C96:C2 - 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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)

(2 18)(3 35)(4 52)(5 69)(6 86)(8 24)(9 41)(10 58)(11 75)(12 92)(14 30)(15 47)(16 64)(17 81)(20 36)(21 53)(22 70)(23 87)(26 42)(27 59)(28 76)(29 93)(32 48)(33 65)(34 82)(38 54)(39 71)(40 88)(44 60)(45 77)(46 94)(50 66)(51 83)(56 72)(57 89)(62 78)(63 95)(68 84)(74 90)(80 96)

G:=sub<Sym(96)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (2,18)(3,35)(4,52)(5,69)(6,86)(8,24)(9,41)(10,58)(11,75)(12,92)(14,30)(15,47)(16,64)(17,81)(20,36)(21,53)(22,70)(23,87)(26,42)(27,59)(28,76)(29,93)(32,48)(33,65)(34,82)(38,54)(39,71)(40,88)(44,60)(45,77)(46,94)(50,66)(51,83)(56,72)(57,89)(62,78)(63,95)(68,84)(74,90)(80,96)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (2,18)(3,35)(4,52)(5,69)(6,86)(8,24)(9,41)(10,58)(11,75)(12,92)(14,30)(15,47)(16,64)(17,81)(20,36)(21,53)(22,70)(23,87)(26,42)(27,59)(28,76)(29,93)(32,48)(33,65)(34,82)(38,54)(39,71)(40,88)(44,60)(45,77)(46,94)(50,66)(51,83)(56,72)(57,89)(62,78)(63,95)(68,84)(74,90)(80,96) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(2,18),(3,35),(4,52),(5,69),(6,86),(8,24),(9,41),(10,58),(11,75),(12,92),(14,30),(15,47),(16,64),(17,81),(20,36),(21,53),(22,70),(23,87),(26,42),(27,59),(28,76),(29,93),(32,48),(33,65),(34,82),(38,54),(39,71),(40,88),(44,60),(45,77),(46,94),(50,66),(51,83),(56,72),(57,89),(62,78),(63,95),(68,84),(74,90),(80,96)]])

72 conjugacy classes

class	1	2A	2B	3	4A	4B	4C	6	8A	8B	8C	8D	8E	8F	12A	12B	16A	···	16H	16I	16J	16K	16L	24A	24B	24C	24D	32A	···	32H	32I	···	32P	48A	···	48H	96A	···	96P
order	1	2	2	3	4	4	4	6	8	8	8	8	8	8	12	12	16	···	16	16	16	16	16	24	24	24	24	32	···	32	32	···	32	48	···	48	96	···	96
size	1	1	6	2	1	1	6	2	1	1	1	1	6	6	2	2	1	···	1	6	6	6	6	2	2	2	2	2	···	2	6	···	6	2	···	2	2	···	2

72 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2
type	+	+	+	+							+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₈	C₈	C₁₆	C₁₆	S₃	D₆	C₄×S₃	S₃×C₈	M₆(2)	S₃×C₁₆	C₉₆⋊C₂
kernel	C₉₆⋊C₂	C₃⋊C₃₂	C₉₆	S₃×C₁₆	C₃⋊C₁₆	S₃×C₈	C₃⋊C₈	C₄×S₃	Dic₃	D₆	C₃₂	C₁₆	C₈	C₄	C₃	C₂	C₁
# reps	1	1	1	1	2	2	4	4	8	8	1	1	2	4	8	8	16

Matrix representation of C₉₆⋊C₂ ►in GL₂(𝔽₁₇) generated by

15	10
1	0

1	2
0	16

G:=sub<GL(2,GF(17))| [15,1,10,0],[1,0,2,16] >;

C₉₆⋊C₂ in GAP, Magma, Sage, TeX

C_{96}\rtimes C_2

% in TeX

G:=Group("C96:C2");

// GroupNames label

G:=SmallGroup(192,6);

// by ID

G=gap.SmallGroup(192,6);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,36,58,80,102,6278]);

// Polycyclic

G:=Group<a,b|a^96=b^2=1,b*a*b=a^17>;

// generators/relations

Export

Subgroup lattice of C₉₆⋊C₂ in TeX

G = C96⋊C2 order 192 = 26·3

6th semidirect product of C96 and C2 acting faithfully

G = C₉₆⋊C₂ order 192 = 2⁶·3

6^th semidirect product of C₉₆ and C₂ acting faithfully