Abstract group 8192.bbs: $C_2^5:C_4.D_4^2$

• Label: 8192.bbs
• Order: \( 2^{13} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: 4
• $\card{\Aut(G)}$: \( 2^{20} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $5$

Group information

Description:	$C_2^5:C_4.D_4^2$
Order:	\(8192\)\(\medspace = 2^{13} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	Group of order \(1048576\)\(\medspace = 2^{20} \)
Composition factors:	$C_2$ x 13
Nilpotency class:	$6$
Derived length:	$3$

This group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).

Group statistics

Order	1	2	4	8
Elements	1	575	5056	2560	8192
Conjugacy classes	1	31	73	14	119
Divisions	1	31	69	12	113
Autjugacy classes	1	26	62	10	99

Dimension	1	2	4	8	16	32
Irr. complex chars.	32	32	26	15	10	4	119
Irr. rational chars.	32	32	14	21	10	4	113

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
Rank:	$5$
Inequivalent generating 5-tuples:	not computed

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:	${\langle a, b, c, d, e, f, g, h, i \mid c^{4}=d^{4}=e^{4}=f^{2}=g^{2}=h^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $68$ $\langle(1,4,18,37)(2,10,34,17)(3,14,42,56)(5,19,48,51)(6,12,35,55)(7,11,36,61) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^4.Q_8)$ . $D_4^2$ (4)	$(C_2^4:D_4)$ . $D_4^2$ (2)	$(C_2^6.C_2^4)$ . $D_4$ (32)	$C_2^7$ . $(D_8:C_2^2)$	all 168
Aut. group:	$\Aut(C_2^6.\SD_{16})$

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

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 700 normal subgroups (626 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2^2$
Commutator:	a subgroup isomorphic to $C_2^6:C_4$
Frattini:	a subgroup isomorphic to $C_2^6:C_4$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5:C_4.D_4^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 4 (order at least 2048) are shown, as well as all normal subgroups of any index.

Order 8192: $C_2^5:C_4.D_4^2$

Order 4096: $C_2^6.C_2^4.C_2^2$ x 24, $C_2^6.C_2^5.C_2$ x 2, $C_2^6.C_2.C_2^5$ x 2, $C_2^5.C_2^5.C_2^2$ x 2, $C_2^6.C_2^6$

Order 2048: $C_2^6.C_2^2\wr C_2$ x 43, $C_2^5.C_2^4.C_2^2$ x 42, $C_2^6.C_2^5$ x 40, $C_2^6.C_2^4.C_2$ x 23, $C_2^5.D_4^2$ x 21, $C_2^6.C_2^3.C_2^2$ x 20, $C_2^6.C_2.C_2^4$ x 20, $C_2^5.C_2^5.C_2$ x 5, $C_2.C_2^6.C_2^4$ x 3, $C_2^5.C_2.C_2^5$ x 2

Order 1024: $C_2^6.C_2^4$ x 65, $C_2^5.C_2^4.C_2$ x 27, $C_2^5.C_2^3.C_2^2$ x 24, $C_2^6.C_2^3.C_2$ x 23, $C_2^6.C_2.C_2^3$ x 14, $C_2^5.C_2.C_2^4$ x 9, $C_2.C_2^6.C_2^3$ x 9, $C_2^6.C_2.D_4$ x 8, $C_2^5.C_2^2\wr C_2$ x 8

Order 512: $C_2^5.C_4.C_2^2$ x 8, $C_2\times C_2^4.C_2^3.C_2$ x 8, $C_2^5.\SD_{16}$ x 8, $C_2^5.C_2^4$ x 6, $C_2\times C_2^5.C_2^3$ x 4, $C_2^5.C_2^4$ x 4, $C_2^6:D_4$ x 4, $C_2^5.C_2^4$ x 4, $C_2^6.C_2^3$ x 4, $C_2^6:C_8$ x 4, $C_2\times C_2^5.C_2^3$ x 4, $C_2^6.C_2^3$ x 4, $C_2\times C_2^5.C_2^3$ x 3, $C_2\times C_2^5.C_2^3$ x 3, $C_2\times C_2^5.C_2^3$ x 3, $C_2^6.C_2^3$ x 3, $C_2\times C_2^5.C_2^3$ x 3, $C_2^6.C_2^3$ x 3, $C_2\times C_2^5.C_2^3$ x 2, $C_2^4.C_2^5$ x 2, $C_2^4.C_2^5$ x 2, $C_2^5.C_2^4$ x 2, $C_2\times C_2^4.C_2^3.C_2$ x 2, $C_2\times C_2^4.C_2^3.C_2$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2\times C_2^5.C_2^3$ x 2, $C_2\times C_2^5.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2\times C_2^5.C_2^3$, $C_2\times C_2^5.C_2^3$, $C_2\times C_2^5.C_2^3$, $C_2^3.C_2^6$, $C_2^2\times C_2^5.C_2^2$, $C_2^6.C_2^3$, $C_2^4.C_2^5$, $C_2\times C_2^4.C_2^4$, $C_2\times C_2^5.C_2^3$, $C_2\times C_2^5.C_2^3$, $C_2\times C_2^4.C_2^4$, $C_2^4.C_2^5$, $C_2^5.C_2^4$, $C_2^6.C_2^3$, $C_2^5.C_2^4$, $C_2^6.C_2^3$, $C_2^6.C_2^3$, $C_2^6.C_2^3$, $C_2^5.C_2^4$, $C_2^6.C_2^3$, $C_2^6.C_2^3$, $C_2^6.C_2^3$, $C_2^6.C_2^3$, $C_2^6.C_2^3$, $C_2^6.C_2^3$, $C_2\times C_2^5.C_2^3$, $C_2^6.C_2^3$

Order 256: $C_2^6:C_4$ x 12, $C_2^5.Q_8$ x 8, $C_2^4.C_4^2$ x 8, $C_2^5.D_4$ x 4, $C_2^4.C_4^2$ x 4, $C_2^5:C_8$ x 4, $C_2^3.C_2^5$ x 4, $C_2^6:C_4$ x 4, $C_2^5.D_4$ x 4, $C_2^5:D_4$ x 3, $C_2^5.D_4$ x 3, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^4:C_4^2$ x 2, $C_2^5:D_4$ x 2, $C_2^4.C_4^2$ x 2, $C_2^5.Q_8$ x 2, $C_2^5.D_4$ x 2, $C_2^5:D_4$ x 2, $D_4:C_2^5$, $C_2^5:D_4$, $C_2^5:D_4$, $C_2^5.D_4$, $C_2^3.C_2^5$

Order 128: $C_2^5:C_4$ x 8, $C_2^5:C_4$ x 6, $D_4\times C_2^4$ x 5, $C_2^5:C_4$ x 5, $C_2^3.C_4^2$ x 4, $C_2^4:D_4$ x 4, $C_2^4.Q_8$ x 4, $C_2^4.D_4$ x 2, $C_2^7$, $C_2^5\times C_4$, $C_2^3\times C_4^2$

Order 64: $C_2^6$ x 7, $C_2^4\times C_4$ x 6, $C_2^4:C_4$ x 4, $D_4\times C_2^3$ x 4, $C_2^4:C_4$ x 2

Order 32: $C_2^5$ x 9, $C_2^3\times C_4$ x 6

Order 16: $C_2^4$ x 11

Order 8: $C_2^3$ x 7

Order 4: $C_2^2$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 8192: $C_2^5:C_4.D_4^2$ ($C_1$)

Order 4096: $C_2^6.C_2^4.C_2^2$ ($C_2$) x 24, $C_2^6.C_2^5.C_2$ ($C_2$) x 2, $C_2^6.C_2.C_2^5$ ($C_2$) x 2, $C_2^5.C_2^5.C_2^2$ ($C_2$) x 2, $C_2^6.C_2^6$ ($C_2$)

Order 2048: $C_2^6.C_2^2\wr C_2$ ($C_2^2$) x 43, $C_2^5.C_2^4.C_2^2$ ($C_2^2$) x 25, $C_2^6.C_2^4.C_2$ ($C_2^2$) x 23, $C_2^6.C_2^3.C_2^2$ ($C_2^2$) x 20, $C_2^6.C_2.C_2^4$ ($C_2^2$) x 20, $C_2^6.C_2^5$ ($C_2^2$) x 16, $C_2^5.C_2^5.C_2$ ($C_2^2$) x 4, $C_2^5.C_2.C_2^5$ ($C_2^2$) x 2, $C_2.C_2^6.C_2^4$ ($C_2^2$) x 2

Order 1024: $C_2^6.C_2^4$ ($C_2^3$) x 41, $C_2^5.C_2^4.C_2$ ($C_2^3$) x 27, $C_2^6.C_2^4$ ($D_4$) x 24, $C_2^5.C_2^3.C_2^2$ ($C_2^3$) x 24, $C_2^6.C_2^3.C_2$ ($C_2^3$) x 23, $C_2^6.C_2.C_2^3$ ($C_2^3$) x 14, $C_2^5.C_2.C_2^4$ ($C_2^3$) x 9, $C_2.C_2^6.C_2^3$ ($C_2^3$) x 9, $C_2^6.C_2.D_4$ ($C_2^3$) x 8, $C_2^5.C_2^2\wr C_2$ ($D_4$) x 8

Order 512: $C_2^5.C_4.C_2^2$ ($C_2\times D_4$) x 8, $C_2\times C_2^4.C_2^3.C_2$ ($C_2^4$) x 8, $C_2^5.\SD_{16}$ ($C_2\times D_4$) x 8, $C_2^6:D_4$ ($C_2\times D_4$) x 4, $C_2^5.C_2^4$ ($C_2\times D_4$) x 4, $C_2^6:C_8$ ($C_2\times D_4$) x 4, $C_2\times C_2^5.C_2^3$ ($C_2\times D_4$) x 3, $C_2\times C_2^5.C_2^3$ ($C_2\times D_4$) x 3, $C_2^5.C_2^4$ ($C_2\times D_4$) x 3, $C_2\times C_2^5.C_2^3$ ($C_2\times D_4$) x 3, $C_2^6.C_2^3$ ($C_2\times D_4$) x 3, $C_2\times C_2^5.C_2^3$ ($C_2\times D_4$) x 3, $C_2^5.C_2^4$ ($C_2^4$) x 3, $C_2^5.C_2^4$ ($C_2\times D_4$) x 3, $C_2^6.C_2^3$ ($C_2\times D_4$) x 3, $C_2^6.C_2^3$ ($C_2\times D_4$) x 3, $C_2\times C_2^5.C_2^3$ ($C_2\times D_4$) x 2, $C_2\times C_2^5.C_2^3$ ($C_2\times D_4$) x 2, $C_2^4.C_2^5$ ($C_2^4$) x 2, $C_2^4.C_2^5$ ($C_2\times D_4$) x 2, $C_2^5.C_2^4$ ($C_2\times D_4$) x 2, $C_2\times C_2^4.C_2^3.C_2$ ($C_2^4$) x 2, $C_2\times C_2^4.C_2^3.C_2$ ($C_2^4$) x 2, $C_2^6.C_2^3$ ($C_2\times D_4$) x 2, $C_2^6.C_2^3$ ($C_2\times D_4$) x 2, $C_2^6.C_2^3$ ($C_2\times D_4$) x 2, $C_2^6.C_2^3$ ($C_2\times D_4$) x 2, $C_2^6.C_2^3$ ($C_2\times D_4$) x 2, $C_2^6.C_2^3$ ($C_2\times D_4$) x 2, $C_2^6.C_2^3$ ($C_2\times D_4$) x 2, $C_2^6.C_2^3$ ($C_2^4$) x 2, $C_2^6.C_2^3$ ($C_2\times D_4$) x 2, $C_2^6.C_2^3$ ($C_2\times D_4$) x 2, $C_2\times C_2^5.C_2^3$ ($C_2\times D_4$) x 2, $C_2\times C_2^5.C_2^3$ ($C_2^4$) x 2, $C_2\times C_2^5.C_2^3$ ($C_2\times D_4$) x 2, $C_2^6.C_2^3$ ($C_2\times D_4$) x 2, $C_2\times C_2^5.C_2^3$ ($C_2^4$), $C_2\times C_2^5.C_2^3$ ($C_2^4$), $C_2^5.C_2^4$ ($C_2^4$), $C_2\times C_2^5.C_2^3$ ($C_2\times D_4$), $C_2\times C_2^5.C_2^3$ ($C_2\times D_4$), $C_2\times C_2^5.C_2^3$ ($C_2\times D_4$), $C_2^3.C_2^6$ ($C_2\times D_4$), $C_2^2\times C_2^5.C_2^2$ ($C_2\times D_4$), $C_2^6.C_2^3$ ($C_2\times D_4$), $C_2^4.C_2^5$ ($C_2\times D_4$), $C_2\times C_2^4.C_2^4$ ($C_2^4$), $C_2\times C_2^5.C_2^3$ ($C_2\times D_4$), $C_2\times C_2^5.C_2^3$ ($C_2^4$), $C_2\times C_2^4.C_2^4$ ($C_2^4$), $C_2^4.C_2^5$ ($C_2\times D_4$), $C_2^5.C_2^4$ ($C_2\times D_4$), $C_2^6.C_2^3$ ($C_2\times D_4$), $C_2^5.C_2^4$ ($C_2\times D_4$), $C_2^6.C_2^3$ ($C_2\times D_4$), $C_2^6.C_2^3$ ($C_2\times D_4$), $C_2^6.C_2^3$ ($C_2\times D_4$), $C_2^5.C_2^4$ ($C_2\times D_4$), $C_2^6.C_2^3$ ($C_2^4$), $C_2^6.C_2^3$ ($C_2\times D_4$), $C_2^6.C_2^3$ ($C_2\times D_4$), $C_2^6.C_2^3$ ($C_2\times D_4$), $C_2^6.C_2^3$ ($C_2\times D_4$), $C_2^6.C_2^3$ ($C_2^4$), $C_2^6.C_2^3$ ($C_2\times D_4$), $C_2^6.C_2^3$ ($C_2\times D_4$), $C_2\times C_2^5.C_2^3$ ($C_2\times D_4$), $C_2\times C_2^5.C_2^3$ ($C_2^4$), $C_2\times C_2^5.C_2^3$ ($C_2\times D_4$), $C_2^6.C_2^3$ ($C_2^4$), $C_2^6.C_2^3$ ($C_2\times D_4$)

Order 256: $C_2^6:C_4$ ($C_2^2\times D_4$) x 9, $C_2^4.C_4^2$ ($C_2^2\times D_4$) x 8, $C_2^5.D_4$ ($C_2^2\wr C_2$) x 4, $C_2^5.Q_8$ ($D_4:C_2^2$) x 4, $C_2^5.Q_8$ ($C_2^2\times D_4$) x 4, $C_2^5:C_8$ ($C_2^2\times D_4$) x 4, $C_2^3.C_2^5$ ($C_2^2\times D_4$) x 4, $C_2^5:D_4$ ($C_2^2\times D_4$) x 3, $C_2^5.D_4$ ($C_2^2\wr C_2$) x 2, $C_2^5.D_4$ ($C_2^2\times D_4$) x 2, $C_2^4:C_4^2$ ($C_2^2\times D_4$) x 2, $C_2^4.C_4^2$ ($C_2^2\times D_4$) x 2, $C_2^4.C_4^2$ ($C_2^2\wr C_2$) x 2, $C_2^4.C_4^2$ ($C_2^2\times D_4$) x 2, $C_2^5.Q_8$ ($C_2^2\times D_4$) x 2, $C_2^5.D_4$ ($C_2^2\times D_4$) x 2, $C_2^5:D_4$ ($C_2^2\times D_4$) x 2, $C_2^6:C_4$ ($D_4:C_2^2$) x 2, $C_2^6:C_4$ ($C_2^2\times D_4$) x 2, $C_2^5.D_4$ ($C_2^2\times D_4$) x 2, $C_2^5.D_4$ ($C_2^2\wr C_2$) x 2, $C_2^6:C_4$ ($C_2^2\wr C_2$) x 2, $D_4:C_2^5$ ($C_2^2\times D_4$), $C_2^5:D_4$ ($C_2^2\times D_4$), $C_2^5:D_4$ ($C_2^2\wr C_2$), $C_2^5:D_4$ ($C_2^2\wr C_2$), $C_2^5:D_4$ ($C_2^2\times D_4$), $C_2^5.D_4$ ($C_2^2\times D_4$), $C_2^5.D_4$ ($C_2^2\wr C_2$), $C_2^3.C_2^5$ ($C_2^2\times D_4$), $C_2^5.D_4$ ($C_2^2\times D_4$), $C_2^5.D_4$ ($C_2^2\wr C_2$), $C_2^6:C_4$ ($C_2^5$)

Order 128: $C_2^5:C_4$ ($C_2^3:D_4$) x 6, $C_2^3.C_4^2$ ($D_4^2$) x 4, $C_2^4.Q_8$ ($D_4^2$) x 4, $C_2^5:C_4$ ($D_4:C_2^3$) x 3, $D_4\times C_2^4$ ($C_2^3:D_4$) x 3, $C_2^5:C_4$ ($D_4^2$) x 2, $C_2^4:D_4$ ($D_8:C_2^2$) x 2, $C_2^4:D_4$ ($D_4^2$) x 2, $C_2^5:C_4$ ($C_2^3:D_4$) x 2, $C_2^5:C_4$ ($C_2\wr C_2^2$) x 2, $C_2^5:C_4$ ($D_4\times C_2^3$) x 2, $C_2^5:C_4$ ($D_4\times C_2^3$), $C_2^7$ ($D_8:C_2^2$), $D_4\times C_2^4$ ($D_4\times C_2^3$), $D_4\times C_2^4$ ($C_2\wr C_2^2$), $C_2^5\times C_4$ ($C_2^3:D_4$), $C_2^4.D_4$ ($D_8:C_2^2$), $C_2^4.D_4$ ($C_2\wr C_2^2$), $C_2^5:C_4$ ($D_8:C_2^2$), $C_2^3\times C_4^2$ ($D_8:C_2^2$)

Order 64: $C_2^4\times C_4$ ($C_2^4:D_4$) x 4, $C_2^4:C_4$ ($D_4^2:C_2$) x 2, $C_2^4:C_4$ ($C_2\times D_4^2$) x 2, $C_2^6$ ($\SD_{16}:D_4$) x 2, $C_2^6$ ($C_2^4:D_4$) x 2, $C_2^4:C_4$ ($\SD_{16}:D_4$) x 2, $C_2^6$ ($D_8:C_2^3$), $C_2^6$ ($C_2^4:D_4$), $C_2^6$ ($C_4^2:C_2^3$), $D_4\times C_2^3$ ($D_8:C_2^3$), $D_4\times C_2^3$ ($D_8:C_2^3$), $D_4\times C_2^3$ ($D_4^2:C_2$), $D_4\times C_2^3$ ($C_2\times D_4^2$), $C_2^4\times C_4$ ($D_8:C_2^3$), $C_2^4\times C_4$ ($C_4^2:C_2^3$)

Order 32: $C_2^3\times C_4$ ($D_4^2:C_2^2$) x 4, $C_2^5$ ($C_4^2.C_2^4$), $C_2^5$ ($C_4.D_4^2$), $C_2^5$ ($C_2^5:C_2^3$), $C_2^5$ ($C_2^5:D_4$), $C_2^5$ ($C_4^2.C_2^4$), $C_2^5$ ($C_4^2:C_2^4$), $C_2^5$ ($(C_2\times D_8):C_2^3$), $C_2^5$ ($C_2^5.D_4$), $C_2^5$ ($C_2^2:D_4^2$), $C_2^3\times C_4$ ($D_4^2:C_2^2$), $C_2^3\times C_4$ ($D_4^2:C_2^2$)

Order 16: $C_2^4$ ($C_2^3.C_2^5.C_2$) x 2, $C_2^4$ ($D_4^2:C_2^3$) x 2, $C_2^4$ ($C_2^3:D_4^2$), $C_2^4$ ($C_2^3.C_2^5.C_2$), $C_2^4$ ($C_2^3.C_2^5.C_2$), $C_2^4$ ($C_2^3.C_2^5.C_2$), $C_2^4$ ($C_2^3.C_2^5.C_2$), $C_2^4$ ($C_2^3.C_2^5.C_2$), $C_2^4$ ($C_2^3.C_2^5.C_2$)

Order 8: $C_2^3$ ($C_2^3.C_2^5.C_2^2$) x 6, $C_2^3$ ($C_2^4.C_2^4.C_2^2$)

Order 4: $C_2^2$ ($C_2^5.C_2.C_2^5$) x 2, $C_2^2$ ($C_2^4.C_2^5.C_2^2$)

Order 2: $C_2$ ($C_2^5.C_2^5.C_2^2$) x 3

Order 1: $C_1$ ($C_2^5:C_4.D_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Character theory

Complex character table

See the $119 \times 119$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $113 \times 113$ rational character table.