Abstract group 3072.bju: $(C_2^5\times D_{12}).C_2^2$

• Label: 3072.bju
• Order: \( 2^{10} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \)
• $\card{Z(G)}$: 16
• $\card{\Aut(G)}$: \( 2^{31} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{25} \cdot 3 \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $7$

Group information

Description:	$(C_2^5\times D_{12}).C_2^2$
Order:	\(3072\)\(\medspace = 2^{10} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	Group of order \(19327352832\)\(\medspace = 2^{31} \cdot 3^{2} \)
Composition factors:	$C_2$ x 10, $C_3$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and rational.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	895	2	640	446	512	320	256	3072
Conjugacy classes	1	151	1	72	75	32	36	16	384
Divisions	1	151	1	72	75	32	36	16	384
Autjugacy classes	1	12	1	11	7	2	6	1	41

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
Rank:	$7$
Inequivalent generating 7-tuples:	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:	${\langle a, b, c, d, e, f, g, h \mid b^{6}=d^{4}=e^{2}=f^{2}=g^{2}=h^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $28$ $\langle(2,21)(3,11)(4,12,9,5)(6,14,13,8)(7,16)(17,25)(18,26,23,19)(20,28,27,22) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$(C_2^6.D_4)$ . $S_3$	$S_3$ . $(C_2^6.D_4)$ (16)	$C_6$ . $(C_2^6.D_4)$ (12)	$C_6$ . $(D_8.C_2^5)$ (2)	all 98
Aut. group:	$\Aut(C_4\times D_{24})$	$\Aut(C_{12}:Q_{16})$	$\Aut(D_{24}:C_4)$	$\Aut(C_3:(C_4\times Q_{16}))$	all 24

Elements of the group are displayed as permutations of degree 28.

Homology

Abelianization:	$C_{2}^{7} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 43303 normal subgroups (26 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 $C_2^4$
Commutator:	a subgroup isomorphic to $C_2\times C_{12}$
Frattini:	a subgroup isomorphic to $C_2\times C_4$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.C_2^5.C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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 3 (order at least 1024) are shown, as well as normal subgroups of index up to 6 or of order up to 8.

Order 3072: $(C_2^5\times D_{12}).C_2^2$

Order 1536: $(C_2\times C_{24}):C_2^5$ x 96, $C_2^2\times (C_2\times Q_8):C_2^2\times S_3$ x 4, $C_2^2\times C_3:((C_2^3\times D_4).C_2)$ x 4, $C_2^2\times C_3:((C_2^3\times D_4).C_2)$ x 4, $C_2^2\times (C_2\times D_8):C_2\times S_3$ x 4, $C_2^3\times C_8:C_2^2\times S_3$ x 2, $C_2^2\times C_3:((C_2^3\times D_4).C_2)$ x 2, $C_2^2\times C_3:((C_2^3\times D_4).C_2)$ x 2, $C_2^2\times (C_2\times C_8):C_2^2\times S_3$ x 2, $C_2^5\times S_3\times D_4$, $C_3:(C_2^3.C_2^6)$, $C_2^2\times C_3:((C_2^3\times D_4).C_2)$, $C_2^2\times C_3:((C_2^3\times D_4).C_2)$, $C_2^2\times C_3:((C_2^3\times D_4).C_2)$, $C_2^2\times C_2^2:\OD_{16}\times S_3$, $C_6\times C_2.C_2^6.C_2$

Order 1024: $C_4.C_2^5.C_2^3$

Order 768: $(C_2\times C_{24}):C_2^4$ x 1024, $C_2\times C_3:((C_2^3\times D_4).C_2)$ x 192, $C_2\times C_3:((C_2^3\times D_4).C_2)$ x 192, $C_2\times (C_2\times Q_8):C_2^2\times S_3$ x 192, $C_2\times (C_2\times D_8):C_2\times S_3$ x 192, $C_{24}:C_2^5$ x 96, $C_2\times C_3:((C_2^3\times D_4).C_2)$ x 96, $C_2\times C_3:((C_2^3\times D_4).C_2)$ x 96, $C_2\times (C_2\times C_8):C_2^2\times S_3$ x 96, $C_{12}:C_2^6$ x 60, $(C_2\times C_{24}):C_2^4$ x 48, $C_4^2:C_2^2\times D_6$ x 48, $C_2\times C_3:((C_2^3\times D_4).C_2)$ x 48, $C_2\times C_3:((C_2^3\times D_4).C_2)$ x 48, $C_2\times C_3:((C_2^3\times D_4).C_2)$ x 48, $C_2\times C_2^2:\OD_{16}\times S_3$ x 48, $C_2^2\times D_{12}.D_4$ x 8, $C_2^2\times D_6:\SD_{16}$ x 8, $C_2^2\times C_2.(S_3\times D_8)$ x 8, $C_2^2\times D_6:D_8$ x 8, $C_2^2\times (C_2\times C_3:C_8):C_2^2$ x 8, $C_2^2\times D_4:C_4\times S_3$ x 8, $C_2\times C_6\times (C_2\times Q_8):C_2^2$ x 4, $C_2\times C_6\times (C_2\times D_8):C_2$ x 4, $C_2^3\times D_{24}:C_2$ x 4, $C_2^3\times D_8:S_3$ x 4, $C_2^3\times \SD_{16}\times S_3$ x 4, $C_2^3\times D_8\times S_3$ x 4, $C_2^2\times (C_2\times C_6):\SD_{16}$ x 4, $C_2^2\times C_3:((C_2\times Q_8):C_2^2)$ x 4, $C_2^2\times C_2^2:(Q_8:S_3)$ x 4, $C_2^2\times (C_2\times D_{24}):C_2$ x 4, $C_2^2\times (C_2\times C_6\times D_4):C_2$ x 4, $C_2^2\times (C_2\times C_6):D_8$ x 4, $C_2^2\times C_2^2:(C_{24}:C_2)$ x 4, $C_2^2\times C_2^2:C_8\times S_3$ x 4, $C_6\times C_4.C_2^5$ x 2, $C_2\times C_6\times (C_2\times C_8):C_2^2$ x 2, $C_2^3\times C_8:D_6$ x 2, $C_2^3\times C_3:D_8:C_2$ x 2, $C_2^3\times Q_8:D_6$ x 2, $C_2^3\times \OD_{16}\times S_3$ x 2, $C_2^2\times C_2^2:(D_6.C_2^2)$ x 2, $C_2^2\times C_2^3:D_6:C_2$ x 2, $C_2^2\times C_2^2:Q_8\times S_3$ x 2, $C_2^3:D_4\times D_6$ x 2, $C_2^2\times C_3:((C_2\times C_8):C_2^2)$ x 2, $C_2^2\times C_2^2.(C_6.D_4)$ x 2, $C_2^2\times C_4:C_4:D_6$ x 2, $C_2^2\times C_3:(C_2^2:\OD_{16})$ x 2, $C_2^4\times C_6\times D_4$, $C_6\times C_2^4.C_2^3$, $C_2\times C_6\times C_2^2:\OD_{16}$, $C_3:(C_4.C_2^6)$, $C_2^5\times D_{12}$, $C_2^5\times C_4\times S_3$, $C_2^2\times (C_2^2\times C_6).C_2^3$, $C_2^2\times C_2^4:S_3:C_2$, $C_2^2\times (C_2^2\times C_4\times S_3):C_2$, $C_2^2\times (C_2\times C_4):C_4\times S_3$, $C_2^2\times C_2^2:(C_3:\OD_{16})$

Order 512: $C_2^6.D_4$

Order 8: $C_2\times C_4$ x 128, $C_2^3$ x 99, $D_4$ x 32

Order 6: $S_3$ x 16, $C_6$ x 15

Order 4: $C_2^2$ x 59, $C_4$ x 16

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

All normal subgroups of index up to 6 or order up to 8 are shown.

Order 3072: $(C_2^5\times D_{12}).C_2^2$ ($C_1$)

Order 1536: $(C_2\times C_{24}):C_2^5$ ($C_2$) x 96, $C_2^2\times (C_2\times Q_8):C_2^2\times S_3$ ($C_2$) x 4, $C_2^2\times C_3:((C_2^3\times D_4).C_2)$ ($C_2$) x 4, $C_2^2\times C_3:((C_2^3\times D_4).C_2)$ ($C_2$) x 4, $C_2^2\times (C_2\times D_8):C_2\times S_3$ ($C_2$) x 4, $C_2^3\times C_8:C_2^2\times S_3$ ($C_2$) x 2, $C_2^2\times C_3:((C_2^3\times D_4).C_2)$ ($C_2$) x 2, $C_2^2\times C_3:((C_2^3\times D_4).C_2)$ ($C_2$) x 2, $C_2^2\times (C_2\times C_8):C_2^2\times S_3$ ($C_2$) x 2, $C_2^5\times S_3\times D_4$ ($C_2$), $C_3:(C_2^3.C_2^6)$ ($C_2$), $C_2^2\times C_3:((C_2^3\times D_4).C_2)$ ($C_2$), $C_2^2\times C_3:((C_2^3\times D_4).C_2)$ ($C_2$), $C_2^2\times C_3:((C_2^3\times D_4).C_2)$ ($C_2$), $C_2^2\times C_2^2:\OD_{16}\times S_3$ ($C_2$), $C_6\times C_2.C_2^6.C_2$ ($C_2$)

Order 768: $(C_2\times C_{24}):C_2^4$ ($C_2^2$) x 1024, $C_2\times C_3:((C_2^3\times D_4).C_2)$ ($C_2^2$) x 192, $C_2\times C_3:((C_2^3\times D_4).C_2)$ ($C_2^2$) x 192, $C_2\times (C_2\times Q_8):C_2^2\times S_3$ ($C_2^2$) x 192, $C_2\times (C_2\times D_8):C_2\times S_3$ ($C_2^2$) x 192, $C_{24}:C_2^5$ ($C_2^2$) x 96, $C_2\times C_3:((C_2^3\times D_4).C_2)$ ($C_2^2$) x 96, $C_2\times C_3:((C_2^3\times D_4).C_2)$ ($C_2^2$) x 96, $C_2\times (C_2\times C_8):C_2^2\times S_3$ ($C_2^2$) x 96, $C_{12}:C_2^6$ ($C_2^2$) x 60, $(C_2\times C_{24}):C_2^4$ ($C_2^2$) x 48, $C_4^2:C_2^2\times D_6$ ($C_2^2$) x 48, $C_2\times C_3:((C_2^3\times D_4).C_2)$ ($C_2^2$) x 48, $C_2\times C_3:((C_2^3\times D_4).C_2)$ ($C_2^2$) x 48, $C_2\times C_3:((C_2^3\times D_4).C_2)$ ($C_2^2$) x 48, $C_2\times C_2^2:\OD_{16}\times S_3$ ($C_2^2$) x 48, $C_2^2\times D_{12}.D_4$ ($C_2^2$) x 8, $C_2^2\times D_6:\SD_{16}$ ($C_2^2$) x 8, $C_2^2\times C_2.(S_3\times D_8)$ ($C_2^2$) x 8, $C_2^2\times D_6:D_8$ ($C_2^2$) x 8, $C_2^2\times (C_2\times C_3:C_8):C_2^2$ ($C_2^2$) x 8, $C_2^2\times D_4:C_4\times S_3$ ($C_2^2$) x 8, $C_2\times C_6\times (C_2\times Q_8):C_2^2$ ($C_2^2$) x 4, $C_2\times C_6\times (C_2\times D_8):C_2$ ($C_2^2$) x 4, $C_2^3\times D_{24}:C_2$ ($C_2^2$) x 4, $C_2^3\times D_8:S_3$ ($C_2^2$) x 4, $C_2^3\times \SD_{16}\times S_3$ ($C_2^2$) x 4, $C_2^3\times D_8\times S_3$ ($C_2^2$) x 4, $C_2^2\times (C_2\times C_6):\SD_{16}$ ($C_2^2$) x 4, $C_2^2\times C_3:((C_2\times Q_8):C_2^2)$ ($C_2^2$) x 4, $C_2^2\times C_2^2:(Q_8:S_3)$ ($C_2^2$) x 4, $C_2^2\times (C_2\times D_{24}):C_2$ ($C_2^2$) x 4, $C_2^2\times (C_2\times C_6\times D_4):C_2$ ($C_2^2$) x 4, $C_2^2\times (C_2\times C_6):D_8$ ($C_2^2$) x 4, $C_2^2\times C_2^2:(C_{24}:C_2)$ ($C_2^2$) x 4, $C_2^2\times C_2^2:C_8\times S_3$ ($C_2^2$) x 4, $C_6\times C_4.C_2^5$ ($C_2^2$) x 2, $C_2\times C_6\times (C_2\times C_8):C_2^2$ ($C_2^2$) x 2, $C_2^3\times C_8:D_6$ ($C_2^2$) x 2, $C_2^3\times C_3:D_8:C_2$ ($C_2^2$) x 2, $C_2^3\times Q_8:D_6$ ($C_2^2$) x 2, $C_2^3\times \OD_{16}\times S_3$ ($C_2^2$) x 2, $C_2^2\times C_2^2:(D_6.C_2^2)$ ($C_2^2$) x 2, $C_2^2\times C_2^3:D_6:C_2$ ($C_2^2$) x 2, $C_2^2\times C_2^2:Q_8\times S_3$ ($C_2^2$) x 2, $C_2^3:D_4\times D_6$ ($C_2^2$) x 2, $C_2^2\times C_3:((C_2\times C_8):C_2^2)$ ($C_2^2$) x 2, $C_2^2\times C_2^2.(C_6.D_4)$ ($C_2^2$) x 2, $C_2^2\times C_4:C_4:D_6$ ($C_2^2$) x 2, $C_2^2\times C_3:(C_2^2:\OD_{16})$ ($C_2^2$) x 2, $C_2^4\times C_6\times D_4$ ($C_2^2$), $C_6\times C_2^4.C_2^3$ ($C_2^2$), $C_2\times C_6\times C_2^2:\OD_{16}$ ($C_2^2$), $C_3:(C_4.C_2^6)$ ($C_2^2$), $C_2^5\times D_{12}$ ($C_2^2$), $C_2^5\times C_4\times S_3$ ($C_2^2$), $C_2^2\times (C_2^2\times C_6).C_2^3$ ($C_2^2$), $C_2^2\times C_2^4:S_3:C_2$ ($C_2^2$), $C_2^2\times (C_2^2\times C_4\times S_3):C_2$ ($C_2^2$), $C_2^2\times (C_2\times C_4):C_4\times S_3$ ($C_2^2$), $C_2^2\times C_2^2:(C_3:\OD_{16})$ ($C_2^2$)

Order 512: $C_2^6.D_4$ ($S_3$)

Order 384: ? (unidentified group of order 8) x 11907

Order 256: ? (unidentified group of order 12) x 63

Order 192: ? (unidentified group of order 16) x 13299

Order 128: ? (unidentified group of order 24) x 651

Order 96: ? (unidentified group of order 32) x 6675

Order 64: ? (unidentified group of order 48) x 1443

Order 48: ? (unidentified group of order 64) x 3187

Order 32: ? (unidentified group of order 96) x 1011

Order 24: ? (unidentified group of order 128) x 1091

Order 16: ? (unidentified group of order 192) x 563

Order 12: ? (unidentified group of order 256) x 235

Order 8: $C_2\times C_4$ ($C_2^5:D_6$) x 84, $C_2^3$ ($C_{24}:C_2^4$) x 56, $C_2\times C_4$ ($C_{12}:C_2^5$) x 43, $D_4$ ($C_{12}:C_2^5$) x 32, $C_2^3$ ($C_2^5:D_6$) x 28, $C_2^3$ ($C_{12}:C_2^5$) x 15, $C_2\times C_4$ ($D_6\times C_2^5$)

Order 6: $S_3$ ($C_2^6.D_4$) x 16, $C_6$ ($C_2^6.D_4$) x 12, $C_6$ ($D_8.C_2^5$) x 2, $C_6$ ($C_2^6:D_4$)

Order 4: $C_2^2$ ($C_{24}:C_2^5$) x 28, $C_2^2$ ($(C_2\times C_{24}):C_2^4$) x 16, $C_2^2$ ($C_2^6:D_6$) x 14, $C_4$ ($C_2^6:D_6$) x 14, $C_4$ ($C_{12}:C_2^6$) x 2, $C_2^2$ ($C_{12}:C_2^6$)

Order 3: $C_3$ ($C_4.C_2^6.C_2^2$)

Order 2: $C_2$ ($(C_2\times C_{24}):C_2^5$) x 12, $C_2$ ($C_2^3\times C_8:C_2^2\times S_3$) x 2, $C_2$ ($C_6.(D_4\times C_2^5)$)

Order 1: $C_1$ ($(C_2^5\times D_{12}).C_2^2$)

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 2 larger groups in the database.

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

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

The $384 \times 384$ rational character table is not available for this group.