C4:Q16 - GroupNames

G = C₄⋊Q₁₆ order 64 = 2⁶

The semidirect product of C₄ and Q₁₆ acting via Q₁₆/C₈=C₂

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₄⋊₁Q₁₆, C₈.₈D₄, C₄².₈₁C₂², (C₄×C₈).₇C₂, C₄.₃(C₂×D₄), C₄⋊Q₈.₉C₂, (C₂×C₄).₇₈D₄, (C₂×Q₁₆).₃C₂, C₂.₁₀(C₂×Q₁₆), C₂.₇(C₄⋊₁D₄), (C₂×C₈).₇₉C₂², (C₂×C₄).₁₁₉C₂³, C₂².₁₁₅(C₂×D₄), (C₂×Q₈).₂₅C₂², SmallGroup(64,175)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂×C₄ — C₄⋊Q₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₄² — C₄×C₈ — C₄⋊Q₁₆

Lower central

C₁ — C₂ — C₂×C₄ — C₄⋊Q₁₆

Upper central

C₁ — C₂² — C₄² — C₄⋊Q₁₆

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄⋊Q₁₆

Generators and relations for C₄⋊Q₁₆
G = < a,b,c | a⁴=b⁸=1, c²=b⁴, ab=ba, cac^-1=a^-1, cbc^-1=b^-1 >

Subgroups: 97 in 61 conjugacy classes, 33 normal (7 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₄², C₄⋊C₄, C₂×C₈, Q₁₆, C₂×Q₈, C₄×C₈, C₄⋊Q₈, C₂×Q₁₆, C₄⋊Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₄⋊₁D₄, C₂×Q₁₆, C₄⋊Q₁₆

Character table of C₄⋊Q₁₆

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	1	1	2	2	2	2	2	2	8	8	8	8	2	2	2	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	1	-1	1	-1	1	-1	1	-1	-1	-1	1	-1	1	-1	1	1	linear of order 2
ρ₃	1	1	1	1	-1	-1	1	-1	1	-1	1	-1	-1	1	1	1	-1	1	-1	1	-1	-1	linear of order 2
ρ₄	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₅	1	1	1	1	-1	-1	1	-1	1	-1	-1	1	1	-1	1	1	-1	1	-1	1	-1	-1	linear of order 2
ρ₆	1	1	1	1	1	1	1	1	1	1	-1	-1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₇	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	1	1	1	1	linear of order 2
ρ₈	1	1	1	1	-1	-1	1	-1	1	-1	-1	1	-1	1	-1	-1	1	-1	1	-1	1	1	linear of order 2
ρ₉	2	-2	-2	2	0	0	-2	0	2	0	0	0	0	0	2	2	0	-2	0	-2	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	-2	2	0	0	2	0	-2	0	0	0	0	0	0	0	2	0	2	0	-2	-2	orthogonal lifted from D₄
ρ₁₁	2	-2	-2	2	0	0	-2	0	2	0	0	0	0	0	-2	-2	0	2	0	2	0	0	orthogonal lifted from D₄
ρ₁₂	2	-2	-2	2	0	0	2	0	-2	0	0	0	0	0	0	0	-2	0	-2	0	2	2	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	2	-2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	-2	2	-2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	-2	-2	0	2	0	-2	0	0	0	0	0	0	-√2	√2	-√2	-√2	√2	√2	-√2	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₆	2	-2	2	-2	2	0	0	0	0	-2	0	0	0	0	√2	-√2	√2	-√2	-√2	√2	-√2	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₇	2	2	-2	-2	0	2	0	-2	0	0	0	0	0	0	√2	-√2	√2	√2	-√2	-√2	√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₈	2	-2	2	-2	2	0	0	0	0	-2	0	0	0	0	-√2	√2	-√2	√2	√2	-√2	√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₉	2	-2	2	-2	-2	0	0	0	0	2	0	0	0	0	√2	-√2	-√2	-√2	√2	√2	√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₂₀	2	2	-2	-2	0	-2	0	2	0	0	0	0	0	0	-√2	√2	√2	-√2	-√2	√2	√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₂₁	2	2	-2	-2	0	-2	0	2	0	0	0	0	0	0	√2	-√2	-√2	√2	√2	-√2	-√2	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₂₂	2	-2	2	-2	-2	0	0	0	0	2	0	0	0	0	-√2	√2	√2	√2	-√2	-√2	-√2	√2	symplectic lifted from Q₁₆, Schur index 2

Smallest permutation representation of C₄⋊Q₁₆
►Regular action on 64 points

Generators in S₆₄

(1 33 31 14)(2 34 32 15)(3 35 25 16)(4 36 26 9)(5 37 27 10)(6 38 28 11)(7 39 29 12)(8 40 30 13)(17 42 49 58)(18 43 50 59)(19 44 51 60)(20 45 52 61)(21 46 53 62)(22 47 54 63)(23 48 55 64)(24 41 56 57)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(1 45 5 41)(2 44 6 48)(3 43 7 47)(4 42 8 46)(9 49 13 53)(10 56 14 52)(11 55 15 51)(12 54 16 50)(17 40 21 36)(18 39 22 35)(19 38 23 34)(20 37 24 33)(25 59 29 63)(26 58 30 62)(27 57 31 61)(28 64 32 60)

G:=sub<Sym(64)| (1,33,31,14)(2,34,32,15)(3,35,25,16)(4,36,26,9)(5,37,27,10)(6,38,28,11)(7,39,29,12)(8,40,30,13)(17,42,49,58)(18,43,50,59)(19,44,51,60)(20,45,52,61)(21,46,53,62)(22,47,54,63)(23,48,55,64)(24,41,56,57), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,45,5,41)(2,44,6,48)(3,43,7,47)(4,42,8,46)(9,49,13,53)(10,56,14,52)(11,55,15,51)(12,54,16,50)(17,40,21,36)(18,39,22,35)(19,38,23,34)(20,37,24,33)(25,59,29,63)(26,58,30,62)(27,57,31,61)(28,64,32,60)>;

G:=Group( (1,33,31,14)(2,34,32,15)(3,35,25,16)(4,36,26,9)(5,37,27,10)(6,38,28,11)(7,39,29,12)(8,40,30,13)(17,42,49,58)(18,43,50,59)(19,44,51,60)(20,45,52,61)(21,46,53,62)(22,47,54,63)(23,48,55,64)(24,41,56,57), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,45,5,41)(2,44,6,48)(3,43,7,47)(4,42,8,46)(9,49,13,53)(10,56,14,52)(11,55,15,51)(12,54,16,50)(17,40,21,36)(18,39,22,35)(19,38,23,34)(20,37,24,33)(25,59,29,63)(26,58,30,62)(27,57,31,61)(28,64,32,60) );

G=PermutationGroup([[(1,33,31,14),(2,34,32,15),(3,35,25,16),(4,36,26,9),(5,37,27,10),(6,38,28,11),(7,39,29,12),(8,40,30,13),(17,42,49,58),(18,43,50,59),(19,44,51,60),(20,45,52,61),(21,46,53,62),(22,47,54,63),(23,48,55,64),(24,41,56,57)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,45,5,41),(2,44,6,48),(3,43,7,47),(4,42,8,46),(9,49,13,53),(10,56,14,52),(11,55,15,51),(12,54,16,50),(17,40,21,36),(18,39,22,35),(19,38,23,34),(20,37,24,33),(25,59,29,63),(26,58,30,62),(27,57,31,61),(28,64,32,60)]])

C₄⋊Q₁₆ is a maximal subgroup of
C₈.₂₅D₈ C₈.₂₇D₈ (C₂×Q₈).D₄ D₄⋊Q₁₆ Q₈⋊Q₁₆ C₄².₁₉₅C₂³ C₈.₂₈D₈ Q₈⋊₁Q₁₆ C₈.D₈ C₈.SD₁₆ C₈²⋊₅C₂ C₈.₉SD₁₆ C₄².₆₆₇C₂³ C₈.₂D₈ D₈.₄D₄ Q₁₆.₄D₄ C₈.₅D₈ C₄.SD₃₂ C₈.₁₄SD₁₆ C₁₆⋊₅D₄ C₄².₃₆₀D₄ M₄(2)⋊₈D₄ M₄(2).₂₀D₄ C₄².₃₆₇D₄ C₄².₃₈₉C₂³ C₄².₂₆₂D₄ C₄².₂₆₇D₄ C₄².₂₆₈D₄ C₄².₄₀₉C₂³ SD₁₆⋊₃D₄ D₈⋊₁₃D₄ D₄×Q₁₆ D₈○Q₁₆ C₄².₅₂₇C₂³ Q₈⋊₆Q₁₆ C₄².₇₃C₂³
C_4p⋊Q₁₆: C₈⋊₅Q₁₆ C₈⋊₄Q₁₆ C₈⋊₃Q₁₆ C₁₂⋊₄Q₁₆ C₁₂⋊₃Q₁₆ C₂₀⋊₄Q₁₆ C₂₀⋊₃Q₁₆ C₂₈⋊₄Q₁₆ ...
C_8p.D₄: C₄⋊Q₃₂ C₈.₇D₈ C₂₄.₂₆D₄ C₄₀.₂₆D₄ C₅₆.₂₆D₄ ...
C₄⋊Q₁₆ is a maximal quotient of
C_4p⋊Q₁₆: C₈⋊₅Q₁₆ C₈⋊₄Q₁₆ C₈⋊₃Q₁₆ C₁₂⋊₄Q₁₆ C₁₂⋊₃Q₁₆ C₂₀⋊₄Q₁₆ C₂₀⋊₃Q₁₆ C₂₈⋊₄Q₁₆ ...
(C₂×C₈).D_2p: C₈.₇Q₁₆ C₄².₅₉Q₈ C₄².₄₃₁D₄ (C₂×C₄)⋊₆Q₁₆ (C₂×C₄)⋊₂Q₁₆ (C₂×C₈).₆₀D₄ C₂₄.₂₆D₄ C₄₀.₂₆D₄ ...

Matrix representation of C₄⋊Q₁₆ ►in GL₄(𝔽₁₇) generated by

0	16	0	0
1	0	0	0
0	0	16	0
0	0	0	16

0	1	0	0
16	0	0	0
0	0	8	0
0	0	9	15

11	13	0	0
13	6	0	0
0	0	12	15
0	0	13	5

G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,16,0,0,0,0,16],[0,16,0,0,1,0,0,0,0,0,8,9,0,0,0,15],[11,13,0,0,13,6,0,0,0,0,12,13,0,0,15,5] >;

C₄⋊Q₁₆ in GAP, Magma, Sage, TeX

C_4\rtimes Q_{16}

% in TeX

G:=Group("C4:Q16");

// GroupNames label

G:=SmallGroup(64,175);

// by ID

G=gap.SmallGroup(64,175);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,247,362,86,963,117]);

// Polycyclic

G:=Group<a,b,c|a^4=b^8=1,c^2=b^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;

// generators/relations

Export

Character table of C₄⋊Q₁₆ in TeX

G = C4⋊Q16 order 64 = 26

The semidirect product of C4 and Q16 acting via Q16/C8=C2

G = C₄⋊Q₁₆ order 64 = 2⁶

The semidirect product of C₄ and Q₁₆ acting via Q₁₆/C₈=C₂