Abstract group 576.819: $C_4^2:C_{36}$

• Label: 576.819
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3^{2} \)
• Perm deg.: $29$
• Trans deg.: $576$
• Rank: $3$

Group information

Description:	$C_4^2:C_{36}$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_2^8.C_2^2.C_6^2$, of order \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
Nilpotency class:	$2$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	7	2	56	14	6	112	42	336	576
Conjugacy classes	1	7	2	20	14	6	40	42	120	252
Divisions	1	7	1	10	7	1	10	7	10	54
Autjugacy classes	1	2	1	2	2	1	2	2	2	15

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	144	108	0	0	0	0	0	252
Irr. rational chars.	8	12	10	8	6	4	6	54

Minimal presentations

Permutation degree:	$29$
Transitive degree:	$576$
Rank:	$3$
Inequivalent generating triples:	$1638$

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:	$\langle a, b, c \mid a^{4}=b^{4}=c^{36}=[b,c]=1, b^{a}=bc^{18}, c^{a}=b^{2}c^{19} \rangle$
Permutation group:	Degree $29$ $\langle(1,2)(3,5)(4,6)(7,8)(9,10,11,12)(13,14,17,16)(15,19,20,18), (1,3,4,7)(2,5,6,8) \!\cdots\! \rangle$
Direct product:	$C_9$ $\, \times\, $ $(C_4^2:C_4)$
Semidirect product:	$C_4^2$ $\,\rtimes\,$ $C_{36}$	$(C_4\times C_{36})$ $\,\rtimes\,$ $C_4$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6.C_4^2)$ . $C_6$	$(C_4\times C_{12})$ . $C_{12}$	$C_6$ . $(C_4^2:C_6)$	$C_6$ . $(C_4^2:C_6)$	all 30

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

Homology

Abelianization:	$C_{2}^{2} \times C_{36} \simeq C_{2}^{2} \times C_{4} \times C_{9}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 315 subgroups in 207 conjugacy classes, 123 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_{18}$	$G/Z \simeq$ $C_2^3$
Commutator:	$G' \simeq$ $C_2^2$	$G/G' \simeq$ $C_2^2\times C_{36}$
Frattini:	$\Phi \simeq$ $C_2^2\times C_6$	$G/\Phi \simeq$ $C_2^2\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^2:C_{36}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_4^2:C_{36}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 576: $C_4^2:C_{36}$

Order 288: $C_{18}.C_4^2$ x 6, $C_2\times C_4\times C_{36}$

Order 192: $C_4^2:C_{12}$

Order 144: $C_2^2\times C_{36}$ x 7, $C_4\times C_{36}$ x 4

Order 96: $C_6.C_4^2$ x 6, $C_2\times C_4\times C_{12}$

Order 72: $C_2\times C_{36}$ x 24, $C_2^2\times C_{18}$

Order 64: $C_4^2:C_4$

Order 48: $C_2^2\times C_{12}$ x 7, $C_4\times C_{12}$ x 4

Order 36: $C_{36}$ x 10, $C_2\times C_{18}$ x 7

Order 32: $C_2.C_4^2$ x 6, $C_2\times C_4^2$

Order 24: $C_2\times C_{12}$ x 24, $C_2^2\times C_6$

Order 18: $C_{18}$ x 7

Order 16: $C_2^2\times C_4$ x 7, $C_4^2$ x 4

Order 12: $C_{12}$ x 10, $C_2\times C_6$ x 7

Order 9: $C_9$

Order 8: $C_2\times C_4$ x 24, $C_2^3$

Order 6: $C_6$ x 7

Order 4: $C_4$ x 10, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $C_4^2:C_{36}$

Order 288: $C_{18}.C_4^2$, $C_2\times C_4\times C_{36}$

Order 192: $C_4^2:C_{12}$

Order 144: $C_2^2\times C_{36}$ x 2, $C_4\times C_{36}$

Order 96: $C_6.C_4^2$, $C_2\times C_4\times C_{12}$

Order 72: $C_2\times C_{36}$ x 3, $C_2^2\times C_{18}$

Order 64: $C_4^2:C_4$

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

Order 36: $C_2\times C_{18}$ x 2, $C_{36}$ x 2

Order 32: $C_2\times C_4^2$, $C_2.C_4^2$

Order 24: $C_2\times C_{12}$ x 3, $C_2^2\times C_6$

Order 18: $C_{18}$ x 2

Order 16: $C_2^2\times C_4$ x 2, $C_4^2$

Order 12: $C_2\times C_6$ x 2, $C_{12}$ x 2

Order 9: $C_9$

Order 8: $C_2\times C_4$ x 3, $C_2^3$

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $C_4^2:C_{36}$ ($C_1$)

Order 288: $C_{18}.C_4^2$ ($C_2$) x 6, $C_2\times C_4\times C_{36}$ ($C_2$)

Order 192: $C_4^2:C_{12}$ ($C_3$)

Order 144: $C_2^2\times C_{36}$ ($C_2^2$) x 7, $C_4\times C_{36}$ ($C_4$) x 4

Order 96: $C_6.C_4^2$ ($C_6$) x 6, $C_2\times C_4\times C_{12}$ ($C_6$)

Order 72: $C_2\times C_{36}$ ($C_2\times C_4$) x 6, $C_2^2\times C_{18}$ ($C_2^3$)

Order 64: $C_4^2:C_4$ ($C_9$)

Order 48: $C_2^2\times C_{12}$ ($C_2\times C_6$) x 7, $C_4\times C_{12}$ ($C_{12}$) x 4

Order 36: $C_2\times C_{18}$ ($D_4:C_2$) x 6, $C_2\times C_{18}$ ($C_2^2\times C_4$)

Order 32: $C_2.C_4^2$ ($C_{18}$) x 6, $C_2\times C_4^2$ ($C_{18}$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_{12}$) x 6, $C_2^2\times C_6$ ($C_2^2\times C_6$)

Order 18: $C_{18}$ ($C_4^2:C_2$) x 4, $C_{18}$ ($C_4^2:C_2$) x 3

Order 16: $C_2^2\times C_4$ ($C_2\times C_{18}$) x 7, $C_4^2$ ($C_{36}$) x 4

Order 12: $C_2\times C_6$ ($D_4:C_6$) x 6, $C_2\times C_6$ ($C_2^2\times C_{12}$)

Order 9: $C_9$ ($C_4^2:C_4$)

Order 8: $C_2\times C_4$ ($C_2\times C_{36}$) x 6, $C_2^3$ ($C_2^2\times C_{18}$)

Order 6: $C_6$ ($C_4^2:C_6$) x 4, $C_6$ ($C_4^2:C_6$) x 3

Order 4: $C_2^2$ ($D_4:C_{18}$) x 6, $C_2^2$ ($C_2^2\times C_{36}$)

Order 3: $C_3$ ($C_4^2:C_{12}$)

Order 2: $C_2$ ($C_4^2:C_{18}$) x 4, $C_2$ ($C_4^2:C_{18}$) x 3

Order 1: $C_1$ ($C_4^2:C_{36}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $C_4^2:C_{36}$ ($C_1$)

Order 288: $C_{18}.C_4^2$ ($C_2$), $C_2\times C_4\times C_{36}$ ($C_2$)

Order 192: $C_4^2:C_{12}$ ($C_3$)

Order 144: $C_2^2\times C_{36}$ ($C_2^2$) x 2, $C_4\times C_{36}$ ($C_4$)

Order 96: $C_6.C_4^2$ ($C_6$), $C_2\times C_4\times C_{12}$ ($C_6$)

Order 72: $C_2\times C_{36}$ ($C_2\times C_4$), $C_2^2\times C_{18}$ ($C_2^3$)

Order 64: $C_4^2:C_4$ ($C_9$)

Order 48: $C_2^2\times C_{12}$ ($C_2\times C_6$) x 2, $C_4\times C_{12}$ ($C_{12}$)

Order 36: $C_2\times C_{18}$ ($D_4:C_2$), $C_2\times C_{18}$ ($C_2^2\times C_4$)

Order 32: $C_2\times C_4^2$ ($C_{18}$), $C_2.C_4^2$ ($C_{18}$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_{12}$), $C_2^2\times C_6$ ($C_2^2\times C_6$)

Order 18: $C_{18}$ ($C_4^2:C_2$), $C_{18}$ ($C_4^2:C_2$)

Order 16: $C_2^2\times C_4$ ($C_2\times C_{18}$) x 2, $C_4^2$ ($C_{36}$)

Order 12: $C_2\times C_6$ ($D_4:C_6$), $C_2\times C_6$ ($C_2^2\times C_{12}$)

Order 9: $C_9$ ($C_4^2:C_4$)

Order 8: $C_2^3$ ($C_2^2\times C_{18}$), $C_2\times C_4$ ($C_2\times C_{36}$)

Order 6: $C_6$ ($C_4^2:C_6$), $C_6$ ($C_4^2:C_6$)

Order 4: $C_2^2$ ($D_4:C_{18}$), $C_2^2$ ($C_2^2\times C_{36}$)

Order 3: $C_3$ ($C_4^2:C_{12}$)

Order 2: $C_2$ ($C_4^2:C_{18}$), $C_2$ ($C_4^2:C_{18}$)

Order 1: $C_1$ ($C_4^2:C_{36}$)

Series

Derived series

$C_4^2:C_{36}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_4^2:C_{36}$

$\rhd$

$C_{18}.C_4^2$

$\rhd$

$C_2^2\times C_{36}$

$\rhd$

$C_2^2\times C_{18}$

$\rhd$

$C_2\times C_{18}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_4^2:C_{36}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_{18}$

$\lhd$

$C_4^2:C_{36}$

Supergroups

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

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $252 \times 252$ 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 $54 \times 54$ rational character table.