C4^2.155D6 - GroupNames

G = C₄².₁₅₅D₆ order 192 = 2⁶·3

155^th non-split extension by C₄² of D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄².₁₅₅D₆, C₆.₁₃₄2₊¹⁺⁴, C₄⋊C₄.₁₁₃D₆, C₁₂⋊D₄⋊₃₄C₂, C₄⋊D₁₂⋊₁₅C₂, C₄².C₂⋊₁₁S₃, Dic₃⋊₅D₄⋊₃₈C₂, C₄²⋊₂S₃⋊₂₂C₂, D₆.D₄⋊₃₆C₂, C₂.₅₉(D₄○D₁₂), (C₂×C₆).₂₄₁C₂⁴, C₁₂.₁₃₁(C₄○D₄), (C₄×C₁₂).₂₀₀C₂², (C₂×C₁₂).₁₈₉C₂³, D₆⋊C₄.₁₁₂C₂², C₄.₂₀(Q₈⋊₃S₃), (C₂×D₁₂).₁₆₆C₂², C₂².₂₆₂(S₃×C₂³), Dic₃⋊C₄.₁₂₄C₂², (C₂²×S₃).₁₀₆C₂³, C₃⋊₆(C₂².₃₄C₂⁴), (C₄×Dic₃).₁₄₆C₂², (C₂×Dic₃).₂₆₁C₂³, C₆.₁₁₈(C₂×C₄○D₄), (S₃×C₂×C₄).₁₃₁C₂², C₂.₂₅(C₂×Q₈⋊₃S₃), (C₃×C₄².C₂)⋊₁₄C₂, (C₃×C₄⋊C₄).₁₉₆C₂², (C₂×C₄).₅₉₄(C₂²×S₃), SmallGroup(192,1256)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₄².₁₅₅D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×S₃ — S₃×C₂×C₄ — D₆.D₄ — C₄².₁₅₅D₆

Lower central

C₃ — C₂×C₆ — C₄².₁₅₅D₆

Upper central

C₁ — C₂² — C₄².C₂

Generators and relations for C₄².₁₅₅D₆
G = < a,b,c,d | a⁴=b⁴=1, c⁶=d²=a², ab=ba, cac^-1=ab², dad^-1=a^-1b², cbc^-1=a²b, dbd^-1=a²b^-1, dcd^-1=c⁵ >

Subgroups: 736 in 240 conjugacy classes, 95 normal (19 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, C₂×C₄, C₂×C₄, D₄, C₂³, Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₄×S₃, D₁₂, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₂²×S₃, C₂²×S₃, C₄²⋊C₂, C₄×D₄, C₄⋊D₄, C₂².D₄, C₄².C₂, C₄⋊₁D₄, C₄×Dic₃, Dic₃⋊C₄, D₆⋊C₄, C₄×C₁₂, C₃×C₄⋊C₄, S₃×C₂×C₄, S₃×C₂×C₄, C₂×D₁₂, C₂².₃₄C₂⁴, C₄²⋊₂S₃, C₄⋊D₁₂, Dic₃⋊₅D₄, D₆.D₄, C₁₂⋊D₄, C₃×C₄².C₂, C₄².₁₅₅D₆
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄○D₄, C₂⁴, C₂²×S₃, C₂×C₄○D₄, 2₊¹⁺⁴, Q₈⋊₃S₃, S₃×C₂³, C₂².₃₄C₂⁴, C₂×Q₈⋊₃S₃, D₄○D₁₂, C₄².₁₅₅D₆

Smallest permutation representation of C₄².₁₅₅D₆
►On 96 points

Generators in S₉₆

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

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

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	2D	···	2H	3	4A	4B	4C	···	4H	4I	4J	4K	4L	4M	6A	6B	6C	12A	···	12F	12G	12H	12I	12J
order	1	2	2	2	2	···	2	3	4	4	4	···	4	4	4	4	4	4	6	6	6	12	···	12	12	12	12	12
size	1	1	1	1	12	···	12	2	2	2	4	···	4	6	6	6	6	12	2	2	2	4	···	4	8	8	8	8

36 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

C₄○D₄

2₊¹⁺⁴

Q₈⋊₃S₃

D₄○D₁₂

kernel

C₄².₁₅₅D₆

C₄²⋊₂S₃

C₄⋊D₁₂

Dic₃⋊₅D₄

D₆.D₄

C₁₂⋊D₄

C₃×C₄².C₂

C₄².C₂

C₄²

C₄⋊C₄

C₁₂

C₆

C₄

C₂

# reps

Matrix representation of C₄².₁₅₅D₆ ►in GL₈(𝔽₁₃)

0	12	0	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	0	1	2	0
0	0	0	0	12	0	0	2
0	0	0	0	1	0	0	12
0	0	0	0	0	1	1	0

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

0	8	0	0	0	0	0	0
8	0	0	0	0	0	0	0
0	0	1	1	0	0	0	0
0	0	12	0	0	0	0	0
0	0	0	0	0	5	0	0
0	0	0	0	8	0	0	0
0	0	0	0	5	0	0	8
0	0	0	0	0	5	5	0

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

G:=sub<GL(8,GF(13))| [0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,1,0,0,0,0,2,0,0,1,0,0,0,0,0,2,12,0],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,8,5,0,0,0,0,0,5,0,0,5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,5,0,0,5,0,0,0,0,0,8,5,0,0,0,0,0,0,3,5,0,0,0,0,0,3,0,0,8] >;

C₄².₁₅₅D₆ in GAP, Magma, Sage, TeX

C_4^2._{155}D_6

% in TeX

G:=Group("C4^2.155D6");

// GroupNames label

G:=SmallGroup(192,1256);

// by ID

G=gap.SmallGroup(192,1256);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,184,675,570,80,6278]);

// Polycyclic

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

// generators/relations

0	12	0	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	0	1	2	0
0	0	0	0	12	0	0	2
0	0	0	0	1	0	0	12
0	0	0	0	0	1	1	0

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

0	8	0	0	0	0	0	0
8	0	0	0	0	0	0	0
0	0	1	1	0	0	0	0
0	0	12	0	0	0	0	0
0	0	0	0	0	5	0	0
0	0	0	0	8	0	0	0
0	0	0	0	5	0	0	8
0	0	0	0	0	5	5	0

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

0	12	0	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	0	1	2	0
0	0	0	0	12	0	0	2
0	0	0	0	1	0	0	12
0	0	0	0	0	1	1	0

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

0	8	0	0	0	0	0	0
8	0	0	0	0	0	0	0
0	0	1	1	0	0	0	0
0	0	12	0	0	0	0	0
0	0	0	0	0	5	0	0
0	0	0	0	8	0	0	0
0	0	0	0	5	0	0	8
0	0	0	0	0	5	5	0

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

G = C42.155D6 order 192 = 26·3

155th non-split extension by C42 of D6 acting via D6/C3=C22

G = C₄².₁₅₅D₆ order 192 = 2⁶·3

155^th non-split extension by C₄² of D₆ acting via D₆/C₃=C₂²

0	12	0	0	0	0	0	0
1	0	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	0	1	2	0
0	0	0	0	12	0	0	2
0	0	0	0	1	0	0	12
0	0	0	0	0	1	1	0

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

0	8	0	0	0	0	0	0
8	0	0	0	0	0	0	0
0	0	1	1	0	0	0	0
0	0	12	0	0	0	0	0
0	0	0	0	0	5	0	0
0	0	0	0	8	0	0	0
0	0	0	0	5	0	0	8
0	0	0	0	0	5	5	0

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