Abstract group 1632.791: $C_{136}.D_6$

• Label: 1632.791
• Order: \( 2^{5} \cdot 3 \cdot 17 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 17 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 17 \)
• $\card{Z(G)}$: \( 2 \cdot 17 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $36$
• Trans deg.: $816$
• Rank: $3$

Group information

Description:	$C_{136}.D_6$
Order:	\(1632\)\(\medspace = 2^{5} \cdot 3 \cdot 17 \)
Exponent:	\(408\)\(\medspace = 2^{3} \cdot 3 \cdot 17 \)
Automorphism group:	$C_3:((C_2\times C_8).C_2^6)$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_{17}$
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	6	8	12	17	24	34	51	68	102	136	204	408
Elements	1	23	2	24	10	16	12	16	8	368	32	384	160	256	192	128	1632
Conjugacy classes	1	4	1	5	2	4	2	16	2	64	16	80	32	64	32	32	357
Divisions	1	4	1	4	2	2	2	1	1	4	1	4	2	2	2	1	34
Autjugacy classes	1	4	1	4	2	2	2	1	1	4	1	4	2	2	2	1	34

Dimension	1	2	4	8	16	32	64	128
Irr. complex chars.	136	170	51	0	0	0	0	0	357
Irr. rational chars.	8	6	1	2	8	6	1	2	34

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$816$
Rank:	$3$
Inequivalent generating triples:	$412608$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	128
Arbitrary	4	6	26

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=c^{136}=[a,c]=1, b^{6}=c^{68}, b^{a}=b^{5}c^{68}, c^{b}=c^{35} \rangle$
Permutation group:	Degree $36$ $\langle(2,6)(3,9)(8,12)(11,13)(18,19), (1,2)(3,10)(4,12)(5,6)(7,11)(8,14)(9,16) \!\cdots\! \rangle$
Direct product:	$C_{17}$ $\, \times\, $ $(C_8.D_6)$
Semidirect product:	$(D_4:S_3)$ $\,\rtimes\,$ $C_{34}$	$(S_3\times C_8)$ $\,\rtimes\,$ $C_{34}$	$(C_{68}.D_6)$ $\,\rtimes\,$ $C_2$	$C_{51}$ $\,\rtimes\,$ $(D_8:C_2)$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{136}$ . $D_6$	$C_{408}$ . $C_2^2$	$C_{204}$ . $C_2^3$	$Q_8$ . $(S_3\times C_{34})$	all 32

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

Homology

Abelianization:	$C_{2}^{2} \times C_{34} \simeq C_{2}^{3} \times C_{17}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 308 subgroups in 124 conjugacy classes, 54 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{34}$	$G/Z \simeq$ $S_3\times D_4$
Commutator:	$G' \simeq$ $C_{12}$	$G/G' \simeq$ $C_2^2\times C_{34}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $D_6\times C_{34}$
Fitting:	$\operatorname{Fit} \simeq$ $\SD_{16}\times C_{51}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{136}.D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{102}$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$

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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Classes of subgroups up to conjugation

Order 1632: $C_{136}.D_6$

Order 816: $\SD_{16}\times C_{51}$, $C_{51}:Q_{16}$, $D_{12}:C_{34}$, $C_{24}:C_{34}$, $S_3\times C_{136}$, $D_{12}:C_{34}$, $C_{68}.D_6$

Order 544: $D_8:C_{34}$

Order 408: $C_{17}\times D_{12}$ x 2, $S_3\times C_{68}$ x 2, $C_{408}$, $Q_8\times C_{51}$, $D_4\times C_{51}$, $D_6:C_{34}$, $C_6:C_{68}$, $C_{51}:Q_8$, $C_3:C_{136}$

Order 272: $D_4:C_{34}$ x 2, $\SD_{16}\times C_{17}$ x 2, $Q_{16}\times C_{17}$, $D_8\times C_{17}$, $C_2\times C_{136}$

Order 204: $C_{204}$ x 2, $S_3\times C_{34}$ x 2, $C_3:C_{68}$ x 2, $C_2\times C_{102}$

Order 136: $D_4\times C_{17}$ x 4, $C_2\times C_{68}$ x 3, $C_{136}$ x 2, $Q_8\times C_{17}$ x 2

Order 102: $C_{102}$ x 2, $S_3\times C_{17}$ x 2

Order 96: $C_8.D_6$

Order 68: $C_{68}$ x 4, $C_2\times C_{34}$ x 3

Order 51: $C_{51}$

Order 48: $C_{24}:C_2$, $D_{12}:C_2$, $S_3\times C_8$, $D_4:S_3$, $C_3\times \SD_{16}$, $C_3:Q_{16}$, $C_3:D_8$

Order 34: $C_{34}$ x 4

Order 32: $D_8:C_2$

Order 24: $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3:D_4$, $C_6:C_4$, $C_3:Q_8$, $C_{24}$, $C_3\times Q_8$, $C_3\times D_4$, $C_3:C_8$

Order 17: $C_{17}$

Order 16: $\SD_{16}$ x 2, $D_4:C_2$ x 2, $Q_{16}$, $D_8$, $C_2\times C_8$

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

Order 8: $D_4$ x 4, $C_2\times C_4$ x 3, $Q_8$ x 2, $C_8$ x 2

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

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1632: $C_{136}.D_6$

Order 816: $\SD_{16}\times C_{51}$, $C_{51}:Q_{16}$, $D_{12}:C_{34}$, $C_{24}:C_{34}$, $S_3\times C_{136}$, $D_{12}:C_{34}$, $C_{68}.D_6$

Order 544: $D_8:C_{34}$

Order 408: $C_{17}\times D_{12}$ x 2, $S_3\times C_{68}$ x 2, $C_{408}$, $Q_8\times C_{51}$, $D_4\times C_{51}$, $D_6:C_{34}$, $C_6:C_{68}$, $C_{51}:Q_8$, $C_3:C_{136}$

Order 272: $D_4:C_{34}$ x 2, $\SD_{16}\times C_{17}$ x 2, $Q_{16}\times C_{17}$, $D_8\times C_{17}$, $C_2\times C_{136}$

Order 204: $C_{204}$ x 2, $S_3\times C_{34}$ x 2, $C_3:C_{68}$ x 2, $C_2\times C_{102}$

Order 136: $D_4\times C_{17}$ x 4, $C_2\times C_{68}$ x 3, $C_{136}$ x 2, $Q_8\times C_{17}$ x 2

Order 102: $C_{102}$ x 2, $S_3\times C_{17}$ x 2

Order 96: $C_8.D_6$

Order 68: $C_{68}$ x 4, $C_2\times C_{34}$ x 3

Order 51: $C_{51}$

Order 48: $C_{24}:C_2$, $D_{12}:C_2$, $S_3\times C_8$, $D_4:S_3$, $C_3\times \SD_{16}$, $C_3:Q_{16}$, $C_3:D_8$

Order 34: $C_{34}$ x 4

Order 32: $D_8:C_2$

Order 24: $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3:D_4$, $C_6:C_4$, $C_3:Q_8$, $C_{24}$, $C_3\times Q_8$, $C_3\times D_4$, $C_3:C_8$

Order 17: $C_{17}$

Order 16: $\SD_{16}$ x 2, $D_4:C_2$ x 2, $Q_{16}$, $D_8$, $C_2\times C_8$

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

Order 8: $D_4$ x 4, $C_2\times C_4$ x 3, $Q_8$ x 2, $C_8$ x 2

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

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1632: $C_{136}.D_6$ ($C_1$)

Order 816: $\SD_{16}\times C_{51}$ ($C_2$), $C_{51}:Q_{16}$ ($C_2$), $D_{12}:C_{34}$ ($C_2$), $C_{24}:C_{34}$ ($C_2$), $S_3\times C_{136}$ ($C_2$), $D_{12}:C_{34}$ ($C_2$), $C_{68}.D_6$ ($C_2$)

Order 408: $C_{408}$ ($C_2^2$), $Q_8\times C_{51}$ ($C_2^2$), $D_4\times C_{51}$ ($C_2^2$), $C_{17}\times D_{12}$ ($C_2^2$), $S_3\times C_{68}$ ($C_2^2$), $C_{51}:Q_8$ ($C_2^2$), $C_3:C_{136}$ ($C_2^2$)

Order 272: $\SD_{16}\times C_{17}$ ($S_3$)

Order 204: $C_{204}$ ($C_2^3$), $S_3\times C_{34}$ ($D_4$), $C_3:C_{68}$ ($D_4$)

Order 136: $C_{136}$ ($D_6$), $Q_8\times C_{17}$ ($D_6$), $D_4\times C_{17}$ ($D_6$)

Order 102: $C_{102}$ ($C_2\times D_4$)

Order 96: $C_8.D_6$ ($C_{17}$)

Order 68: $C_{68}$ ($C_2\times D_6$)

Order 51: $C_{51}$ ($D_8:C_2$)

Order 48: $C_{24}:C_2$ ($C_{34}$), $D_{12}:C_2$ ($C_{34}$), $S_3\times C_8$ ($C_{34}$), $D_4:S_3$ ($C_{34}$), $C_3\times \SD_{16}$ ($C_{34}$), $C_3:Q_{16}$ ($C_{34}$), $C_3:D_8$ ($C_{34}$)

Order 34: $C_{34}$ ($S_3\times D_4$)

Order 24: $D_{12}$ ($C_2\times C_{34}$), $C_4\times S_3$ ($C_2\times C_{34}$), $C_3:Q_8$ ($C_2\times C_{34}$), $C_{24}$ ($C_2\times C_{34}$), $C_3\times Q_8$ ($C_2\times C_{34}$), $C_3\times D_4$ ($C_2\times C_{34}$), $C_3:C_8$ ($C_2\times C_{34}$)

Order 17: $C_{17}$ ($C_8.D_6$)

Order 16: $\SD_{16}$ ($S_3\times C_{17}$)

Order 12: $D_6$ ($D_4\times C_{17}$), $C_{12}$ ($C_2^2\times C_{34}$), $C_3:C_4$ ($D_4\times C_{17}$)

Order 8: $Q_8$ ($S_3\times C_{34}$), $D_4$ ($S_3\times C_{34}$), $C_8$ ($S_3\times C_{34}$)

Order 6: $C_6$ ($D_4\times C_{34}$)

Order 4: $C_4$ ($D_6\times C_{34}$)

Order 3: $C_3$ ($D_8:C_{34}$)

Order 2: $C_2$ ($D_{12}:C_{34}$)

Order 1: $C_1$ ($C_{136}.D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1632: $C_{136}.D_6$ ($C_1$)

Order 816: $\SD_{16}\times C_{51}$ ($C_2$), $C_{51}:Q_{16}$ ($C_2$), $D_{12}:C_{34}$ ($C_2$), $C_{24}:C_{34}$ ($C_2$), $S_3\times C_{136}$ ($C_2$), $D_{12}:C_{34}$ ($C_2$), $C_{68}.D_6$ ($C_2$)

Order 408: $C_{408}$ ($C_2^2$), $Q_8\times C_{51}$ ($C_2^2$), $D_4\times C_{51}$ ($C_2^2$), $C_{17}\times D_{12}$ ($C_2^2$), $S_3\times C_{68}$ ($C_2^2$), $C_{51}:Q_8$ ($C_2^2$), $C_3:C_{136}$ ($C_2^2$)

Order 272: $\SD_{16}\times C_{17}$ ($S_3$)

Order 204: $C_{204}$ ($C_2^3$), $S_3\times C_{34}$ ($D_4$), $C_3:C_{68}$ ($D_4$)

Order 136: $C_{136}$ ($D_6$), $Q_8\times C_{17}$ ($D_6$), $D_4\times C_{17}$ ($D_6$)

Order 102: $C_{102}$ ($C_2\times D_4$)

Order 96: $C_8.D_6$ ($C_{17}$)

Order 68: $C_{68}$ ($C_2\times D_6$)

Order 51: $C_{51}$ ($D_8:C_2$)

Order 48: $C_{24}:C_2$ ($C_{34}$), $D_{12}:C_2$ ($C_{34}$), $S_3\times C_8$ ($C_{34}$), $D_4:S_3$ ($C_{34}$), $C_3\times \SD_{16}$ ($C_{34}$), $C_3:Q_{16}$ ($C_{34}$), $C_3:D_8$ ($C_{34}$)

Order 34: $C_{34}$ ($S_3\times D_4$)

Order 24: $D_{12}$ ($C_2\times C_{34}$), $C_4\times S_3$ ($C_2\times C_{34}$), $C_3:Q_8$ ($C_2\times C_{34}$), $C_{24}$ ($C_2\times C_{34}$), $C_3\times Q_8$ ($C_2\times C_{34}$), $C_3\times D_4$ ($C_2\times C_{34}$), $C_3:C_8$ ($C_2\times C_{34}$)

Order 17: $C_{17}$ ($C_8.D_6$)

Order 16: $\SD_{16}$ ($S_3\times C_{17}$)

Order 12: $D_6$ ($D_4\times C_{17}$), $C_{12}$ ($C_2^2\times C_{34}$), $C_3:C_4$ ($D_4\times C_{17}$)

Order 8: $Q_8$ ($S_3\times C_{34}$), $D_4$ ($S_3\times C_{34}$), $C_8$ ($S_3\times C_{34}$)

Order 6: $C_6$ ($D_4\times C_{34}$)

Order 4: $C_4$ ($D_6\times C_{34}$)

Order 3: $C_3$ ($D_8:C_{34}$)

Order 2: $C_2$ ($D_{12}:C_{34}$)

Order 1: $C_1$ ($C_{136}.D_6$)

Series

Derived series

$C_{136}.D_6$

$\rhd$

$C_{12}$

$\rhd$

$C_1$

Chief series

$C_{136}.D_6$

$\rhd$

$S_3\times C_{136}$

$\rhd$

$C_{408}$

$\rhd$

$C_{136}$

$\rhd$

$C_{68}$

$\rhd$

$C_{34}$

$\rhd$

$C_{17}$

$\rhd$

$C_1$

Lower central series

$C_{136}.D_6$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_{34}$

$\lhd$

$C_{68}$

$\lhd$

$\SD_{16}\times C_{17}$

Character theory

Complex character table

See the $357 \times 357$ 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 $34 \times 34$ rational character table.