Abstract group 5184.dc: $(C_3\times C_6^3).D_4$

• Label: 5184.dc
• Order: \( 2^{6} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $20$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$(C_3\times C_6^3).D_4$
Order:	\(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^4.Q_8.C_6.C_2^5.C_2^3$, of order \(995328\)\(\medspace = 2^{12} \cdot 3^{5} \)
Composition factors:	$C_2$ x 6, $C_3$ x 4
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	727	80	1512	1136	1728	5184
Conjugacy classes	1	11	14	10	90	24	150
Divisions	1	11	14	6	90	16	138
Autjugacy classes	1	9	3	3	13	2	31

Dimension	1	2	4	8
Irr. complex chars.	16	4	66	64	150
Irr. rational chars.	8	8	50	72	138

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$43680$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	8
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid c^{2}=d^{6}=e^{3}=f^{3}=g^{3}=[d,e]=[d,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,2)(4,5)(7,8)(9,12)(10,11)(13,14)(16,17)(18,19), (1,3,2)(4,6)(7,9,12,8) \!\cdots\! \rangle$
Transitive group:	24T7572				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6^2:S_3^2)$ $\,\rtimes\,$ $C_4$	$(C_6^2.S_3^2)$ $\,\rtimes\,$ $C_4$	$(C_6^2:S_3^2)$ $\,\rtimes\,$ $C_4$	$C_3^2$ $\,\rtimes\,$ $(D_6^2:C_4)$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3\times C_6^3)$ . $D_4$	$(C_6^3:S_3)$ . $C_2^2$	$C_6^2$ . $(S_3^2:C_4)$ (2)	$(C_3^4:C_2^3)$ . $D_4$ (3)	all 21

Elements of the group are displayed as permutations of degree 20.

Homology

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

Subgroups

There are 172486 subgroups in 5933 conjugacy classes, 71 normal (29 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^4.C_2^4.C_2$
Commutator:	$G' \simeq$ $C_3^3:D_6$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2\times C_3^4:D_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6^3$	$G/\operatorname{Fit} \simeq$ $D_4$
Radical:	$R \simeq$ $(C_3\times C_6^3).D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^3:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

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

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 5184: $(C_3\times C_6^3).D_4$

Order 2592: $C_6^2.\SOPlus(4,2)$ x 4, $C_3^4.C_2^4.C_2$, $C_3^4.C_2^4.C_2$, $C_3^4.C_2^3.C_2^2$

Order 1296: $C_6^2:S_3^2$ x 6, $(C_3^2\times C_6^2):C_4$ x 3, $C_6^2.S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^3:S_3$, $C_6^2:S_3^2$, $C_2^2\times C_3^4:C_4$, $C_6^3:C_6$, $C_6^2.S_3^2$, $C_6^3.C_6$

Order 864: $C_6^3:C_2^2$ x 4, $C_6^3.C_2^2$ x 4

Order 648: $C_3^4:C_2^3$ x 8, $C_3^4:C_2^3$ x 5, $C_3^3:(C_4\times S_3)$ x 5, $C_2\times C_3^4:C_4$ x 4, $C_3^4:D_4$ x 4, $C_3^3:D_{12}$ x 4, $C_6\times C_3^2:C_{12}$ x 3, $C_3\times C_6^3$, $C_3^4:C_2^3$

Order 576: $D_6^2:C_4$ x 4

Order 432: $C_6^2:D_6$ x 32, $C_6^3:C_2$ x 14, $C_6^2.D_6$ x 12, $C_6^2.D_6$ x 8, $C_3:D_6^2$ x 4, $C_6^3:C_2$ x 4, $D_6:C_6^2$ x 4, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_6^2:C_{12}$ x 4, $C_6^2:C_{12}$ x 4

Order 324: $C_3^3:D_6$ x 13, $C_3^2\times C_6^2$ x 4, $C_3^2:C_6^2$ x 4, $C_3^2:S_3^2$ x 4, $C_3^3:C_{12}$ x 4, $C_3^4:C_4$ x 2

Order 288: $C_6^2.D_4$ x 16, $D_6^2:C_2$ x 10, $C_2^3.S_3^2$ x 10, $C_2\times C_6^2:C_4$ x 7, $C_6^2.C_2^3$, $C_6^2:C_2^3$

Order 216: $C_6^2:S_3$ x 164, $C_6:S_3^2$ x 32, $C_6.S_3^2$ x 32, $C_6^2:C_6$ x 16, $C_6^2:C_6$ x 16, $C_3^2:D_{12}$ x 16, $C_3^2:D_{12}$ x 16, $C_6^3$ x 14, $C_6^2.C_6$ x 12, $C_6.C_6^2$ x 12, $C_6^2:C_6$ x 4, $S_3\times C_6^2$ x 4

Order 162: $C_3^3:S_3$ x 5, $C_3^3\times C_6$ x 4, $C_3^2\wr C_2$ x 2

Order 144: $D_6:D_6$ x 72, $C_6^2:C_2^2$ x 45, $C_2^2.S_3^2$ x 26, $C_6^2:C_4$ x 24, $C_6.D_{12}$ x 20, $C_6^2:C_4$ x 16, $C_6^2:C_2^2$ x 16, $C_6:D_{12}$ x 16, $D_6^2$ x 10, $C_6^2:C_4$ x 10, $C_{12}:D_6$ x 8, $C_{12}:D_6$ x 3, $C_6.D_{12}$ x 2, $C_6^2:C_2^2$ x 2, $C_4:C_6^2$, $C_6:D_{12}$, $C_6^2:C_4$, $C_6^2:C_4$

Order 108: $C_3^2:D_6$ x 312, $C_3\times C_6^2$ x 82, $C_3:S_3^2$ x 32, $C_3^2:D_6$ x 16, $C_3^2\times D_6$ x 16, $C_3^2:C_{12}$ x 16, $C_3^2:C_{12}$ x 16

Order 96: $D_6.D_4$ x 4, $D_4\times D_6$ x 4

Order 81: $C_3^4$

Order 72: $C_6:D_6$ x 533, $S_3\times D_6$ x 68, $C_6.D_6$ x 68, $C_6\wr C_2$ x 64, $C_3:D_{12}$ x 64, $C_6:C_{12}$ x 48, $C_2\times C_6^2$ x 45, $C_2\times C_3^2:C_4$ x 40, $C_6\times D_6$ x 16, $C_6^2:C_2$ x 8, $C_{12}:S_3$ x 8, $D_4\times C_3^2$ x 4, $C_3:D_{12}$ x 4, $C_6\times C_{12}$ x 3, $C_6.D_6$ x 3

Order 64: $C_2^4:C_4$

Order 54: $C_3^2:S_3$ x 96, $C_3^2\times C_6$ x 82, $C_3^2:C_6$ x 8, $S_3\times C_3^2$ x 8

Order 48: $S_3\times D_4$ x 32, $C_2^2\times D_6$ x 18, $C_4\times D_6$ x 12, $C_6:D_4$ x 8, $D_6:C_4$ x 8, $C_6\times D_4$ x 4, $C_2\times D_{12}$ x 4, $C_2^2:C_{12}$ x 4, $C_6.D_4$ x 4

Order 36: $C_6:S_3$ x 1014, $C_6^2$ x 266, $C_3:C_{12}$ x 64, $C_6\times S_3$ x 64, $S_3^2$ x 64, $C_3^2:C_4$ x 20, $C_3\times C_{12}$ x 4, $C_3^2:C_4$ x 4

Order 32: $C_2^3:C_4$ x 4, $C_2^3:C_4$ x 2, $C_2^2\times D_4$

Order 27: $C_3^3$ x 14

Order 24: $C_2\times D_6$ x 200, $C_3:D_4$ x 32, $C_4\times S_3$ x 32, $C_2^2\times C_6$ x 18, $D_{12}$ x 16, $C_3\times D_4$ x 16, $C_2\times C_{12}$ x 12, $C_6:C_4$ x 12

Order 18: $C_3:S_3$ x 308, $C_3\times C_6$ x 264, $C_3\times S_3$ x 32

Order 16: $C_2\times D_4$ x 8, $C_2^2:C_4$ x 6, $C_2^2\times C_4$ x 3, $C_2^4$ x 2

Order 12: $D_6$ x 360, $C_2\times C_6$ x 98, $C_{12}$ x 16, $C_3:C_4$ x 16

Order 9: $C_3^2$ x 44

Order 8: $C_2^3$ x 15, $C_2\times C_4$ x 12, $D_4$ x 8

Order 6: $S_3$ x 104, $C_6$ x 90

Order 4: $C_2^2$ x 25, $C_4$ x 6

Order 3: $C_3$ x 14

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5184: $(C_3\times C_6^3).D_4$

Order 2592: $C_3^4.C_2^4.C_2$, $C_3^4.C_2^4.C_2$, $C_3^4.C_2^3.C_2^2$, $C_6^2.\SOPlus(4,2)$

Order 1296: $C_6^2:S_3^2$ x 4, $C_6^2.S_3^2$ x 2, $(C_3^2\times C_6^2):C_4$ x 2, $C_6^3:S_3$, $C_6^2:S_3^2$, $C_2^2\times C_3^4:C_4$, $C_6^3:C_6$, $C_6^2.S_3^2$, $C_6^3.C_6$, $C_6^2.S_3^2$

Order 864: $C_6^3:C_2^2$, $C_6^3.C_2^2$

Order 648: $C_3^4:C_2^3$ x 7, $C_3^4:C_2^3$ x 3, $C_3^3:(C_4\times S_3)$ x 3, $C_2\times C_3^4:C_4$ x 2, $C_3^4:D_4$ x 2, $C_6\times C_3^2:C_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3\times C_6^3$, $C_3^4:C_2^3$

Order 576: $D_6^2:C_4$

Order 432: $C_6^2:D_6$ x 4, $C_6^3:C_2$ x 3, $C_6^2.D_6$ x 2, $C_3:D_6^2$, $C_6^3:C_2$, $D_6:C_6^2$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2:C_{12}$, $C_6^2:C_{12}$, $C_6^2.D_6$

Order 324: $C_3^3:D_6$ x 10, $C_3^2\times C_6^2$ x 4, $C_3^2:C_6^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^3:C_{12}$ x 2, $C_3^4:C_4$

Order 288: $D_6^2:C_2$ x 2, $C_2\times C_6^2:C_4$ x 2, $C_2^3.S_3^2$ x 2, $C_6^2.C_2^3$, $C_6^2.D_4$, $C_6^2:C_2^3$

Order 216: $C_6^2:S_3$ x 21, $C_6^3$ x 3, $C_6:S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6^2:C_6$ x 2, $C_6^2.C_6$ x 2, $C_6^2:C_6$ x 2, $C_6.C_6^2$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$ x 2, $C_6^2:C_6$, $S_3\times C_6^2$

Order 162: $C_3^3\times C_6$ x 4, $C_3^3:S_3$ x 4, $C_3^2\wr C_2$

Order 144: $C_6^2:C_2^2$ x 11, $D_6:D_6$ x 8, $C_{12}:D_6$ x 4, $C_2^2.S_3^2$ x 4, $C_6^2:C_4$ x 4, $C_6^2:C_4$ x 2, $C_6.D_{12}$ x 2, $D_6^2$ x 2, $C_6^2:C_4$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}:D_6$ x 2, $C_6^2:C_2^2$ x 2, $C_6:D_{12}$ x 2, $C_6.D_{12}$, $C_4:C_6^2$, $C_6:D_{12}$, $C_6^2:C_4$, $C_6^2:C_4$

Order 108: $C_3^2:D_6$ x 30, $C_3\times C_6^2$ x 12, $C_3^2:D_6$ x 2, $C_3^2\times D_6$ x 2, $C_3:S_3^2$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:C_{12}$ x 2

Order 96: $D_6.D_4$, $D_4\times D_6$

Order 81: $C_3^4$

Order 72: $C_6:D_6$ x 74, $C_2\times C_6^2$ x 11, $S_3\times D_6$ x 6, $C_6.D_6$ x 6, $C_2\times C_3^2:C_4$ x 4, $C_6^2:C_2$ x 4, $C_6\wr C_2$ x 4, $C_6:C_{12}$ x 4, $C_3:D_{12}$ x 4, $C_{12}:S_3$ x 3, $C_6\times D_6$ x 2, $D_4\times C_3^2$ x 2, $C_6\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_3:D_{12}$ x 2

Order 64: $C_2^4:C_4$

Order 54: $C_3^2\times C_6$ x 12, $C_3^2:S_3$ x 12, $C_3^2:C_6$, $S_3\times C_3^2$

Order 48: $C_2^2\times D_6$ x 4, $S_3\times D_4$ x 4, $C_6:D_4$ x 2, $C_4\times D_6$ x 2, $C_6\times D_4$, $C_2\times D_{12}$, $C_2^2:C_{12}$, $C_6.D_4$, $D_6:C_4$

Order 36: $C_6:S_3$ x 104, $C_6^2$ x 42, $C_3:C_{12}$ x 4, $C_6\times S_3$ x 4, $S_3^2$ x 4, $C_3^2:C_4$ x 2, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2

Order 32: $C_2^3:C_4$ x 2, $C_2^3:C_4$, $C_2^2\times D_4$

Order 27: $C_3^3$ x 3

Order 24: $C_2\times D_6$ x 25, $C_3:D_4$ x 4, $C_2^2\times C_6$ x 4, $C_4\times S_3$ x 3, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2, $D_{12}$ x 2, $C_3\times D_4$ x 2

Order 18: $C_3\times C_6$ x 41, $C_3:S_3$ x 41, $C_3\times S_3$ x 2

Order 16: $C_2\times D_4$ x 6, $C_2^2:C_4$ x 4, $C_2^2\times C_4$ x 3, $C_2^4$ x 2

Order 12: $D_6$ x 34, $C_2\times C_6$ x 14, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 10

Order 8: $C_2^3$ x 12, $C_2\times C_4$ x 7, $D_4$ x 4

Order 6: $C_6$ x 13, $S_3$ x 13

Order 4: $C_2^2$ x 18, $C_4$ x 3

Order 3: $C_3$ x 3

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5184: $(C_3\times C_6^3).D_4$ ($C_1$)

Order 2592: $C_6^2.\SOPlus(4,2)$ ($C_2$) x 4, $C_3^4.C_2^4.C_2$ ($C_2$), $C_3^4.C_2^4.C_2$ ($C_2$), $C_3^4.C_2^3.C_2^2$ ($C_2$)

Order 1296: $C_6^2:S_3^2$ ($C_2^2$) x 2, $C_6^2:S_3^2$ ($C_4$) x 2, $(C_3^2\times C_6^2):C_4$ ($C_2^2$) x 2, $C_6^2.S_3^2$ ($C_2^2$) x 2, $C_6^3:S_3$ ($C_2^2$), $C_6^2:S_3^2$ ($C_4$), $C_6^2.S_3^2$ ($C_4$)

Order 648: $C_3^4:C_2^3$ ($D_4$) x 3, $C_3^4:C_2^3$ ($C_2\times C_4$) x 2, $C_3^4:C_2^3$ ($C_2\times C_4$) x 2, $C_3^3:(C_4\times S_3)$ ($C_2\times C_4$) x 2, $C_3\times C_6^3$ ($D_4$), $C_3^4:C_2^3$ ($C_2^3$)

Order 324: $C_3^2\times C_6^2$ ($C_2^2:C_4$) x 2, $C_3^3:D_6$ ($C_2^2:C_4$) x 2, $C_3^2\times C_6^2$ ($C_2\times D_4$), $C_3^3:D_6$ ($C_2\times D_4$), $C_3^3:D_6$ ($C_2^2\times C_4$)

Order 162: $C_3^3:S_3$ ($C_2^3:C_4$) x 2, $C_3^3\times C_6$ ($C_2^3:C_4$)

Order 81: $C_3^4$ ($C_2^4:C_4$)

Order 72: $C_2\times C_6^2$ ($\SOPlus(4,2)$) x 4

Order 36: $C_6^2$ ($S_3^2:C_4$) x 8, $C_6^2$ ($S_3^2:C_2^2$) x 4

Order 18: $C_3\times C_6$ ($C_6^2.D_4$) x 4

Order 9: $C_3^2$ ($D_6^2:C_4$) x 4

Order 8: $C_2^3$ ($C_3^4:D_4$)

Order 4: $C_2^2$ ($(C_3^3\times C_6).D_4$) x 2, $C_2^2$ ($C_2\times C_3^4:D_4$)

Order 2: $C_2$ ($C_3^4.C_2^4.C_2$)

Order 1: $C_1$ ($(C_3\times C_6^3).D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5184: $(C_3\times C_6^3).D_4$ ($C_1$)

Order 2592: $C_3^4.C_2^4.C_2$ ($C_2$), $C_3^4.C_2^4.C_2$ ($C_2$), $C_3^4.C_2^3.C_2^2$ ($C_2$), $C_6^2.\SOPlus(4,2)$ ($C_2$)

Order 1296: $C_6^3:S_3$ ($C_2^2$), $C_6^2:S_3^2$ ($C_4$), $C_6^2:S_3^2$ ($C_2^2$), $C_6^2:S_3^2$ ($C_4$), $C_6^2.S_3^2$ ($C_4$), $(C_3^2\times C_6^2):C_4$ ($C_2^2$), $C_6^2.S_3^2$ ($C_2^2$)

Order 648: $C_3^4:C_2^3$ ($D_4$) x 3, $C_3^4:C_2^3$ ($C_2\times C_4$) x 2, $C_3\times C_6^3$ ($D_4$), $C_3^4:C_2^3$ ($C_2^3$), $C_3^4:C_2^3$ ($C_2\times C_4$), $C_3^3:(C_4\times S_3)$ ($C_2\times C_4$)

Order 324: $C_3^2\times C_6^2$ ($C_2^2:C_4$) x 2, $C_3^3:D_6$ ($C_2^2:C_4$) x 2, $C_3^2\times C_6^2$ ($C_2\times D_4$), $C_3^3:D_6$ ($C_2\times D_4$), $C_3^3:D_6$ ($C_2^2\times C_4$)

Order 162: $C_3^3\times C_6$ ($C_2^3:C_4$), $C_3^3:S_3$ ($C_2^3:C_4$)

Order 81: $C_3^4$ ($C_2^4:C_4$)

Order 72: $C_2\times C_6^2$ ($\SOPlus(4,2)$)

Order 36: $C_6^2$ ($S_3^2:C_4$) x 2, $C_6^2$ ($S_3^2:C_2^2$)

Order 18: $C_3\times C_6$ ($C_6^2.D_4$)

Order 9: $C_3^2$ ($D_6^2:C_4$)

Order 8: $C_2^3$ ($C_3^4:D_4$)

Order 4: $C_2^2$ ($(C_3^3\times C_6).D_4$) x 2, $C_2^2$ ($C_2\times C_3^4:D_4$)

Order 2: $C_2$ ($C_3^4.C_2^4.C_2$)

Order 1: $C_1$ ($(C_3\times C_6^3).D_4$)

Series

Derived series

$(C_3\times C_6^3).D_4$

$\rhd$

$(C_3\times C_6^3).D_4$

$\rhd$

$C_3^3:D_6$

$\rhd$

$C_3^3:D_6$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$(C_3\times C_6^3).D_4$

$\rhd$

$(C_3\times C_6^3).D_4$

$\rhd$

$C_3^4.C_2^3.C_2^2$

$\rhd$

$C_3^4.C_2^3.C_2^2$

$\rhd$

$C_6^3:S_3$

$\rhd$

$C_6^3:S_3$

$\rhd$

$C_3^4:C_2^3$

$\rhd$

$C_3^4:C_2^3$

$\rhd$

$C_3^3:D_6$

$\rhd$

$C_3^3:D_6$

$\rhd$

$C_3^3\times C_6$

$\rhd$

$C_3^3\times C_6$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$(C_3\times C_6^3).D_4$

$\rhd$

$(C_3\times C_6^3).D_4$

$\rhd$

$C_3^3:D_6$

$\rhd$

$C_3^3:D_6$

$\rhd$

$C_3^3\times C_6$

$\rhd$

$C_3^3\times C_6$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

$\lhd$

$C_2^3$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 8 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 $150 \times 150$ 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 $138 \times 138$ rational character table (warning: may be slow to load).