C2^2xC16 - GroupNames

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

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

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

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

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

C₂²×C₁₆ is a maximal subgroup of
C₂².₇M₅(2) M₅(2)⋊₇C₄ C₈.₇C₄² C₈.₈C₄² C₈.₉C₄² C₂²⋊C₃₂ C₁₆○₂M₅(2) (C₂×D₄).₅C₈ C₂³.₂₄D₈ C₄⋊C₄.₇C₈ C₂³.₂₅D₈ C₁₆⋊₉D₄ C₁₆⋊₇D₄ C₁₆.₁₉D₄ C₁₆⋊₈D₄
C₂²×C₁₆ is a maximal quotient of
C₄².₁₃C₈ D₄○C₃₂

64 conjugacy classes

class	1	2A	···	2G	4A	···	4H	8A	···	8P	16A	···	16AF
order	1	2	···	2	4	···	4	8	···	8	16	···	16
size	1	1	···	1	1	···	1	1	···	1	1	···	1

64 irreducible representations

dim	1	1	1	1	1	1	1	1
type	+	+	+
image	C₁	C₂	C₂	C₄	C₄	C₈	C₈	C₁₆
kernel	C₂²×C₁₆	C₂×C₁₆	C₂²×C₈	C₂×C₈	C₂²×C₄	C₂×C₄	C₂³	C₂²
# reps	1	6	1	6	2	12	4	32

Matrix representation of C₂²×C₁₆ ►in GL₃(𝔽₁₇) generated by

16	0	0
0	16	0
0	0	16

1	0	0
0	1	0
0	0	16

5	0	0
0	9	0
0	0	1

G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,1,0,0,0,16],[5,0,0,0,9,0,0,0,1] >;

C₂²×C₁₆ in GAP, Magma, Sage, TeX

C_2^2\times C_{16}

% in TeX

G:=Group("C2^2xC16");

// GroupNames label

G:=SmallGroup(64,183);

// by ID

G=gap.SmallGroup(64,183);

# by ID

G:=PCGroup([6,-2,2,2,-2,-2,-2,48,69,88]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₂²×C₁₆ in TeX

G = C22×C16 order 64 = 26

Abelian group of type [2,2,16]

G = C₂²×C₁₆ order 64 = 2⁶