C80:2C4 - GroupNames

(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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)

(1 16)(2 63 50 79)(3 30 19 62)(4 77 68 45)(5 44 37 28)(6 11)(7 58 55 74)(8 25 24 57)(9 72 73 40)(10 39 42 23)(12 53 60 69)(13 20 29 52)(14 67 78 35)(15 34 47 18)(17 48 65 64)(21 76)(22 43 70 59)(26 71)(27 38 75 54)(31 66)(32 33 80 49)(36 61)(41 56)(46 51)

G:=sub<Sym(80)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,16)(2,63,50,79)(3,30,19,62)(4,77,68,45)(5,44,37,28)(6,11)(7,58,55,74)(8,25,24,57)(9,72,73,40)(10,39,42,23)(12,53,60,69)(13,20,29,52)(14,67,78,35)(15,34,47,18)(17,48,65,64)(21,76)(22,43,70,59)(26,71)(27,38,75,54)(31,66)(32,33,80,49)(36,61)(41,56)(46,51)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,16)(2,63,50,79)(3,30,19,62)(4,77,68,45)(5,44,37,28)(6,11)(7,58,55,74)(8,25,24,57)(9,72,73,40)(10,39,42,23)(12,53,60,69)(13,20,29,52)(14,67,78,35)(15,34,47,18)(17,48,65,64)(21,76)(22,43,70,59)(26,71)(27,38,75,54)(31,66)(32,33,80,49)(36,61)(41,56)(46,51) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,16),(2,63,50,79),(3,30,19,62),(4,77,68,45),(5,44,37,28),(6,11),(7,58,55,74),(8,25,24,57),(9,72,73,40),(10,39,42,23),(12,53,60,69),(13,20,29,52),(14,67,78,35),(15,34,47,18),(17,48,65,64),(21,76),(22,43,70,59),(26,71),(27,38,75,54),(31,66),(32,33,80,49),(36,61),(41,56),(46,51)]])

38 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	4D	4E	4F	5	8A	8B	8C	8D	10	16A	16B	16C	16D	16E	16F	16G	16H	20A	20B	40A	40B	40C	40D	80A	···	80H
order	1	2	2	2	4	4	4	4	4	4	5	8	8	8	8	10	16	16	16	16	16	16	16	16	20	20	40	40	40	40	80	···	80
size	1	1	5	5	2	10	40	40	40	40	4	2	2	10	10	4	2	2	2	2	10	10	10	10	4	4	4	4	4	4	4	···	4

38 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4
type	+	+	+			-	+	-	+	+	-	+	+
image	C₁	C₂	C₂	C₄	C₄	Q₈	D₄	Q₁₆	D₈	D₁₆	Q₃₂	F₅	C₂×F₅	C₄⋊F₅	D₅.D₈	C₈₀⋊₂C₄
kernel	C₈₀⋊₂C₄	D₅×C₁₆	D₅.D₈	C₅⋊₂C₁₆	C₈₀	C₅⋊₂C₈	C₄×D₅	Dic₅	D₁₀	D₅	D₅	C₁₆	C₈	C₄	C₂	C₁
# reps	1	1	2	2	2	1	1	2	2	4	4	1	1	2	4	8

Matrix representation of C₈₀⋊₂C₄ ►in GL₄(𝔽₂₄₁) generated by

195	62	195	0
0	195	62	195
46	46	0	108
133	179	179	133

0	195	62	195
108	0	46	46
46	46	0	108
195	62	195	0

G:=sub<GL(4,GF(241))| [195,0,46,133,62,195,46,179,195,62,0,179,0,195,108,133],[0,108,46,195,195,0,46,62,62,46,0,195,195,46,108,0] >;

C₈₀⋊₂C₄ in GAP, Magma, Sage, TeX

C_{80}\rtimes_2C_4

% in TeX

G:=Group("C80:2C4");

// GroupNames label

G:=SmallGroup(320,187);

// by ID

G=gap.SmallGroup(320,187);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,675,192,1684,102,6278,3156]);

// Polycyclic

G:=Group<a,b|a^80=b^4=1,b*a*b^-1=a^63>;

// generators/relations

Export

Subgroup lattice of C₈₀⋊₂C₄ in TeX

G = C80⋊2C4 order 320 = 26·5

2nd semidirect product of C80 and C4 acting faithfully

G = C₈₀⋊₂C₄ order 320 = 2⁶·5

2^nd semidirect product of C₈₀ and C₄ acting faithfully