C2^2.D16 - GroupNames

G = C₂².D₁₆ order 128 = 2⁷

3^rd non-split extension by C₂² of D₁₆ acting via D₁₆/D₈=C₂

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C₂².₄D₁₆, C₂³.₄₈D₈, C₁₆⋊₃C₄⋊₆C₂, (C₂×C₈).₇₃D₄, (C₂×C₄).₄₁D₈, C₂.₈(C₂×D₁₆), C₂.D₁₆⋊₇C₂, C₂²⋊C₁₆⋊₆C₂, C₈⋊₇D₄.₅C₂, C₈.₆₉(C₄○D₄), (C₂×D₈).₉C₂², (C₂×C₁₆).₁₀C₂², (C₂×C₈).₅₃₂C₂³, C₂².₁₁₈(C₂×D₈), (C₂²×C₄).₃₅₂D₄, C₂.D₈.₁₇C₂², C₂.₁₇(Q₃₂⋊C₂), C₄.₁₄(C₈.C₂²), (C₂²×C₈).₁₂₉C₂², C₄.₃₉(C₂².D₄), C₂.₁₂(C₂².D₈), (C₂×C₂.D₈)⋊₁₆C₂, (C₂×C₄).₈₀₀(C₂×D₄), SmallGroup(128,964)

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

Derived series

C₁ — C₂×C₈ — C₂².D₁₆

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₂.D₈ — C₂×C₂.D₈ — C₂².D₁₆

Lower central

C₁ — C₂ — C₄ — C₂×C₈ — C₂².D₁₆

Upper central

C₁ — C₂² — C₂²×C₄ — C₂²×C₈ — C₂².D₁₆

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — C₂².D₁₆

Generators and relations for C₂².D₁₆
G = < a,b,c,d | a²=b²=c¹⁶=d²=1, cac^-1=dad=ab=ba, bc=cb, bd=db, dcd=bc^-1 >

Subgroups: 212 in 75 conjugacy classes, 32 normal (20 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₁₆, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, D₈, C₂²×C₄, C₂²×C₄, C₂×D₄, D₄⋊C₄, C₂.D₈, C₂.D₈, C₂.D₈, C₂×C₁₆, C₂×C₄⋊C₄, C₄⋊D₄, C₂²×C₈, C₂×D₈, C₂²⋊C₁₆, C₂.D₁₆, C₁₆⋊₃C₄, C₂×C₂.D₈, C₈⋊₇D₄, C₂².D₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, D₈, C₂×D₄, C₄○D₄, D₁₆, C₂².D₄, C₂×D₈, C₈.C₂², C₂².D₈, C₂×D₁₆, Q₃₂⋊C₂, C₂².D₁₆

Character table of C₂².D₁₆

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	1	1	2	2	16	2	2	4	8	8	8	8	16	2	2	2	2	4	4	4	4	4	4	4	4	4	4
ρ₁	1	1	1	1	1	1	1	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	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	-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	-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	-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	-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	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	1	1	-1	-1	1	-1	-1	linear of order 2
ρ₉	2	2	2	2	-2	-2	0	2	2	-2	0	0	0	0	0	-2	-2	-2	-2	2	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	2	2	0	2	2	2	0	0	0	0	0	-2	-2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	-2	0	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	-√2	√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₂	2	2	2	2	2	2	0	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	-√2	√2	-√2	√2	orthogonal lifted from D₈
ρ₁₃	2	2	2	2	-2	-2	0	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	√2	-√2	√2	√2	-√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₄	2	2	2	2	2	2	0	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	√2	-√2	√2	-√2	orthogonal lifted from D₈
ρ₁₅	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	√2	-√2	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	orthogonal lifted from D₁₆
ρ₁₆	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	-√2	√2	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	orthogonal lifted from D₁₆
ρ₁₇	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	√2	-√2	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	orthogonal lifted from D₁₆
ρ₁₈	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	-√2	√2	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	orthogonal lifted from D₁₆
ρ₁₉	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	-√2	√2	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	orthogonal lifted from D₁₆
ρ₂₀	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	-√2	√2	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	orthogonal lifted from D₁₆
ρ₂₁	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	-√2	√2	-√2	√2	√2	-√2	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	orthogonal lifted from D₁₆
ρ₂₂	2	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	√2	-√2	√2	-√2	√2	-√2	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	ζ₁₆⁵-ζ₁₆³	ζ₁₆¹⁵-ζ₁₆⁹	-ζ₁₆⁵+ζ₁₆³	-ζ₁₆¹⁵+ζ₁₆⁹	orthogonal lifted from D₁₆
ρ₂₃	2	-2	-2	2	0	0	0	2	-2	0	0	-2i	0	2i	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₄	2	-2	-2	2	0	0	0	2	-2	0	2i	0	-2i	0	0	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₅	2	-2	-2	2	0	0	0	2	-2	0	-2i	0	2i	0	0	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₆	2	-2	-2	2	0	0	0	2	-2	0	0	2i	0	-2i	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₇	4	-4	-4	4	0	0	0	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₈	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	-2√2	-2√2	2√2	2√2	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₃₂⋊C₂, Schur index 2
ρ₂₉	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	2√2	2√2	-2√2	-2√2	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₃₂⋊C₂, Schur index 2

Smallest permutation representation of C₂².D₁₆
►On 64 points

Generators in S₆₄

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

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

(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)

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

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

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

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

Matrix representation of C₂².D₁₆ ►in GL₄(𝔽₁₇) generated by

1	0	0	0
0	1	0	0
0	0	16	15
0	0	0	1

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

13	11	0	0
6	13	0	0
0	0	13	0
0	0	4	4

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

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

C₂².D₁₆ in GAP, Magma, Sage, TeX

C_2^2.D_{16}

% in TeX

G:=Group("C2^2.D16");

// GroupNames label

G:=SmallGroup(128,964);

// by ID

G=gap.SmallGroup(128,964);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,58,1684,438,242,4037,1027,124]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^16=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,b*d=d*b,d*c*d=b*c^-1>;

// generators/relations

Export

Character table of C₂².D₁₆ in TeX

G = C22.D16 order 128 = 27

3rd non-split extension by C22 of D16 acting via D16/D8=C2

G = C₂².D₁₆ order 128 = 2⁷

3^rd non-split extension by C₂² of D₁₆ acting via D₁₆/D₈=C₂