C2xQ8.15D6 - GroupNames

G = C₂×Q₈.₁₅D₆ order 192 = 2⁶·3

Direct product of C₂ and Q₈.₁₅D₆

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂×Q₈.₁₅D₆, C₆.₁₀C₂⁵, D₆.₅C₂⁴, C₁₂.₄₅C₂⁴, C₆⋊₁2_-¹⁺⁴, D₁₂.₃₆C₂³, Dic₃.₅C₂⁴, Dic₆.₃₆C₂³, (C₂×Q₈)⋊₃₈D₆, C₂.₁₁(S₃×C₂⁴), C₄.₄₅(S₃×C₂³), (S₃×Q₈)⋊₁₃C₂², (C₆×Q₈)⋊₄₄C₂², (C₂²×Q₈)⋊₁₇S₃, C₃⋊D₄.₆C₂³, C₃⋊₁(C₂×2_-¹⁺⁴), C₄○D₁₂⋊₂₃C₂², (C₄×S₃).₁₈C₂³, (C₂×C₆).₃₃₀C₂⁴, (C₂²×C₄).₃₀₇D₆, C₂².₉(S₃×C₂³), Q₈.₃₉(C₂²×S₃), (C₃×Q₈).₂₉C₂³, (C₂×C₁₂).₅₆₆C₂³, Q₈⋊₃S₃⋊₁₂C₂², (C₂×D₁₂).₂₈₃C₂², (C₂²×C₆).₄₃₇C₂³, C₂³.₂₅₂(C₂²×S₃), (C₂²×S₃).₂₄₈C₂³, (C₂²×C₁₂).₃₀₂C₂², (C₂×Dic₃).₂₉₈C₂³, (C₂×Dic₆).₃₁₂C₂², (Q₈×C₂×C₆)⋊₁₁C₂, (C₂×S₃×Q₈)⋊₂₀C₂, (C₂×C₄○D₁₂)⋊₃₅C₂, (C₂×Q₈⋊₃S₃)⋊₂₀C₂, (S₃×C₂×C₄).₁₇₂C₂², (C₂×C₄).₂₅₂(C₂²×S₃), (C₂×C₃⋊D₄).₁₅₀C₂², SmallGroup(192,1519)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₂×Q₈.₁₅D₆

Chief series

C₁ — C₃ — C₆ — D₆ — C₂²×S₃ — S₃×C₂×C₄ — C₂×S₃×Q₈ — C₂×Q₈.₁₅D₆

Lower central

C₃ — C₆ — C₂×Q₈.₁₅D₆

Upper central

C₁ — C₂² — C₂²×Q₈

Generators and relations for C₂×Q₈.₁₅D₆
G = < a,b,c,d,e | a²=b⁴=1, c²=d⁶=e²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=ebe^-1=b^-1, bd=db, dcd^-1=ece^-1=b²c, ede^-1=d⁵ >

Subgroups: 1512 in 794 conjugacy classes, 447 normal (11 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₂³, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, Dic₆, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₃×Q₈, C₂²×S₃, C₂²×C₆, C₂²×Q₈, C₂²×Q₈, C₂×C₄○D₄, 2_-¹⁺⁴, C₂×Dic₆, S₃×C₂×C₄, C₂×D₁₂, C₄○D₁₂, S₃×Q₈, Q₈⋊₃S₃, C₂×C₃⋊D₄, C₂²×C₁₂, C₆×Q₈, C₂×2_-¹⁺⁴, C₂×C₄○D₁₂, C₂×S₃×Q₈, C₂×Q₈⋊₃S₃, Q₈.₁₅D₆, Q₈×C₂×C₆, C₂×Q₈.₁₅D₆
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₂⁴, C₂²×S₃, 2_-¹⁺⁴, C₂⁵, S₃×C₂³, C₂×2_-¹⁺⁴, Q₈.₁₅D₆, S₃×C₂⁴, C₂×Q₈.₁₅D₆

Smallest permutation representation of C₂×Q₈.₁₅D₆
►On 96 points

Generators in S₉₆

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

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

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

(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)

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	···	2M	3	4A	···	4L	4M	···	4T	6A	···	6G	12A	···	12L
order	1	2	2	2	2	2	2	···	2	3	4	···	4	4	···	4	6	···	6	12	···	12
size	1	1	1	1	2	2	6	···	6	2	2	···	2	6	···	6	2	···	2	4	···	4

54 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₆	D₆	2_-¹⁺⁴	Q₈.₁₅D₆
kernel	C₂×Q₈.₁₅D₆	C₂×C₄○D₁₂	C₂×S₃×Q₈	C₂×Q₈⋊₃S₃	Q₈.₁₅D₆	Q₈×C₂×C₆	C₂²×Q₈	C₂²×C₄	C₂×Q₈	C₆	C₂
# reps	1	6	4	4	16	1	1	3	12	2	4

Matrix representation of C₂×Q₈.₁₅D₆ ►in GL₆(𝔽₁₃)

12	0	0	0	0	0
0	12	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	12	0
0	0	0	0	0	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	7	12	4	0
0	0	1	6	0	4
0	0	4	0	6	1
0	0	0	4	12	7

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	12	0
0	0	0	0	0	12
0	0	1	0	0	0
0	0	0	1	0	0

1	12	0	0	0	0
1	0	0	0	0	0
0	0	3	10	8	8
0	0	3	0	5	3
0	0	8	8	10	3
0	0	5	3	10	0

1	0	0	0	0	0
1	12	0	0	0	0
0	0	2	9	0	0
0	0	11	11	0	0
0	0	0	0	11	4
0	0	0	0	2	2

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,1,4,0,0,0,12,6,0,4,0,0,4,0,6,12,0,0,0,4,1,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,12,0,0],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,3,3,8,5,0,0,10,0,8,3,0,0,8,5,10,10,0,0,8,3,3,0],[1,1,0,0,0,0,0,12,0,0,0,0,0,0,2,11,0,0,0,0,9,11,0,0,0,0,0,0,11,2,0,0,0,0,4,2] >;

C₂×Q₈.₁₅D₆ in GAP, Magma, Sage, TeX

C_2\times Q_8._{15}D_6

% in TeX

G:=Group("C2xQ8.15D6");

// GroupNames label

G:=SmallGroup(192,1519);

// by ID

G=gap.SmallGroup(192,1519);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,297,136,1684,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^6=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=e*c*e^-1=b^2*c,e*d*e^-1=d^5>;

// generators/relations

G = C2×Q8.15D6 order 192 = 26·3

Direct product of C2 and Q8.15D6

G = C₂×Q₈.₁₅D₆ order 192 = 2⁶·3

Direct product of C₂ and Q₈.₁₅D₆