C4^2.176D6 - GroupNames

G = C₄².₁₇₆D₆ order 192 = 2⁶·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄².₁₇₆D₆, C₆.₃₈2_-¹⁺⁴, C₄:Q₈:₁₄S₃, C₄:C₄.₁₂₅D₆, (Q₈xDic₃):₂₃C₂, (C₂xQ₈).₁₇₂D₆, Dic₃.Q₈:₄₂C₂, (C₂xC₆).₂₇₅C₂⁴, C₄²:₂S₃.₉C₂, D₆:₃Q₈.₁₃C₂, C₁₂.₆Q₈:₂₅C₂, C₁₂.₁₃₈(C₄oD₄), C₄.₄₂(D₄:₂S₃), (C₂xC₁₂).₁₀₈C₂³, (C₄xC₁₂).₂₁₆C₂², D₆:C₄.₁₅₄C₂², (C₆xQ₈).₁₄₂C₂², Dic₃:C₄.₆₃C₂², C₄:Dic₃.₂₅₄C₂², C₂².₂₉₆(S₃xC₂³), (C₂²xS₃).₁₂₀C₂³, C₂.₃₉(Q₈.₁₅D₆), C₃:₇(C₂².₃₅C₂⁴), (C₄xDic₃).₁₆₄C₂², (C₂xDic₃).₂₇₃C₂³, (C₃xC₄:Q₈):₁₇C₂, C₄:C₄:S₃.₄C₂, C₆.₁₀₁(C₂xC₄oD₄), C₂.₆₅(C₂xD₄:₂S₃), (S₃xC₂xC₄).₁₄₈C₂², (C₃xC₄:C₄).₂₁₈C₂², (C₂xC₄).₂₂₁(C₂²xS₃), SmallGroup(192,1290)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₄².₁₇₆D₆

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xS₃ — S₃xC₂xC₄ — D₆:₃Q₈ — C₄².₁₇₆D₆

Lower central

C₃ — C₂xC₆ — C₄².₁₇₆D₆

Upper central

C₁ — C₂² — C₄:Q₈

Generators and relations for C₄².₁₇₆D₆
G = < a,b,c,d | a⁴=b⁴=1, c⁶=a², d²=a²b², ab=ba, cac^-1=dad^-1=a^-1, cbc^-1=b^-1, dbd^-1=a²b^-1, dcd^-1=b²c⁵ >

Subgroups: 400 in 192 conjugacy classes, 95 normal (19 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, C₂xC₄, C₂xC₄, Q₈, C₂³, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₄², C₄², C₂²:C₄, C₄:C₄, C₄:C₄, C₂²xC₄, C₂xQ₈, C₄xS₃, C₂xDic₃, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₃xQ₈, C₂²xS₃, C₄²:C₂, C₄xQ₈, C₂²:Q₈, C₄².C₂, C₄²:₂C₂, C₄:Q₈, C₄xDic₃, C₄xDic₃, Dic₃:C₄, C₄:Dic₃, D₆:C₄, C₄xC₁₂, C₃xC₄:C₄, S₃xC₂xC₄, C₆xQ₈, C₂².₃₅C₂⁴, C₁₂.₆Q₈, C₄²:₂S₃, Dic₃.Q₈, C₄:C₄:S₃, Q₈xDic₃, D₆:₃Q₈, C₃xC₄:Q₈, C₄².₁₇₆D₆
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄oD₄, C₂⁴, C₂²xS₃, C₂xC₄oD₄, 2_-¹⁺⁴, D₄:₂S₃, S₃xC₂³, C₂².₃₅C₂⁴, C₂xD₄:₂S₃, Q₈.₁₅D₆, C₄².₁₇₆D₆

Smallest permutation representation of C₄².₁₇₆D₆
►On 96 points

Generators in S₉₆

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

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

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

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

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

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

36 conjugacy classes

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

36 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+		-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₆	D₆	D₆	C₄oD₄	2_-¹⁺⁴	D₄:₂S₃	Q₈.₁₅D₆
kernel	C₄².₁₇₆D₆	C₁₂.₆Q₈	C₄²:₂S₃	Dic₃.Q₈	C₄:C₄:S₃	Q₈xDic₃	D₆:₃Q₈	C₃xC₄:Q₈	C₄:Q₈	C₄²	C₄:C₄	C₂xQ₈	C₁₂	C₆	C₄	C₂
# reps	1	1	1	4	4	2	2	1	1	1	4	2	4	2	2	4

Matrix representation of C₄².₁₇₆D₆ ►in GL₆(F₁₃)

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

8	0	0	0	0	0
8	5	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	11	0	0	0	0
0	12	0	0	0	0
0	0	1	6	0	4
0	0	7	7	9	4
0	0	0	4	12	7
0	0	9	4	6	6

12	2	0	0	0	0
12	1	0	0	0	0
0	0	6	6	4	9
0	0	12	7	0	9
0	0	9	4	6	6
0	0	0	4	12	7

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

C₄².₁₇₆D₆ in GAP, Magma, Sage, TeX

C_4^2._{176}D_6

% in TeX

G:=Group("C4^2.176D6");

// GroupNames label

G:=SmallGroup(192,1290);

// by ID

G=gap.SmallGroup(192,1290);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,100,675,297,136,6278]);

// Polycyclic

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

// generators/relations

G = C42.176D6 order 192 = 26·3

176th non-split extension by C42 of D6 acting via D6/C3=C22

G = C₄².₁₇₆D₆ order 192 = 2⁶·3

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