Abstract group 1080.178: $C_{30}.C_6^2$

• Label: 1080.178
• Order: \( 2^{3} \cdot 3^{3} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{3} \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{3} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{3} \)
• Perm deg.: $21$
• Trans deg.: $540$
• Rank: $2$

Group information

Description:	$C_{30}.C_6^2$
Order:	\(1080\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \)
Exponent:	\(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$(D_5\times \He_3).C_2^5$, of order \(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \)
Composition factors:	$C_2$ x 3, $C_3$ x 3, $C_5$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	9	10	12	15	18	30	36	45	90
Elements	1	13	8	10	4	104	18	12	80	32	234	96	180	72	216	1080
Conjugacy classes	1	3	8	1	2	24	18	6	8	16	54	48	18	36	108	351
Divisions	1	3	4	1	1	12	3	2	4	4	9	8	3	3	6	64
Autjugacy classes	1	3	2	1	1	6	1	2	2	2	3	4	1	1	2	32

Dimension	1	2	4	6	8	12	16	24	48
Irr. complex chars.	108	243	0	0	0	0	0	0	0	351
Irr. rational chars.	4	17	6	12	9	3	4	6	3	64

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$540$
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	3	6	14

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{18}=c^{30}=[a,b]=[b,c]=1, c^{a}=b^{9}c^{19} \rangle$
Permutation group:	Degree $21$ $\langle(2,3)(4,5)(6,7)(8,9), (7,9), (10,11,13)(12,14,16)(15,17,18)(19,20,21), (10,12,15,11,14,17,13,16,18) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 10 & 0 \\ 0 & 10 \end{array}\right), \left(\begin{array}{rr} 82 & 54 \\ 90 & 28 \end{array}\right), \left(\begin{array}{rr} 1 & 33 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 28 & 27 \\ 45 & 1 \end{array}\right), \left(\begin{array}{rr} 34 & 55 \\ 66 & 1 \end{array}\right), \left(\begin{array}{rr} 91 & 9 \\ 81 & 19 \end{array}\right), \left(\begin{array}{rr} 67 & 0 \\ 0 & 67 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/99\Z)$
Direct product:	$C_3$ $\, \times\, $ $C_9$ $\, \times\, $ $(C_5:D_4)$
Semidirect product:	$(C_9\times D_{10})$ $\,\rtimes\,$ $C_6$ (3)	$(C_3\times C_{45})$ $\,\rtimes\,$ $D_4$	$(C_6\times C_{18})$ $\,\rtimes\,$ $D_5$	$C_{45}$ $\,\rtimes\,$ $(C_3\times D_4)$ (3)	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{30}$ . $C_6^2$	$C_6$ . $(C_9\times D_{10})$ (3)	$(C_6\times C_{30})$ . $C_6$	$C_{90}$ . $(C_2\times C_6)$ (3)	all 26

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

Homology

Abelianization:	$C_{6} \times C_{18} \simeq C_{2}^{2} \times C_{3} \times C_{9}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 400 subgroups in 160 conjugacy classes, 90 normal (36 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3\times C_{18}$	$G/Z \simeq$ $D_{10}$
Commutator:	$G' \simeq$ $C_{10}$	$G/G' \simeq$ $C_6\times C_{18}$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_3^2\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times C_{90}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{30}.C_6^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 1080: $C_{30}.C_6^2$

Order 540: $C_6\times C_{90}$, $C_{30}:C_{18}$, $C_{15}:C_{36}$

Order 360: $C_{45}:D_4$ x 3, $C_6^2:D_5$

Order 270: $C_3\times C_{90}$ x 2, $C_{15}:C_{18}$

Order 216: $C_{12}:C_{18}$

Order 180: $C_9\times D_{10}$ x 3, $C_5:C_{36}$ x 3, $C_2\times C_{90}$ x 3, $C_6\times C_{30}$, $C_3^2\times D_{10}$, $C_{15}:C_{12}$

Order 135: $C_3\times C_{45}$

Order 120: $C_{15}:D_4$ x 4

Order 108: $C_6\times C_{18}$ x 2, $C_3\times C_{36}$

Order 90: $C_{90}$ x 6, $C_9\times D_5$ x 3, $C_3\times C_{30}$ x 2, $C_3^2\times D_5$

Order 72: $D_4\times C_9$ x 3, $D_4\times C_3^2$

Order 60: $C_5:C_{12}$ x 4, $C_2\times C_{30}$ x 4, $C_3\times D_{10}$ x 4

Order 54: $C_3\times C_{18}$ x 3

Order 45: $C_{45}$ x 3, $C_3\times C_{15}$

Order 40: $C_5:D_4$

Order 36: $C_2\times C_{18}$ x 6, $C_{36}$ x 3, $C_6^2$ x 2, $C_3\times C_{12}$

Order 30: $C_{30}$ x 8, $C_3\times D_5$ x 4

Order 27: $C_3\times C_9$

Order 24: $C_3\times D_4$ x 4

Order 20: $C_2\times C_{10}$, $D_{10}$, $C_5:C_4$

Order 18: $C_{18}$ x 9, $C_3\times C_6$ x 3

Order 15: $C_{15}$ x 4

Order 12: $C_2\times C_6$ x 8, $C_{12}$ x 4

Order 10: $C_{10}$ x 2, $D_5$

Order 9: $C_9$ x 3, $C_3^2$

Order 8: $D_4$

Order 6: $C_6$ x 12

Order 5: $C_5$

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1080: $C_{30}.C_6^2$

Order 540: $C_6\times C_{90}$, $C_{30}:C_{18}$, $C_{15}:C_{36}$

Order 360: $C_6^2:D_5$, $C_{45}:D_4$

Order 270: $C_3\times C_{90}$ x 2, $C_{15}:C_{18}$

Order 216: $C_{12}:C_{18}$

Order 180: $C_9\times D_{10}$, $C_6\times C_{30}$, $C_3^2\times D_{10}$, $C_5:C_{36}$, $C_{15}:C_{12}$, $C_2\times C_{90}$

Order 135: $C_3\times C_{45}$

Order 120: $C_{15}:D_4$ x 2

Order 108: $C_6\times C_{18}$ x 2, $C_3\times C_{36}$

Order 90: $C_{90}$ x 2, $C_3\times C_{30}$ x 2, $C_3^2\times D_5$, $C_9\times D_5$

Order 72: $D_4\times C_3^2$, $D_4\times C_9$

Order 60: $C_5:C_{12}$ x 2, $C_2\times C_{30}$ x 2, $C_3\times D_{10}$ x 2

Order 54: $C_3\times C_{18}$ x 3

Order 45: $C_3\times C_{15}$, $C_{45}$

Order 40: $C_5:D_4$

Order 36: $C_2\times C_{18}$ x 2, $C_6^2$ x 2, $C_3\times C_{12}$, $C_{36}$

Order 30: $C_{30}$ x 4, $C_3\times D_5$ x 2

Order 27: $C_3\times C_9$

Order 24: $C_3\times D_4$ x 2

Order 20: $C_2\times C_{10}$, $D_{10}$, $C_5:C_4$

Order 18: $C_3\times C_6$ x 3, $C_{18}$ x 3

Order 15: $C_{15}$ x 2

Order 12: $C_2\times C_6$ x 4, $C_{12}$ x 2

Order 10: $C_{10}$ x 2, $D_5$

Order 9: $C_3^2$, $C_9$

Order 8: $D_4$

Order 6: $C_6$ x 6

Order 5: $C_5$

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1080: $C_{30}.C_6^2$ ($C_1$)

Order 540: $C_6\times C_{90}$ ($C_2$), $C_{30}:C_{18}$ ($C_2$), $C_{15}:C_{36}$ ($C_2$)

Order 360: $C_{45}:D_4$ ($C_3$) x 3, $C_6^2:D_5$ ($C_3$)

Order 270: $C_3\times C_{90}$ ($C_2^2$)

Order 180: $C_9\times D_{10}$ ($C_6$) x 3, $C_5:C_{36}$ ($C_6$) x 3, $C_2\times C_{90}$ ($C_6$) x 3, $C_6\times C_{30}$ ($C_6$), $C_3^2\times D_{10}$ ($C_6$), $C_{15}:C_{12}$ ($C_6$)

Order 135: $C_3\times C_{45}$ ($D_4$)

Order 120: $C_{15}:D_4$ ($C_9$) x 3, $C_{15}:D_4$ ($C_3^2$)

Order 108: $C_6\times C_{18}$ ($D_5$)

Order 90: $C_{90}$ ($C_2\times C_6$) x 3, $C_3\times C_{30}$ ($C_2\times C_6$)

Order 60: $C_5:C_{12}$ ($C_{18}$) x 3, $C_2\times C_{30}$ ($C_{18}$) x 3, $C_3\times D_{10}$ ($C_{18}$) x 3, $C_5:C_{12}$ ($C_3\times C_6$), $C_2\times C_{30}$ ($C_3\times C_6$), $C_3\times D_{10}$ ($C_3\times C_6$)

Order 54: $C_3\times C_{18}$ ($D_{10}$)

Order 45: $C_{45}$ ($C_3\times D_4$) x 3, $C_3\times C_{15}$ ($C_3\times D_4$)

Order 40: $C_5:D_4$ ($C_3\times C_9$)

Order 36: $C_2\times C_{18}$ ($C_3\times D_5$) x 3, $C_6^2$ ($C_3\times D_5$)

Order 30: $C_{30}$ ($C_2\times C_{18}$) x 3, $C_{30}$ ($C_6^2$)

Order 27: $C_3\times C_9$ ($C_5:D_4$)

Order 20: $C_2\times C_{10}$ ($C_3\times C_{18}$), $D_{10}$ ($C_3\times C_{18}$), $C_5:C_4$ ($C_3\times C_{18}$)

Order 18: $C_{18}$ ($C_3\times D_{10}$) x 3, $C_3\times C_6$ ($C_3\times D_{10}$)

Order 15: $C_{15}$ ($D_4\times C_9$) x 3, $C_{15}$ ($D_4\times C_3^2$)

Order 12: $C_2\times C_6$ ($C_9\times D_5$) x 3, $C_2\times C_6$ ($C_3^2\times D_5$)

Order 10: $C_{10}$ ($C_6\times C_{18}$)

Order 9: $C_9$ ($C_{15}:D_4$) x 3, $C_3^2$ ($C_{15}:D_4$)

Order 6: $C_6$ ($C_9\times D_{10}$) x 3, $C_6$ ($C_3^2\times D_{10}$)

Order 5: $C_5$ ($C_{12}:C_{18}$)

Order 4: $C_2^2$ ($C_{15}:C_{18}$)

Order 3: $C_3$ ($C_{45}:D_4$) x 3, $C_3$ ($C_6^2:D_5$)

Order 2: $C_2$ ($C_{30}:C_{18}$)

Order 1: $C_1$ ($C_{30}.C_6^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1080: $C_{30}.C_6^2$ ($C_1$)

Order 540: $C_6\times C_{90}$ ($C_2$), $C_{30}:C_{18}$ ($C_2$), $C_{15}:C_{36}$ ($C_2$)

Order 360: $C_6^2:D_5$ ($C_3$), $C_{45}:D_4$ ($C_3$)

Order 270: $C_3\times C_{90}$ ($C_2^2$)

Order 180: $C_9\times D_{10}$ ($C_6$), $C_6\times C_{30}$ ($C_6$), $C_3^2\times D_{10}$ ($C_6$), $C_5:C_{36}$ ($C_6$), $C_{15}:C_{12}$ ($C_6$), $C_2\times C_{90}$ ($C_6$)

Order 135: $C_3\times C_{45}$ ($D_4$)

Order 120: $C_{15}:D_4$ ($C_3^2$), $C_{15}:D_4$ ($C_9$)

Order 108: $C_6\times C_{18}$ ($D_5$)

Order 90: $C_{90}$ ($C_2\times C_6$), $C_3\times C_{30}$ ($C_2\times C_6$)

Order 60: $C_5:C_{12}$ ($C_3\times C_6$), $C_5:C_{12}$ ($C_{18}$), $C_2\times C_{30}$ ($C_3\times C_6$), $C_2\times C_{30}$ ($C_{18}$), $C_3\times D_{10}$ ($C_3\times C_6$), $C_3\times D_{10}$ ($C_{18}$)

Order 54: $C_3\times C_{18}$ ($D_{10}$)

Order 45: $C_3\times C_{15}$ ($C_3\times D_4$), $C_{45}$ ($C_3\times D_4$)

Order 40: $C_5:D_4$ ($C_3\times C_9$)

Order 36: $C_2\times C_{18}$ ($C_3\times D_5$), $C_6^2$ ($C_3\times D_5$)

Order 30: $C_{30}$ ($C_2\times C_{18}$), $C_{30}$ ($C_6^2$)

Order 27: $C_3\times C_9$ ($C_5:D_4$)

Order 20: $C_2\times C_{10}$ ($C_3\times C_{18}$), $D_{10}$ ($C_3\times C_{18}$), $C_5:C_4$ ($C_3\times C_{18}$)

Order 18: $C_3\times C_6$ ($C_3\times D_{10}$), $C_{18}$ ($C_3\times D_{10}$)

Order 15: $C_{15}$ ($D_4\times C_3^2$), $C_{15}$ ($D_4\times C_9$)

Order 12: $C_2\times C_6$ ($C_3^2\times D_5$), $C_2\times C_6$ ($C_9\times D_5$)

Order 10: $C_{10}$ ($C_6\times C_{18}$)

Order 9: $C_3^2$ ($C_{15}:D_4$), $C_9$ ($C_{15}:D_4$)

Order 6: $C_6$ ($C_9\times D_{10}$), $C_6$ ($C_3^2\times D_{10}$)

Order 5: $C_5$ ($C_{12}:C_{18}$)

Order 4: $C_2^2$ ($C_{15}:C_{18}$)

Order 3: $C_3$ ($C_6^2:D_5$), $C_3$ ($C_{45}:D_4$)

Order 2: $C_2$ ($C_{30}:C_{18}$)

Order 1: $C_1$ ($C_{30}.C_6^2$)

Series

Derived series

$C_{30}.C_6^2$

$\rhd$

$C_{10}$

$\rhd$

$C_1$

Chief series

$C_{30}.C_6^2$

$\rhd$

$C_{30}:C_{18}$

$\rhd$

$C_3\times C_{90}$

$\rhd$

$C_{90}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{30}.C_6^2$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

Upper central series

$C_1$

$\lhd$

$C_3\times C_{18}$

$\lhd$

$C_6\times C_{18}$

Character theory

Complex character table

See the $351 \times 351$ 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 $64 \times 64$ rational character table.