Abstract group 960.1188: $C_{12}.(C_2\times D_{20})$

• Label: 960.1188
• Order: \( 2^{6} \cdot 3 \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $28$
• Trans deg.: $480$
• Rank: $3$

Group information

Description:	$C_{12}.(C_2\times D_{20})$
Order:	\(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^7\times S_3\times F_5$, of order \(15360\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_5$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	24	30	60
Elements	1	15	2	48	4	6	320	60	24	8	192	160	24	96	960
Conjugacy classes	1	5	1	10	2	3	12	10	8	2	28	8	6	24	120
Divisions	1	5	1	8	1	3	4	5	4	1	8	2	3	4	50
Autjugacy classes	1	4	1	7	1	3	4	4	4	1	7	2	3	4	46

Dimension	1	2	4	8	16	32
Irr. complex chars.	16	60	44	0	0	0	120
Irr. rational chars.	8	12	12	11	5	2	50

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$480$
Rank:	$3$
Inequivalent generating triples:	$16128$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	14

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{8}=c^{60}=[a,b]=1, c^{a}=b^{4}c^{41}, c^{b}=c^{59} \rangle$
Permutation group:	Degree $28$ $\langle(1,2,5,9,6,10,11,14)(3,7,4,8,12,15,13,16)(17,18)(19,20)(22,23)(24,25), (1,3) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{20}:C_8)$ $\,\rtimes\,$ $S_3$	$(C_{60}:C_8)$ $\,\rtimes\,$ $C_2$	$(C_{12}.D_{20})$ $\,\rtimes\,$ $C_2$ (2)	$C_5$ $\,\rtimes\,$ $(C_4^2.D_6)$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(S_3\times C_{20})$ . $Q_8$ (2)	$C_{20}$ . $(S_3\times Q_8)$	$(S_3\times C_{20})$ . $D_4$ (2)	$(C_4\times S_3)$ . $D_{20}$ (2)	all 61

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

Homology

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

Subgroups

There are 872 subgroups in 228 conjugacy classes, 96 normal (80 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $S_3\times D_{10}$
Commutator:	$G' \simeq$ $C_2\times C_{30}$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $S_3\times D_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_4\times C_{60}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{12}.(C_2\times D_{20})$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{30}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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

Classes of subgroups up to automorphism

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

Normal subgroups

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.

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

Subgroup information

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

Classes of subgroups up to conjugation

Order 960: $C_{12}.(C_2\times D_{20})$

Order 480: $C_{12}.D_{20}$ x 2, $C_{20}:C_{24}$, $C_5\times C_4^2:S_3$, $C_{30}:\OD_{16}$, $C_{60}.C_2^3$, $C_{60}:C_8$

Order 320: $C_4^2.D_{10}$

Order 240: $C_{12}.D_{10}$ x 2, $C_{30}:C_8$ x 2, $D_6:C_{20}$ x 2, $C_{30}.D_4$ x 2, $C_{10}:C_{24}$ x 2, $C_{15}:\OD_{16}$ x 2, $C_4\times C_{60}$, $C_{12}:C_{20}$, $D_6\times C_{20}$

Order 192: $C_4^2.D_6$

Order 160: $C_{20}:C_8$ x 4, $C_4^2:C_{10}$, $C_{10}:\OD_{16}$, $C_{20}.C_2^3$

Order 120: $S_3\times C_{20}$ x 4, $C_2\times C_{60}$ x 3, $C_6:C_{20}$ x 3, $C_{15}:C_8$ x 2, $C_5:C_{24}$ x 2, $C_{10}\times D_6$

Order 96: $C_{12}.D_4$ x 2, $C_4^2:S_3$, $C_4:C_{24}$, $C_{12}:C_8$, $C_8:D_6$, $C_8\times D_6$

Order 80: $C_{10}:C_8$ x 6, $C_4:C_{20}$ x 2, $C_2^2:C_{20}$ x 2, $C_4\times C_{20}$ x 2, $C_5:\OD_{16}$ x 2, $C_2^2\times C_{20}$

Order 64: $C_4^2.C_2^2$

Order 60: $C_{60}$ x 4, $S_3\times C_{10}$ x 4, $C_3:C_{20}$ x 4, $C_2\times C_{30}$

Order 48: $C_6:C_8$ x 2, $C_{24}:C_2$ x 2, $S_3\times C_8$ x 2, $C_2\times C_{24}$ x 2, $D_6:C_4$ x 2, $C_6.D_4$ x 2, $C_4\times D_6$, $C_4\times C_{12}$, $C_{12}:C_4$

Order 40: $C_2\times C_{20}$ x 10, $C_5:C_8$ x 4, $C_2^2\times C_{10}$

Order 32: $C_4:C_8$ x 4, $C_2\times \OD_{16}$, $C_2^2\times C_8$, $C_4^2:C_2$

Order 30: $C_{30}$ x 3, $C_5\times S_3$ x 2

Order 24: $C_4\times S_3$ x 4, $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_{24}$ x 2, $C_3:C_8$ x 2, $C_2\times D_6$

Order 20: $C_{20}$ x 8, $C_2\times C_{10}$ x 5

Order 16: $C_2\times C_8$ x 6, $\OD_{16}$ x 2, $C_4:C_4$ x 2, $C_2^2:C_4$ x 2, $C_4^2$ x 2, $C_2^2\times C_4$

Order 15: $C_{15}$

Order 12: $D_6$ x 4, $C_{12}$ x 4, $C_3:C_4$ x 4, $C_2\times C_6$

Order 10: $C_{10}$ x 5

Order 8: $C_2\times C_4$ x 10, $C_8$ x 4, $C_2^3$

Order 6: $C_6$ x 3, $S_3$ x 2

Order 5: $C_5$

Order 4: $C_4$ x 8, $C_2^2$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 960: $C_{12}.(C_2\times D_{20})$

Order 480: $C_{12}.D_{20}$ x 2, $C_{20}:C_{24}$, $C_5\times C_4^2:S_3$, $C_{30}:\OD_{16}$, $C_{60}.C_2^3$, $C_{60}:C_8$

Order 320: $C_4^2.D_{10}$

Order 240: $C_{30}:C_8$ x 2, $D_6:C_{20}$ x 2, $C_{30}.D_4$ x 2, $C_{10}:C_{24}$ x 2, $C_4\times C_{60}$, $C_{12}.D_{10}$, $C_{12}:C_{20}$, $D_6\times C_{20}$, $C_{15}:\OD_{16}$

Order 192: $C_4^2.D_6$

Order 160: $C_{20}:C_8$ x 4, $C_4^2:C_{10}$, $C_{10}:\OD_{16}$, $C_{20}.C_2^3$

Order 120: $C_2\times C_{60}$ x 3, $C_6:C_{20}$ x 3, $C_{15}:C_8$ x 2, $S_3\times C_{20}$ x 2, $C_5:C_{24}$ x 2, $C_{10}\times D_6$

Order 96: $C_{12}.D_4$ x 2, $C_4^2:S_3$, $C_4:C_{24}$, $C_{12}:C_8$, $C_8:D_6$, $C_8\times D_6$

Order 80: $C_{10}:C_8$ x 5, $C_4:C_{20}$ x 2, $C_2^2:C_{20}$ x 2, $C_4\times C_{20}$ x 2, $C_2^2\times C_{20}$, $C_5:\OD_{16}$

Order 64: $C_4^2.C_2^2$

Order 60: $C_{60}$ x 4, $S_3\times C_{10}$ x 3, $C_3:C_{20}$ x 3, $C_2\times C_{30}$

Order 48: $C_6:C_8$ x 2, $C_2\times C_{24}$ x 2, $D_6:C_4$ x 2, $C_6.D_4$ x 2, $C_{24}:C_2$, $S_3\times C_8$, $C_4\times D_6$, $C_4\times C_{12}$, $C_{12}:C_4$

Order 40: $C_2\times C_{20}$ x 8, $C_5:C_8$ x 4, $C_2^2\times C_{10}$

Order 32: $C_4:C_8$ x 4, $C_2\times \OD_{16}$, $C_2^2\times C_8$, $C_4^2:C_2$

Order 30: $C_{30}$ x 3, $C_5\times S_3$

Order 24: $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_4\times S_3$ x 2, $C_{24}$ x 2, $C_3:C_8$ x 2, $C_2\times D_6$

Order 20: $C_{20}$ x 7, $C_2\times C_{10}$ x 4

Order 16: $C_2\times C_8$ x 5, $C_4:C_4$ x 2, $C_2^2:C_4$ x 2, $C_4^2$ x 2, $\OD_{16}$, $C_2^2\times C_4$

Order 15: $C_{15}$

Order 12: $C_{12}$ x 4, $D_6$ x 3, $C_3:C_4$ x 3, $C_2\times C_6$

Order 10: $C_{10}$ x 4

Order 8: $C_2\times C_4$ x 8, $C_8$ x 4, $C_2^3$

Order 6: $C_6$ x 3, $S_3$

Order 5: $C_5$

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 960: $C_{12}.(C_2\times D_{20})$ ($C_1$)

Order 480: $C_{12}.D_{20}$ ($C_2$) x 2, $C_{20}:C_{24}$ ($C_2$), $C_5\times C_4^2:S_3$ ($C_2$), $C_{30}:\OD_{16}$ ($C_2$), $C_{60}.C_2^3$ ($C_2$), $C_{60}:C_8$ ($C_2$)

Order 240: $C_{30}:C_8$ ($C_2^2$) x 2, $D_6:C_{20}$ ($C_4$) x 2, $C_{30}.D_4$ ($C_4$) x 2, $C_{10}:C_{24}$ ($C_2^2$) x 2, $C_4\times C_{60}$ ($C_2^2$), $C_{12}:C_{20}$ ($C_2^2$), $D_6\times C_{20}$ ($C_2^2$)

Order 160: $C_{20}:C_8$ ($S_3$)

Order 120: $C_6:C_{20}$ ($C_2\times C_4$) x 3, $C_2\times C_{60}$ ($C_2\times C_4$) x 2, $S_3\times C_{20}$ ($Q_8$) x 2, $S_3\times C_{20}$ ($D_4$) x 2, $C_{10}\times D_6$ ($C_2\times C_4$), $C_2\times C_{60}$ ($C_2^3$)

Order 96: $C_4^2:S_3$ ($D_5$)

Order 80: $C_{10}:C_8$ ($D_6$) x 2, $C_4\times C_{20}$ ($D_6$)

Order 60: $S_3\times C_{10}$ ($C_4:C_4$) x 2, $C_3:C_{20}$ ($C_4:C_4$) x 2, $C_{60}$ ($C_2\times Q_8$), $C_{60}$ ($C_2\times D_4$), $C_2\times C_{30}$ ($C_2^2\times C_4$)

Order 48: $D_6:C_4$ ($C_5:C_4$) x 2, $C_6.D_4$ ($C_5:C_4$) x 2, $C_4\times D_6$ ($D_{10}$), $C_4\times C_{12}$ ($D_{10}$), $C_{12}:C_4$ ($D_{10}$)

Order 40: $C_2\times C_{20}$ ($C_4\times S_3$) x 2, $C_2\times C_{20}$ ($C_2\times D_6$)

Order 30: $C_{30}$ ($\OD_{16}:C_2$) x 2, $C_{30}$ ($C_2^2.D_4$)

Order 24: $C_6:C_4$ ($C_{10}:C_4$) x 3, $C_2\times C_{12}$ ($C_{10}:C_4$) x 2, $C_4\times S_3$ ($D_{20}$) x 2, $C_4\times S_3$ ($C_5:Q_8$) x 2, $C_2\times C_{12}$ ($C_2\times D_{10}$), $C_2\times D_6$ ($C_{10}:C_4$)

Order 20: $C_2\times C_{10}$ ($C_4\times D_6$), $C_{20}$ ($S_3\times Q_8$), $C_{20}$ ($S_3\times D_4$)

Order 16: $C_4^2$ ($S_3\times D_5$)

Order 15: $C_{15}$ ($C_4^2.C_2^2$)

Order 12: $D_6$ ($C_{20}:C_4$) x 2, $C_3:C_4$ ($C_{20}:C_4$) x 2, $C_2\times C_6$ ($C_2^2.D_{10}$), $C_{12}$ ($C_2\times D_{20}$), $C_{12}$ ($C_{10}:Q_8$)

Order 10: $C_{10}$ ($D_6.D_4$), $C_{10}$ ($C_8.D_6$), $C_{10}$ ($C_8.D_6$)

Order 8: $C_2\times C_4$ ($C_6.D_{10}$) x 2, $C_2\times C_4$ ($S_3\times D_{10}$)

Order 6: $C_6$ ($C_{20}.C_2^3$) x 2, $C_6$ ($C_2^2.D_{20}$)

Order 5: $C_5$ ($C_4^2.D_6$)

Order 4: $C_2^2$ ($D_6.D_{10}$), $C_4$ ($S_3\times D_{20}$), $C_4$ ($C_{12}.D_{10}$)

Order 3: $C_3$ ($C_4^2.D_{10}$)

Order 2: $C_2$ ($D_6.D_{20}$), $C_2$ ($C_{60}.C_2^3$), $C_2$ ($C_{60}.C_2^3$)

Order 1: $C_1$ ($C_{12}.(C_2\times D_{20})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 960: $C_{12}.(C_2\times D_{20})$ ($C_1$)

Order 480: $C_{12}.D_{20}$ ($C_2$) x 2, $C_{20}:C_{24}$ ($C_2$), $C_5\times C_4^2:S_3$ ($C_2$), $C_{30}:\OD_{16}$ ($C_2$), $C_{60}.C_2^3$ ($C_2$), $C_{60}:C_8$ ($C_2$)

Order 240: $C_{30}:C_8$ ($C_2^2$) x 2, $D_6:C_{20}$ ($C_4$) x 2, $C_{30}.D_4$ ($C_4$) x 2, $C_{10}:C_{24}$ ($C_2^2$) x 2, $C_4\times C_{60}$ ($C_2^2$), $C_{12}:C_{20}$ ($C_2^2$), $D_6\times C_{20}$ ($C_2^2$)

Order 160: $C_{20}:C_8$ ($S_3$)

Order 120: $C_6:C_{20}$ ($C_2\times C_4$) x 3, $C_2\times C_{60}$ ($C_2\times C_4$) x 2, $C_{10}\times D_6$ ($C_2\times C_4$), $C_2\times C_{60}$ ($C_2^3$), $S_3\times C_{20}$ ($Q_8$), $S_3\times C_{20}$ ($D_4$)

Order 96: $C_4^2:S_3$ ($D_5$)

Order 80: $C_{10}:C_8$ ($D_6$) x 2, $C_4\times C_{20}$ ($D_6$)

Order 60: $C_{60}$ ($C_2\times Q_8$), $C_{60}$ ($C_2\times D_4$), $C_2\times C_{30}$ ($C_2^2\times C_4$), $S_3\times C_{10}$ ($C_4:C_4$), $C_3:C_{20}$ ($C_4:C_4$)

Order 48: $D_6:C_4$ ($C_5:C_4$) x 2, $C_6.D_4$ ($C_5:C_4$) x 2, $C_4\times D_6$ ($D_{10}$), $C_4\times C_{12}$ ($D_{10}$), $C_{12}:C_4$ ($D_{10}$)

Order 40: $C_2\times C_{20}$ ($C_4\times S_3$) x 2, $C_2\times C_{20}$ ($C_2\times D_6$)

Order 30: $C_{30}$ ($\OD_{16}:C_2$) x 2, $C_{30}$ ($C_2^2.D_4$)

Order 24: $C_6:C_4$ ($C_{10}:C_4$) x 3, $C_2\times C_{12}$ ($C_{10}:C_4$) x 2, $C_2\times C_{12}$ ($C_2\times D_{10}$), $C_4\times S_3$ ($D_{20}$), $C_4\times S_3$ ($C_5:Q_8$), $C_2\times D_6$ ($C_{10}:C_4$)

Order 20: $C_2\times C_{10}$ ($C_4\times D_6$), $C_{20}$ ($S_3\times Q_8$), $C_{20}$ ($S_3\times D_4$)

Order 16: $C_4^2$ ($S_3\times D_5$)

Order 15: $C_{15}$ ($C_4^2.C_2^2$)

Order 12: $C_2\times C_6$ ($C_2^2.D_{10}$), $D_6$ ($C_{20}:C_4$), $C_{12}$ ($C_2\times D_{20}$), $C_{12}$ ($C_{10}:Q_8$), $C_3:C_4$ ($C_{20}:C_4$)

Order 10: $C_{10}$ ($D_6.D_4$), $C_{10}$ ($C_8.D_6$), $C_{10}$ ($C_8.D_6$)

Order 8: $C_2\times C_4$ ($C_6.D_{10}$) x 2, $C_2\times C_4$ ($S_3\times D_{10}$)

Order 6: $C_6$ ($C_{20}.C_2^3$) x 2, $C_6$ ($C_2^2.D_{20}$)

Order 5: $C_5$ ($C_4^2.D_6$)

Order 4: $C_2^2$ ($D_6.D_{10}$), $C_4$ ($S_3\times D_{20}$), $C_4$ ($C_{12}.D_{10}$)

Order 3: $C_3$ ($C_4^2.D_{10}$)

Order 2: $C_2$ ($D_6.D_{20}$), $C_2$ ($C_{60}.C_2^3$), $C_2$ ($C_{60}.C_2^3$)

Order 1: $C_1$ ($C_{12}.(C_2\times D_{20})$)

Series

Derived series

$C_{12}.(C_2\times D_{20})$

$\rhd$

$C_2\times C_{30}$

$\rhd$

$C_1$

Chief series

$C_{12}.(C_2\times D_{20})$

$\rhd$

$C_{60}.C_2^3$

$\rhd$

$D_6\times C_{20}$

$\rhd$

$C_2\times C_{60}$

$\rhd$

$C_{60}$

$\rhd$

$C_{30}$

$\rhd$

$C_{15}$

$\rhd$

$C_5$

$\rhd$

$C_1$

Lower central series

$C_{12}.(C_2\times D_{20})$

$\rhd$

$C_2\times C_{30}$

$\rhd$

$C_{15}$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_4^2$

Character theory

Complex character table

See the $120 \times 120$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $50 \times 50$ rational character table.