_{15}+{\zeta }_{15}^4$	$1-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3-{\zeta }_{15}^4-{\zeta }_{15}^7$	$-1+{\zeta }_{15}+{\zeta }_{15}^2-{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^5+{\zeta }_{15}^7$	$1-{\zeta }_{15}-{\zeta }_{15}^4+{\zeta }_{15}^5$
120.39.2b3	C	$2$	$2$	$0$	$0$	$2{\zeta }_{15}^{-5}$	$2{\zeta }_{15}^5$	${\zeta }_{15}^{-3}+{\zeta }_{15}^3$	${\zeta }_{15}^{-6}+{\zeta }_{15}^6$	$0$	$0$	${\zeta }_{15}^{-6}+{\zeta }_{15}^6$	${\zeta }_{15}^{-3}+{\zeta }_{15}^3$	$-1+{\zeta }_{15}+{\zeta }_{15}^2-{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^5+{\zeta }_{15}^7$	$1-{\zeta }_{15}-{\zeta }_{15}^4+{\zeta }_{15}^5$	${\zeta }_{15}+{\zeta }_{15}^4$	$1-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3-{\zeta }_{15}^4-{\zeta }_{15}^7$
120.39.2b4	C	$2$	$2$	$0$	$0$	$2{\zeta }_{15}^5$	$2{\zeta }_{15}^{-5}$	${\zeta }_{15}^{-3}+{\zeta }_{15}^3$	${\zeta }_{15}^{-6}+{\zeta }_{15}^6$	$0$	$0$	${\zeta }_{15}^{-6}+{\zeta }_{15}^6$	${\zeta }_{15}^{-3}+{\zeta }_{15}^3$	$1-{\zeta }_{15}-{\zeta }_{15}^4+{\zeta }_{15}^5$	$-1+{\zeta }_{15}+{\zeta }_{15}^2-{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^5+{\zeta }_{15}^7$	$1-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3-{\zeta }_{15}^4-{\zeta }_{15}^7$	${\zeta }_{15}+{\zeta }_{15}^4$
120.39.3a	R	$3$	$-1$	$3$	$-1$	$0$	$0$	$3$	$3$	$0$	$0$	$-1$	$-1$	$0$	$0$	$0$	$0$
120.39.3b	R	$3$	$-1$	$-3$	$1$	$0$	$0$	$3$	$3$	$0$	$0$	$-1$	$-1$	$0$	$0$	$0$	$0$
120.39.6a1	R	$6$	$-2$	$0$	$0$	$0$	$0$	$3{\zeta }_5^{-2}+3{\zeta }_5^2$	$3{\zeta }_5^{-1}+3{\zeta }_5$	$0$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$0$	$0$	$0$	$0$
120.39.6a2	R	$6$	$-2$	$0$	$0$	$0$	$0$	$3{\zeta }_5^{-1}+3{\zeta }_5$	$3{\zeta }_5^{-2}+3{\zeta }_5^2$	$0$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$	$0$	$0$	$0$	$0$

Rational character table

		1A	2A	2B	2C	3A	5A	6A	10A	15A
	Size	1	3	5	15	8	4	40	12	32
	2 P	1A	1A	1A	1A	3A	5A	3A	5A	15A
	3 P	1A	2A	2B	2C	1A	5A	2B	10A	5A
	5 P	1A	2A	2B	2C	3A	1A	6A	2A	3A
120.39.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
120.39.1b		$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$
120.39.1c		$2$	$2$	$2$	$2$	$-1$	$2$	$-1$	$2$	$-1$
120.39.1d		$2$	$2$	$-2$	$-2$	$-1$	$2$	$1$	$2$	$-1$
120.39.2a		$4$	$4$	$0$	$0$	$4$	$-1$	$0$	$-1$	$-1$
120.39.2b		$8$	$8$	$0$	$0$	$-4$	$-2$	$0$	$-2$	$1$
120.39.3a		$3$	$-1$	$3$	$-1$	$0$	$3$	$0$	$-1$	$0$
120.39.3b		$3$	$-1$	$-3$	$1$	$0$	$3$	$0$	$-1$	$0$
120.39.6a		$12$	$-4$	$0$	$0$	$0$	$-3$	$0$	$1$	$0$