Abstract group 3072.z: $C_4^3:C_{48}$

• Label: 3072.z
• Order: \( 2^{10} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{5} \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $32$
• Trans deg.: $192$
• Rank: $2$

Group information

Description:	$C_4^3:C_{48}$
Order:	\(3072\)\(\medspace = 2^{10} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_4^2.(C_2^3\times C_{12}).C_2^6$, of order \(98304\)\(\medspace = 2^{15} \cdot 3 \)
Composition factors:	$C_2$ x 10, $C_3$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	12	16	24	48
Elements	1	15	32	240	96	256	384	512	512	1024	3072
Conjugacy classes	1	7	2	56	6	64	24	64	32	64	320
Divisions	1	7	1	32	3	16	6	8	4	4	82
Autjugacy classes	1	5	1	12	2	6	3	2	2	1	35

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	192	0	64	0	64	0	0	0	0	320
Irr. rational chars.	4	10	4	10	14	8	16	4	12	82

Minimal presentations

Permutation degree:	$32$
Transitive degree:	$192$
Rank:	$2$
Inequivalent generating pairs:	$24$

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

Presentation:	$\langle a, b, c, d \mid a^{48}=b^{4}=c^{4}=d^{4}=[a,b]=[b,c]=[b,d]=[c,d]=1, c^{a}=d, d^{a}=c^{3}d \rangle$
Permutation group:	Degree $32$ $\langle(1,2,3,4)(5,8,7,6)(29,30,31,32), (1,2,3,4)(9,12,11,10)(13,16,17,23)(14,19,20,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 24 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 31 & 8 \\ 8 & 7 \end{array}\right), \left(\begin{array}{rr} 27 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 13 & 6 \\ 2 & 3 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 28 & 11 \\ 25 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 16 & 17 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
Direct product:	$C_4$ $\, \times\, $ $(C_4^2:C_{48})$
Semidirect product:	$C_4^3$ $\,\rtimes\,$ $C_{48}$	$(C_4\wr C_3)$ $\,\rtimes\,$ $C_{16}$	$(C_4^3:C_{16})$ $\,\rtimes\,$ $C_3$	$C_4^2$ $\,\rtimes\,$ $(C_4\times C_{48})$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^4$ . $C_{12}$	$(C_4^3\times C_8)$ . $C_6$	$(C_4^3.C_8)$ . $C_6$	$(C_4^3:C_6)$ . $C_8$	all 59

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{32}\Z)$.

Homology

Abelianization:	$C_{4} \times C_{48} \simeq C_{4} \times C_{16} \times C_{3}$
Schur multiplier:	$C_{4}^{2}$
Commutator length:	$1$

Subgroups

There are 4950 subgroups in 1112 conjugacy classes, 109 normal (43 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4\times C_8$	$G/Z \simeq$ $C_4^2:C_6$
Commutator:	$G' \simeq$ $C_4^2$	$G/G' \simeq$ $C_4\times C_{48}$
Frattini:	$\Phi \simeq$ $C_2^3\times C_8$	$G/\Phi \simeq$ $C_2^2\times A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^3:C_{16}$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_4^3:C_{48}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_2^3:C_{24}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^3:C_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 3072: $C_4^3:C_{48}$

Order 1536: $C_4^3.C_{24}$ x 2, $C_4^3:C_{24}$

Order 1024: $C_4^3:C_{16}$

Order 768: $C_4^2:C_{48}$ x 4, $C_4^3.C_{12}$ x 2, $C_4\times A_4\times C_{16}$, $C_4^3:C_{12}$

Order 512: $(C_2\times C_4^2):C_{16}$ x 2, $C_4^3.C_8$ x 2, $C_4^3.C_8$ x 2, $C_4^3\times C_8$

Order 384: $C_4^2:C_{24}$ x 4, $C_4^3:C_6$ x 3, $C_2^3:C_{48}$ x 2, $C_4\times C_8\times A_4$

Order 256: $C_4^2:C_{16}$ x 8, $C_4^2:C_{16}$ x 8, $C_2\times C_4^2\times C_8$ x 6, $C_2^3.\OD_{32}$ x 4, $C_2^3.\OD_{32}$ x 4, $C_2^2\times C_4\times C_{16}$ x 2, $C_4^4$

Order 192: $C_4\wr C_3$ x 6, $A_4\times C_{16}$ x 4, $C_2^3:C_{24}$ x 2, $A_4\times C_4^2$, $C_2^4.A_4$, $C_4\times C_{48}$

Order 128: $C_4^2.C_8$ x 32, $C_4^2\times C_8$ x 28, $C_2^2\times C_4\times C_8$ x 10, $C_2\times C_4\times C_{16}$ x 8, $C_2^2.\OD_{32}$ x 8, $C_2\times C_4^3$ x 7, $C_2^3\times C_{16}$ x 4

Order 96: $C_8\times A_4$ x 4, $C_4^2:C_6$ x 3, $C_2^3:C_{12}$ x 3, $C_2\times C_{48}$ x 2, $C_4\times C_{24}$

Order 64: $C_2\times C_4\times C_8$ x 80, $C_4^3$ x 32, $C_4:C_{16}$ x 32, $C_2^2\times C_{16}$ x 24, $C_2^2\times C_4^2$ x 13, $C_4\times C_{16}$ x 6, $C_2^3\times C_8$ x 4

Order 48: $C_4\times A_4$ x 6, $C_{48}$ x 4, $C_2\times C_{24}$ x 2, $C_2^2\times A_4$, $C_4^2:C_3$, $C_4\times C_{12}$

Order 32: $C_2\times C_4^2$ x 106, $C_4\times C_8$ x 94, $C_2^2\times C_8$ x 32, $C_2\times C_{16}$ x 28, $C_2^3\times C_4$ x 7

Order 24: $C_{24}$ x 4, $C_2\times C_{12}$ x 3, $C_2\times A_4$ x 3

Order 16: $C_4^2$ x 124, $C_2^2\times C_4$ x 62, $C_2\times C_8$ x 52, $C_{16}$ x 8, $C_2^4$

Order 12: $C_{12}$ x 6, $C_2\times C_6$, $A_4$

Order 8: $C_2\times C_4$ x 106, $C_8$ x 16, $C_2^3$ x 7

Order 6: $C_6$ x 3

Order 4: $C_4$ x 32, $C_2^2$ x 13

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3072: $C_4^3:C_{48}$

Order 1536: $C_4^3.C_{24}$, $C_4^3:C_{24}$

Order 1024: $C_4^3:C_{16}$

Order 768: $C_4^3.C_{12}$ x 2, $C_4\times A_4\times C_{16}$, $C_4^3:C_{12}$, $C_4^2:C_{48}$

Order 512: $C_4^3.C_8$ x 2, $(C_2\times C_4^2):C_{16}$, $C_4^3.C_8$, $C_4^3\times C_8$

Order 384: $C_4^2:C_{24}$ x 2, $C_4^3:C_6$ x 2, $C_2^3:C_{48}$, $C_4\times C_8\times A_4$

Order 256: $C_2\times C_4^2\times C_8$ x 6, $C_4^2:C_{16}$ x 4, $C_4^2:C_{16}$ x 2, $C_2^3.\OD_{32}$ x 2, $C_2^3.\OD_{32}$ x 2, $C_2^2\times C_4\times C_{16}$ x 2, $C_4^4$

Order 192: $C_4\wr C_3$ x 3, $C_2^3:C_{24}$ x 2, $A_4\times C_4^2$, $C_2^4.A_4$, $A_4\times C_{16}$, $C_4\times C_{48}$

Order 128: $C_4^2\times C_8$ x 11, $C_2^2\times C_4\times C_8$ x 9, $C_4^2.C_8$ x 8, $C_2\times C_4\times C_{16}$ x 6, $C_2\times C_4^3$ x 5, $C_2^3\times C_{16}$ x 2, $C_2^2.\OD_{32}$ x 2

Order 96: $C_8\times A_4$ x 2, $C_4^2:C_6$ x 2, $C_2^3:C_{12}$ x 2, $C_2\times C_{48}$, $C_4\times C_{24}$

Order 64: $C_2\times C_4\times C_8$ x 34, $C_4^3$ x 12, $C_2^2\times C_4^2$ x 9, $C_2^2\times C_{16}$ x 8, $C_4\times C_{16}$ x 5, $C_4:C_{16}$ x 4, $C_2^3\times C_8$ x 4

Order 48: $C_4\times A_4$ x 3, $C_2\times C_{24}$ x 2, $C_2^2\times A_4$, $C_4^2:C_3$, $C_4\times C_{12}$, $C_{48}$

Order 32: $C_2\times C_4^2$ x 37, $C_4\times C_8$ x 24, $C_2^2\times C_8$ x 18, $C_2\times C_{16}$ x 8, $C_2^3\times C_4$ x 5

Order 24: $C_2\times C_{12}$ x 2, $C_{24}$ x 2, $C_2\times A_4$ x 2

Order 16: $C_4^2$ x 31, $C_2^2\times C_4$ x 27, $C_2\times C_8$ x 22, $C_{16}$ x 2, $C_2^4$

Order 12: $C_{12}$ x 3, $C_2\times C_6$, $A_4$

Order 8: $C_2\times C_4$ x 37, $C_8$ x 6, $C_2^3$ x 5

Order 6: $C_6$ x 2

Order 4: $C_4$ x 12, $C_2^2$ x 9

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3072: $C_4^3:C_{48}$ ($C_1$)

Order 1536: $C_4^3.C_{24}$ ($C_2$) x 2, $C_4^3:C_{24}$ ($C_2$)

Order 1024: $C_4^3:C_{16}$ ($C_3$)

Order 768: $C_4^2:C_{48}$ ($C_4$) x 4, $C_4^3.C_{12}$ ($C_2^2$), $C_4^3.C_{12}$ ($C_4$), $C_4^3:C_{12}$ ($C_4$)

Order 512: $C_4^3.C_8$ ($C_6$) x 2, $C_4^3\times C_8$ ($C_6$)

Order 384: $C_4^2:C_{24}$ ($C_2\times C_4$) x 2, $C_4^2:C_{24}$ ($C_8$) x 2, $C_4^3:C_6$ ($C_8$) x 2, $C_4^3:C_6$ ($C_2\times C_4$)

Order 256: $C_4^2:C_{16}$ ($C_{12}$) x 4, $C_4^4$ ($C_{12}$), $C_2^2\times C_4\times C_{16}$ ($A_4$), $C_2\times C_4^2\times C_8$ ($C_2\times C_6$), $C_2\times C_4^2\times C_8$ ($C_{12}$)

Order 192: $C_4\wr C_3$ ($C_{16}$) x 4, $C_2^4.A_4$ ($C_2\times C_8$), $C_4\wr C_3$ ($C_2\times C_8$), $C_4\wr C_3$ ($C_4^2$)

Order 128: $C_2\times C_4^3$ ($C_{24}$) x 2, $C_4^2\times C_8$ ($C_2\times C_{12}$) x 2, $C_4^2\times C_8$ ($C_{24}$) x 2, $C_2^3\times C_{16}$ ($C_2\times A_4$) x 2, $C_2\times C_4^3$ ($C_2\times C_{12}$), $C_2^2\times C_4\times C_8$ ($C_2\times A_4$)

Order 96: $C_4^2:C_6$ ($C_2\times C_{16}$) x 2, $C_4^2:C_6$ ($C_4\times C_8$)

Order 64: $C_4^3$ ($C_{48}$) x 4, $C_2^2\times C_{16}$ ($C_4\times A_4$) x 4, $C_4^3$ ($C_2\times C_{24}$), $C_4^3$ ($C_4\times C_{12}$), $C_2^3\times C_8$ ($C_2^2\times A_4$), $C_2^3\times C_8$ ($C_4\times A_4$), $C_2^2\times C_4^2$ ($C_4\times A_4$), $C_2^2\times C_4^2$ ($C_2\times C_{24}$)

Order 48: $C_4^2:C_3$ ($C_4\times C_{16}$)

Order 32: $C_2^3\times C_4$ ($C_8\times A_4$) x 2, $C_2^2\times C_8$ ($C_8\times A_4$) x 2, $C_2^2\times C_8$ ($C_2^3:C_{12}$) x 2, $C_2\times C_4^2$ ($C_2\times C_{48}$) x 2, $C_2^3\times C_4$ ($C_2^3:C_{12}$), $C_4\times C_8$ ($C_4^2:C_6$), $C_2\times C_4^2$ ($C_4\times C_{24}$)

Order 16: $C_2^2\times C_4$ ($A_4\times C_{16}$) x 4, $C_2\times C_8$ ($C_4^2:C_{12}$), $C_2\times C_8$ ($C_2^4.A_4$), $C_4^2$ ($C_4^2:C_{12}$), $C_4^2$ ($C_4\times C_{48}$), $C_2^4$ ($C_2^3:C_{24}$), $C_2^2\times C_4$ ($A_4\times C_4^2$), $C_2^2\times C_4$ ($C_2^3:C_{24}$)

Order 8: $C_2^3$ ($C_2^3:C_{48}$) x 2, $C_2\times C_4$ ($C_4^2:C_{24}$) x 2, $C_8$ ($C_4^2:C_{24}$) x 2, $C_8$ ($C_4^3:C_6$) x 2, $C_2^3$ ($C_4\times C_8\times A_4$), $C_2\times C_4$ ($C_2\times C_4^2:C_{12}$)

Order 4: $C_4$ ($C_4^2:C_{48}$) x 4, $C_2^2$ ($C_4\times A_4\times C_{16}$), $C_2^2$ ($C_4^3.C_{12}$), $C_4$ ($C_4^3.C_{12}$), $C_4$ ($C_2^4.C_2^4.C_3$)

Order 2: $C_2$ ($C_4^3.C_{24}$) x 2, $C_2$ ($C_4^3:C_{24}$)

Order 1: $C_1$ ($C_4^3:C_{48}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3072: $C_4^3:C_{48}$ ($C_1$)

Order 1536: $C_4^3.C_{24}$ ($C_2$), $C_4^3:C_{24}$ ($C_2$)

Order 1024: $C_4^3:C_{16}$ ($C_3$)

Order 768: $C_4^3.C_{12}$ ($C_2^2$), $C_4^3.C_{12}$ ($C_4$), $C_4^3:C_{12}$ ($C_4$), $C_4^2:C_{48}$ ($C_4$)

Order 512: $C_4^3.C_8$ ($C_6$), $C_4^3\times C_8$ ($C_6$)

Order 384: $C_4^2:C_{24}$ ($C_2\times C_4$), $C_4^2:C_{24}$ ($C_8$), $C_4^3:C_6$ ($C_2\times C_4$), $C_4^3:C_6$ ($C_8$)

Order 256: $C_4^4$ ($C_{12}$), $C_4^2:C_{16}$ ($C_{12}$), $C_2^2\times C_4\times C_{16}$ ($A_4$), $C_2\times C_4^2\times C_8$ ($C_2\times C_6$), $C_2\times C_4^2\times C_8$ ($C_{12}$)

Order 192: $C_2^4.A_4$ ($C_2\times C_8$), $C_4\wr C_3$ ($C_2\times C_8$), $C_4\wr C_3$ ($C_4^2$), $C_4\wr C_3$ ($C_{16}$)

Order 128: $C_2\times C_4^3$ ($C_2\times C_{12}$), $C_2\times C_4^3$ ($C_{24}$), $C_4^2\times C_8$ ($C_2\times C_{12}$), $C_4^2\times C_8$ ($C_{24}$), $C_2^3\times C_{16}$ ($C_2\times A_4$), $C_2^2\times C_4\times C_8$ ($C_2\times A_4$)

Order 96: $C_4^2:C_6$ ($C_4\times C_8$), $C_4^2:C_6$ ($C_2\times C_{16}$)

Order 64: $C_4^3$ ($C_2\times C_{24}$), $C_4^3$ ($C_4\times C_{12}$), $C_4^3$ ($C_{48}$), $C_2^3\times C_8$ ($C_2^2\times A_4$), $C_2^3\times C_8$ ($C_4\times A_4$), $C_2^2\times C_4^2$ ($C_4\times A_4$), $C_2^2\times C_4^2$ ($C_2\times C_{24}$), $C_2^2\times C_{16}$ ($C_4\times A_4$)

Order 48: $C_4^2:C_3$ ($C_4\times C_{16}$)

Order 32: $C_2^3\times C_4$ ($C_8\times A_4$), $C_2^3\times C_4$ ($C_2^3:C_{12}$), $C_2^2\times C_8$ ($C_8\times A_4$), $C_2^2\times C_8$ ($C_2^3:C_{12}$), $C_4\times C_8$ ($C_4^2:C_6$), $C_2\times C_4^2$ ($C_2\times C_{48}$), $C_2\times C_4^2$ ($C_4\times C_{24}$)

Order 16: $C_2\times C_8$ ($C_4^2:C_{12}$), $C_2\times C_8$ ($C_2^4.A_4$), $C_4^2$ ($C_4^2:C_{12}$), $C_4^2$ ($C_4\times C_{48}$), $C_2^4$ ($C_2^3:C_{24}$), $C_2^2\times C_4$ ($A_4\times C_4^2$), $C_2^2\times C_4$ ($A_4\times C_{16}$), $C_2^2\times C_4$ ($C_2^3:C_{24}$)

Order 8: $C_2^3$ ($C_2^3:C_{48}$), $C_2^3$ ($C_4\times C_8\times A_4$), $C_2\times C_4$ ($C_4^2:C_{24}$), $C_2\times C_4$ ($C_2\times C_4^2:C_{12}$), $C_8$ ($C_4^2:C_{24}$), $C_8$ ($C_4^3:C_6$)

Order 4: $C_2^2$ ($C_4\times A_4\times C_{16}$), $C_2^2$ ($C_4^3.C_{12}$), $C_4$ ($C_4^3.C_{12}$), $C_4$ ($C_2^4.C_2^4.C_3$), $C_4$ ($C_4^2:C_{48}$)

Order 2: $C_2$ ($C_4^3.C_{24}$), $C_2$ ($C_4^3:C_{24}$)

Order 1: $C_1$ ($C_4^3:C_{48}$)

Series

Derived series

$C_4^3:C_{48}$

$\rhd$

$C_4^2$

$\rhd$

$C_1$

Chief series

$C_4^3:C_{48}$

$\rhd$

$C_4^3.C_{24}$

$\rhd$

$C_4^3.C_{12}$

$\rhd$

$C_2\times C_4^2\times C_8$

$\rhd$

$C_4^2\times C_8$

$\rhd$

$C_4^3$

$\rhd$

$C_2\times C_4^2$

$\rhd$

$C_4^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_4^3:C_{48}$

$\rhd$

$C_4^2$

Upper central series

$C_1$

$\lhd$

$C_4\times C_8$

Supergroups

This group is a maximal subgroup of 2 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 $320 \times 320$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $82 \times 82$ rational character table.