Abstract group 24000000.bf: $C_5^6:((C_2\times C_4^3).A_4)$

• Label: 24000000.bf
• Order: \( 2^{9} \cdot 3 \cdot 5^{6} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \cdot 5^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $30$
• Trans deg.: $30$
• Rank: $3$

Group information

Description:	$C_5^6:((C_2\times C_4^3).A_4)$
Order:	\(24000000\)\(\medspace = 2^{9} \cdot 3 \cdot 5^{6} \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_5^6.C_2^3.C_2^4.C_6.C_4.C_2^2$, of order \(192000000\)\(\medspace = 2^{12} \cdot 3 \cdot 5^{6} \)
Composition factors:	$C_2$ x 9, $C_3$, $C_5$ x 6
Derived length:	$4$

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

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30
Elements	1	30575	80000	2976400	15624	2800000	1703800	8000000	1920000	3273600	3200000	24000000
Conjugacy classes	1	9	2	38	28	6	60	8	8	76	4	240
Divisions	1	9	1	28	28	3	60	2	4	54	2	192
Autjugacy classes	1	6	1	25	12	2	21	3	2	30	1	104

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$30$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j \mid c^{4}=f^{4}=g^{10}=h^{5}=i^{5}= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(1,15,26,4,11,30,5,13,28,2,12,29)(3,14,27)(6,18,25,10,16,21,8,17,23,9,19,22) \!\cdots\! \rangle$
Transitive group:	30T3460	30T3606			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_5^6$ $\,\rtimes\,$ $((C_2\times C_4^3).A_4)$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_5^6:(C_4^3.A_4))$ . $C_2$	$(C_5^6:(C_4^3.A_4))$ . $C_2$	$(C_5^6.C_2^4.C_2^3)$ . $A_4$ (3)	$(C_5^6.C_2^4.C_2^4)$ . $C_6$ (3)	all 20

Elements of the group are displayed as permutations of degree 30.

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{2}^{3} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 47 normal subgroups (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_5^6:((C_2\times C_4^3).A_4)$
Commutator:	$G' \simeq$ $C_5^6.C_2.Q_8^2$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_5^6:((C_2\times C_4^3).A_4)$
Fitting:	$\operatorname{Fit} \simeq$ $C_5^6$	$G/\operatorname{Fit} \simeq$ $(C_2\times C_4^3).A_4$
Radical:	$R \simeq$ $C_5^6:((C_2\times C_4^3).A_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5^6$	$G/\operatorname{soc} \simeq$ $(C_2\times C_4^3).A_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^3.C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^6$

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 16 (order at least 1500000) are shown, as well as all normal subgroups of any index.

Order 24000000: $C_5^6:((C_2\times C_4^3).A_4)$

Order 12000000: $C_5^6:(C_4^3.A_4)$, $C_5^6:((C_2^2\times C_4^2).A_4)$, $C_5^6:(C_4^3.A_4)$

Order 8000000: $C_5^6.C_2^4.C_2^5$

Order 6000000: $C_5^6:C_2^4.(C_2\times A_4)$ x 2, $C_5^6:(C_4^3:C_6)$ x 2, $C_5^6:(C_2^2\times Q_8).A_4$, $C_5^3.F_5^3:C_6$

Order 4000000: $C_5^6.C_4^3.C_2^2$ x 4, $C_5^6.C_2^4.C_2^4$ x 4, $C_5^6.C_2^4.C_2^4$ x 2, $C_5^6.C_2^4.C_2^4$ x 2, $C_5^6.C_2^4.C_2^4$ x 2, $C_5^6.C_4^3.C_2^2$ x 2, $C_5^6.C_4^3.C_2^2$ x 2, $C_5^6.C_4^3.C_2^2$ x 2, $C_5^6.C_2^4.C_2^4$, $C_5^6.C_4^3.C_2^2$, $C_5^6.C_4^3.C_2^2$

Order 3000000: $C_5^6:C_4\wr C_3$ x 3, $C_5^6:(C_2^3:\SL(2,3))$ x 2, $C_5^6:(C_4^2:C_{12})$ x 2, $C_5^6:(C_2^2\times C_4).A_4$ x 2, $C_5^6:C_2^4.A_4$ x 2, $C_5^6:(C_2^2\times Q_8).C_6$ x 2, $C_5^6:C_2^4.A_4$, $C_5^6:C_4\wr C_3$

Order 2000000: $C_5^6.C_2.Q_8:D_4$ x 12, $C_5^6.C_2^4.C_2^3$ x 8, $C_5^6.C_2^3.C_2^4$ x 8, $C_5^6.C_2^3.C_2^4$ x 8, $C_5^6.C_2^3.C_2^4$ x 8, $C_5^6.C_2^3.C_2^4$ x 8, $C_5^6.C_2^3.C_2^4$ x 8, $C_5^6.C_2^3.C_2^4$ x 8, $C_5^6.C_2^3.C_2^4$ x 8, $C_5^6.C_2^3.C_2^4$ x 8, $C_5^6.C_2^3.C_2^4$ x 8, $C_5^6.C_2^3.C_2^4$ x 8, $C_5^6.C_2^3.C_2^4$ x 8, $C_5^6.C_2.Q_8:D_4$ x 8, $C_5^6.C_2^3.C_2^4$ x 8, $C_5^6.C_2^4.C_2^3$ x 4, $C_5^6.C_2.Q_8:D_4$ x 4, $C_5^6.C_2^4.C_2^3$ x 4, $C_5^6.C_2.Q_8:D_4$ x 4, $C_5^6.C_2^3.C_2^4$ x 4, $C_5^6.C_4^3.C_2$ x 4, $C_5^6.C_2.Q_8:D_4$ x 4, $C_5^6.C_2^3.C_2^4$ x 4, $C_5^6.C_2.Q_8:D_4$ x 4, $C_5^6.C_2^4.C_2^3$ x 4, $C_5^6.C_2^4.C_2^3$ x 4, $C_5^6.C_2^4.C_2^3$ x 4, $C_5^6.C_2.Q_8^2$ x 4, $C_5^6.C_2^3.C_2^4$ x 4, $C_5^6.C_4^3.C_2$ x 4, $C_5^6.C_2^3.C_2^4$ x 4, $C_5^6.C_2^4.C_2^3$ x 4, $C_5^6.C_2^3.C_2^4$ x 4, $C_5^6.C_2^2.C_2^5$ x 2, $C_5^6.C_2^4.C_2^3$ x 2, $C_5^6.C_2.Q_8^2$ x 2, $C_5^6.C_2^3.C_2^4$ x 2, $C_5^6.C_2^3.C_2^4$ x 2, $C_5^6.C_2^3.C_2^4$ x 2, $C_5^6.C_4^3.C_2$ x 2, $C_5^5.D_{10}.C_2^5$ x 2, $C_5^6.C_4^3.C_2$ x 2, $C_5^6.C_2^3.C_2^4$, $C_5^6.C_4^3.C_2$, $C_5^6.C_2^4.C_2^3$, $C_5^6.C_4^3.C_2$, $C_5^6.C_4^3.C_2$, $C_5^6.C_4^3.C_2$, $C_5^6.C_2^4.C_2^3$

Order 1500000: $C_5^6:(C_4^2:C_6)$ x 4, $C_5^6.C_4^2.C_6$ x 2, $C_5^6:(C_4^2:C_6)$ x 2, $C_5^6:(Q_8:A_4)$ x 2, $C_5^6:(C_4^2:C_6)$, $C_5^3.D_5^3:C_{12}$

Order 1000000: $C_5^6.C_4^3$ x 3, $C_5^6.C_2.C_4:Q_8$ x 2, $C_5^6.C_2^3.C_2^3$ x 2, $C_5^6.C_2^4.C_2^2$ x 2, $C_5^6.C_2^4.C_2^2$ x 2, $C_5^6:(D_4:C_2^3)$ x 2, $C_5^6.C_4^3$, $C_5^5.D_{10}.C_2^4$

Order 500000: $C_5^5.D_{10}.C_2^3$ x 3, $C_5^6:(C_2^2\times Q_8)$ x 2, $C_5^6:(C_2^3\times C_4)$

Order 250000: $C_5^6.C_2^3.C_2$, $C_5^6:(C_2^2\times C_4)$, $C_5^6:C_2^4$

Order 125000: $C_5^6:C_2^3$ x 2, $C_5^6.C_2^3$

Order 62500: $C_5^5.D_{10}$, $C_5^6.C_4$

Order 31250: $C_5^6:C_2$

Order 15625: $C_5^6$

Order 1536: $(C_2\times C_4^3).A_4$

Order 512: $C_4^3.C_2^3$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 16 (order at least 1500000) are shown, as well as all normal subgroups of any index.

Order 24000000: $C_5^6:((C_2\times C_4^3).A_4)$

Order 12000000: $C_5^6:(C_4^3.A_4)$, $C_5^6:((C_2^2\times C_4^2).A_4)$, $C_5^6:(C_4^3.A_4)$

Order 8000000: $C_5^6.C_2^4.C_2^5$

Order 6000000: $C_5^6:(C_2^2\times Q_8).A_4$, $C_5^3.F_5^3:C_6$, $C_5^6:C_2^4.(C_2\times A_4)$, $C_5^6:(C_4^3:C_6)$

Order 4000000: $C_5^6.C_2^4.C_2^4$ x 2, $C_5^6.C_4^3.C_2^2$ x 2, $C_5^6.C_2^4.C_2^4$ x 2, $C_5^6.C_4^3.C_2^2$ x 2, $C_5^6.C_2^4.C_2^4$, $C_5^6.C_2^4.C_2^4$, $C_5^6.C_2^4.C_2^4$, $C_5^6.C_4^3.C_2^2$, $C_5^6.C_4^3.C_2^2$, $C_5^6.C_4^3.C_2^2$, $C_5^6.C_4^3.C_2^2$

Order 3000000: $C_5^6:C_4\wr C_3$ x 2, $C_5^6:(C_2^3:\SL(2,3))$, $C_5^6:(C_4^2:C_{12})$, $C_5^6:(C_2^2\times C_4).A_4$, $C_5^6:C_2^4.A_4$, $C_5^6:C_2^4.A_4$, $C_5^6:C_4\wr C_3$, $C_5^6:(C_2^2\times Q_8).C_6$

Order 2000000: $C_5^6.C_2.Q_8:D_4$ x 6, $C_5^6.C_2^4.C_2^3$ x 3, $C_5^6.C_2^3.C_2^4$ x 3, $C_5^6.C_2^3.C_2^4$ x 3, $C_5^6.C_2^3.C_2^4$ x 3, $C_5^6.C_2^3.C_2^4$ x 3, $C_5^6.C_2^3.C_2^4$ x 3, $C_5^6.C_2^3.C_2^4$ x 3, $C_5^6.C_2^3.C_2^4$ x 3, $C_5^6.C_2.Q_8:D_4$ x 3, $C_5^6.C_2^4.C_2^3$ x 2, $C_5^6.C_2.Q_8^2$ x 2, $C_5^6.C_2^3.C_2^4$ x 2, $C_5^6.C_2^3.C_2^4$ x 2, $C_5^6.C_2^4.C_2^3$ x 2, $C_5^6.C_2.Q_8:D_4$ x 2, $C_5^6.C_2^3.C_2^4$ x 2, $C_5^6.C_2^3.C_2^4$ x 2, $C_5^6.C_2.Q_8:D_4$ x 2, $C_5^6.C_2^3.C_2^4$ x 2, $C_5^6.C_2^4.C_2^3$ x 2, $C_5^6.C_2^4.C_2^3$ x 2, $C_5^6.C_2.Q_8^2$ x 2, $C_5^6.C_4^3.C_2$ x 2, $C_5^5.D_{10}.C_2^5$ x 2, $C_5^6.C_2^3.C_2^4$ x 2, $C_5^6.C_4^3.C_2$ x 2, $C_5^6.C_2^3.C_2^4$ x 2, $C_5^6.C_2^2.C_2^5$, $C_5^6.C_2^3.C_2^4$, $C_5^6.C_2^4.C_2^3$, $C_5^6.C_2.Q_8:D_4$, $C_5^6.C_2^3.C_2^4$, $C_5^6.C_2^3.C_2^4$, $C_5^6.C_4^3.C_2$, $C_5^6.C_4^3.C_2$, $C_5^6.C_2^3.C_2^4$, $C_5^6.C_2^3.C_2^4$, $C_5^6.C_2.Q_8:D_4$, $C_5^6.C_2^4.C_2^3$, $C_5^6.C_4^3.C_2$, $C_5^6.C_4^3.C_2$, $C_5^6.C_4^3.C_2$, $C_5^6.C_2^4.C_2^3$, $C_5^6.C_2^4.C_2^3$, $C_5^6.C_4^3.C_2$, $C_5^6.C_2^3.C_2^4$, $C_5^6.C_2^4.C_2^3$, $C_5^6.C_2^3.C_2^4$

Order 1500000: $C_5^6.C_4^2.C_6$, $C_5^6:(C_4^2:C_6)$, $C_5^3.D_5^3:C_{12}$, $C_5^6:(C_4^2:C_6)$, $C_5^6:(C_4^2:C_6)$, $C_5^6:(Q_8:A_4)$

Order 1000000: $C_5^6.C_4^3$ x 2, $C_5^6.C_4^3$, $C_5^6.C_2.C_4:Q_8$, $C_5^6.C_2^3.C_2^3$, $C_5^6.C_2^4.C_2^2$, $C_5^6.C_2^4.C_2^2$, $C_5^5.D_{10}.C_2^4$, $C_5^6:(D_4:C_2^3)$

Order 500000: $C_5^5.D_{10}.C_2^3$ x 2, $C_5^6:(C_2^2\times Q_8)$, $C_5^6:(C_2^3\times C_4)$

Order 250000: $C_5^6.C_2^3.C_2$, $C_5^6:(C_2^2\times C_4)$, $C_5^6:C_2^4$

Order 125000: $C_5^6.C_2^3$, $C_5^6:C_2^3$

Order 62500: $C_5^5.D_{10}$, $C_5^6.C_4$

Order 31250: $C_5^6:C_2$

Order 15625: $C_5^6$

Order 1536: $(C_2\times C_4^3).A_4$

Order 512: $C_4^3.C_2^3$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 24000000: $C_5^6:((C_2\times C_4^3).A_4)$ ($C_1$)

Order 12000000: $C_5^6:(C_4^3.A_4)$ ($C_2$), $C_5^6:((C_2^2\times C_4^2).A_4)$ ($C_2$), $C_5^6:(C_4^3.A_4)$ ($C_2$)

Order 8000000: $C_5^6.C_2^4.C_2^5$ ($C_3$)

Order 6000000: $C_5^6:(C_2^2\times Q_8).A_4$ ($C_2^2$)

Order 4000000: $C_5^6.C_2^4.C_2^4$ ($C_6$), $C_5^6.C_4^3.C_2^2$ ($C_6$), $C_5^6.C_4^3.C_2^2$ ($C_6$)

Order 2000000: $C_5^6.C_2^4.C_2^3$ ($A_4$) x 2, $C_5^6.C_2^4.C_2^3$ ($A_4$) x 2, $C_5^6.C_2.Q_8^2$ ($C_2\times C_6$), $C_5^6.C_2^4.C_2^3$ ($A_4$)

Order 1000000: $C_5^6.C_4^3$ ($C_2\times A_4$) x 3, $C_5^6.C_2.C_4:Q_8$ ($C_2\times A_4$) x 2, $C_5^6.C_2^3.C_2^3$ ($C_2\times A_4$) x 2, $C_5^6.C_2^4.C_2^2$ ($C_2\times A_4$) x 2, $C_5^6.C_2^4.C_2^2$ ($C_2\times A_4$) x 2, $C_5^6:(D_4:C_2^3)$ ($C_2\times A_4$) x 2, $C_5^6.C_4^3$ ($C_2\times A_4$), $C_5^5.D_{10}.C_2^4$ ($C_2\times A_4$)

Order 500000: $C_5^5.D_{10}.C_2^3$ ($C_2^2\times A_4$) x 3, $C_5^6:(C_2^2\times Q_8)$ ($C_2^2\times A_4$) x 2, $C_5^6:(C_2^3\times C_4)$ ($C_2^2:A_4$)

Order 250000: $C_5^6.C_2^3.C_2$ ($C_2^3:A_4$), $C_5^6:(C_2^2\times C_4)$ ($C_2^3:A_4$), $C_5^6:C_2^4$ ($C_2^3:A_4$)

Order 125000: $C_5^6:C_2^3$ ($(C_2^2\times C_4):A_4$) x 2, $C_5^6.C_2^3$ ($C_2^2\wr C_3$)

Order 62500: $C_5^5.D_{10}$ ($(C_2^3\times C_4):A_4$), $C_5^6.C_4$ ($(C_2\times C_4^2):A_4$)

Order 31250: $C_5^6:C_2$ ($(C_2^2\times C_4^2):A_4$)

Order 15625: $C_5^6$ ($(C_2\times C_4^3).A_4$)

Order 1: $C_1$ ($C_5^6:((C_2\times C_4^3).A_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 24000000: $C_5^6:((C_2\times C_4^3).A_4)$ ($C_1$)

Order 12000000: $C_5^6:(C_4^3.A_4)$ ($C_2$), $C_5^6:((C_2^2\times C_4^2).A_4)$ ($C_2$), $C_5^6:(C_4^3.A_4)$ ($C_2$)

Order 8000000: $C_5^6.C_2^4.C_2^5$ ($C_3$)

Order 6000000: $C_5^6:(C_2^2\times Q_8).A_4$ ($C_2^2$)

Order 4000000: $C_5^6.C_2^4.C_2^4$ ($C_6$), $C_5^6.C_4^3.C_2^2$ ($C_6$), $C_5^6.C_4^3.C_2^2$ ($C_6$)

Order 2000000: $C_5^6.C_2.Q_8^2$ ($C_2\times C_6$), $C_5^6.C_2^4.C_2^3$ ($A_4$), $C_5^6.C_2^4.C_2^3$ ($A_4$), $C_5^6.C_2^4.C_2^3$ ($A_4$)

Order 1000000: $C_5^6.C_4^3$ ($C_2\times A_4$) x 2, $C_5^6.C_4^3$ ($C_2\times A_4$), $C_5^6.C_2.C_4:Q_8$ ($C_2\times A_4$), $C_5^6.C_2^3.C_2^3$ ($C_2\times A_4$), $C_5^6.C_2^4.C_2^2$ ($C_2\times A_4$), $C_5^6.C_2^4.C_2^2$ ($C_2\times A_4$), $C_5^5.D_{10}.C_2^4$ ($C_2\times A_4$), $C_5^6:(D_4:C_2^3)$ ($C_2\times A_4$)

Order 500000: $C_5^5.D_{10}.C_2^3$ ($C_2^2\times A_4$) x 2, $C_5^6:(C_2^2\times Q_8)$ ($C_2^2\times A_4$), $C_5^6:(C_2^3\times C_4)$ ($C_2^2:A_4$)

Order 250000: $C_5^6.C_2^3.C_2$ ($C_2^3:A_4$), $C_5^6:(C_2^2\times C_4)$ ($C_2^3:A_4$), $C_5^6:C_2^4$ ($C_2^3:A_4$)

Order 125000: $C_5^6.C_2^3$ ($C_2^2\wr C_3$), $C_5^6:C_2^3$ ($(C_2^2\times C_4):A_4$)

Order 62500: $C_5^5.D_{10}$ ($(C_2^3\times C_4):A_4$), $C_5^6.C_4$ ($(C_2\times C_4^2):A_4$)

Order 31250: $C_5^6:C_2$ ($(C_2^2\times C_4^2):A_4$)

Order 15625: $C_5^6$ ($(C_2\times C_4^3).A_4$)

Order 1: $C_1$ ($C_5^6:((C_2\times C_4^3).A_4)$)

Series

Derived series

$C_5^6:((C_2\times C_4^3).A_4)$

$\rhd$

$C_5^6.C_2.Q_8^2$

$\rhd$

$C_5^6.C_2^3$

$\rhd$

$C_5^6$

$\rhd$

$C_1$

Chief series

$C_5^6:((C_2\times C_4^3).A_4)$

$\rhd$

$C_5^6:((C_2^2\times C_4^2).A_4)$

$\rhd$

$C_5^6:(C_2^2\times Q_8).A_4$

$\rhd$

$C_5^6.C_2.Q_8^2$

$\rhd$

$C_5^5.D_{10}.C_2^3$

$\rhd$

$C_5^6.C_2^3$

$\rhd$

$C_5^6:C_2$

$\rhd$

$C_5^6$

$\rhd$

$C_1$

Lower central series

$C_5^6:((C_2\times C_4^3).A_4)$

$\rhd$

$C_5^6.C_2.Q_8^2$

Upper central series

$C_1$

Supergroups

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

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

Character theory

Complex character table

The $240 \times 240$ character table is not available for this group.

Rational character table

The $192 \times 192$ rational character table is not available for this group.