C2xC12.10D4 - GroupNames

G = C₂xC₁₂.₁₀D₄ order 192 = 2⁶·3

Direct product of C₂ and C₁₂.₁₀D₄

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

Aliases: C₂xC₁₂.₁₀D₄, (C₆xQ₈).₁₀C₄, C₁₂.₂₀₉(C₂xD₄), (C₂xC₁₂).₁₉₄D₄, (C₂xQ₈).₁₉₁D₆, (C₂xQ₈).₉Dic₃, (C₂²xQ₈).₉S₃, C₆:₂(C₄.₁₀D₄), (C₂²xC₁₂).₁₀C₄, (C₂²xC₄).₁₇₁D₆, (C₂²xC₄).₈Dic₃, C₁₂.₃₅(C₂²:C₄), (C₂xC₁₂).₄₇₆C₂³, (C₆xQ₈).₂₀₂C₂², C₂³.₃₈(C₂xDic₃), C₄.₁₂(C₆.D₄), C₄.Dic₃.₄₆C₂², C₂².₇(C₂²xDic₃), (C₂²xC₁₂).₂₀₂C₂², C₂².₃₆(C₆.D₄), (Q₈xC₂xC₆).₃C₂, C₃:₃(C₂xC₄.₁₀D₄), C₄.₉₃(C₂xC₃:D₄), C₆.₇₉(C₂xC₂²:C₄), (C₂xC₁₂).₁₂₁(C₂xC₄), (C₂xC₄).₂₆(C₂xDic₃), (C₂xC₄).₁₉₉(C₃:D₄), (C₂xC₆).₁₉₈(C₂²xC₄), (C₂²xC₆).₁₃₇(C₂xC₄), (C₂xC₄).₁₃₀(C₂²xS₃), C₂.₁₅(C₂xC₆.D₄), (C₂xC₄.Dic₃).₂₈C₂, (C₂xC₆).₁₁₆(C₂²:C₄), SmallGroup(192,785)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₂xC₁₂.₁₀D₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂xC₁₂ — C₄.Dic₃ — C₂xC₄.Dic₃ — C₂xC₁₂.₁₀D₄

Lower central

C₃ — C₆ — C₂xC₆ — C₂xC₁₂.₁₀D₄

Upper central

C₁ — C₂² — C₂²xC₄ — C₂²xQ₈

Generators and relations for C₂xC₁₂.₁₀D₄
G = < a,b,c,d | a²=b¹²=1, c⁴=b⁶, d²=b⁹, ab=ba, ac=ca, ad=da, cbc^-1=b^-1, dbd^-1=b⁵, dcd^-1=b⁹c³ >

Subgroups: 264 in 146 conjugacy classes, 71 normal (21 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₆, C₈, C₂xC₄, C₂xC₄, C₂xC₄, Q₈, C₂³, C₁₂, C₁₂, C₂xC₆, C₂xC₆, C₂xC₈, M₄(2), C₂²xC₄, C₂²xC₄, C₂xQ₈, C₂xQ₈, C₃:C₈, C₂xC₁₂, C₂xC₁₂, C₂xC₁₂, C₃xQ₈, C₂²xC₆, C₄.₁₀D₄, C₂xM₄(2), C₂²xQ₈, C₂xC₃:C₈, C₄.Dic₃, C₄.Dic₃, C₂²xC₁₂, C₂²xC₁₂, C₆xQ₈, C₆xQ₈, C₂xC₄.₁₀D₄, C₁₂.₁₀D₄, C₂xC₄.Dic₃, Q₈xC₂xC₆, C₂xC₁₂.₁₀D₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, C₂³, Dic₃, D₆, C₂²:C₄, C₂²xC₄, C₂xD₄, C₂xDic₃, C₃:D₄, C₂²xS₃, C₄.₁₀D₄, C₂xC₂²:C₄, C₆.D₄, C₂²xDic₃, C₂xC₃:D₄, C₂xC₄.₁₀D₄, C₁₂.₁₀D₄, C₂xC₆.D₄, C₂xC₁₂.₁₀D₄

Smallest permutation representation of C₂xC₁₂.₁₀D₄
►On 96 points

Generators in S₉₆

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

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

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	4D	4E	4F	4G	4H	6A	···	6G	8A	···	8H	12A	···	12L
order	1	2	2	2	2	2	3	4	4	4	4	4	4	4	4	6	···	6	8	···	8	12	···	12
size	1	1	1	1	2	2	2	2	2	2	2	4	4	4	4	2	···	2	12	···	12	4	···	4

42 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4
type	+	+	+	+			+	+	-	+	-	+		-
image	C₁	C₂	C₂	C₂	C₄	C₄	S₃	D₄	Dic₃	D₆	Dic₃	D₆	C₃:D₄	C₄.₁₀D₄	C₁₂.₁₀D₄
kernel	C₂xC₁₂.₁₀D₄	C₁₂.₁₀D₄	C₂xC₄.Dic₃	Q₈xC₂xC₆	C₂²xC₁₂	C₆xQ₈	C₂²xQ₈	C₂xC₁₂	C₂²xC₄	C₂²xC₄	C₂xQ₈	C₂xQ₈	C₂xC₄	C₆	C₂
# reps	1	4	2	1	4	4	1	4	2	1	2	2	8	2	4

Matrix representation of C₂xC₁₂.₁₀D₄ ►in GL₆(F₇₃)

72	0	0	0	0	0
0	72	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	72	0
0	0	0	0	0	72

0	72	0	0	0	0
1	72	0	0	0	0
0	0	0	1	0	0
0	0	72	0	0	0
0	0	0	0	0	72
0	0	0	0	1	0

70	45	0	0	0	0
42	3	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	72	0	0	0

70	45	0	0	0	0
42	3	0	0	0	0
0	0	0	0	50	45
0	0	0	0	45	23
0	0	45	23	0	0
0	0	23	28	0	0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0],[70,42,0,0,0,0,45,3,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[70,42,0,0,0,0,45,3,0,0,0,0,0,0,0,0,45,23,0,0,0,0,23,28,0,0,50,45,0,0,0,0,45,23,0,0] >;

C₂xC₁₂.₁₀D₄ in GAP, Magma, Sage, TeX

C_2\times C_{12}._{10}D_4

% in TeX

G:=Group("C2xC12.10D4");

// GroupNames label

G:=SmallGroup(192,785);

// by ID

G=gap.SmallGroup(192,785);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,184,297,136,1684,6278]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^9*c^3>;

// generators/relations

G = C2xC12.10D4 order 192 = 26·3

Direct product of C2 and C12.10D4

G = C₂xC₁₂.₁₀D₄ order 192 = 2⁶·3

Direct product of C₂ and C₁₂.₁₀D₄