C11xC8:C2^2 - GroupNames

G = C₁₁xC₈:C₂² order 352 = 2⁵·11

Direct product of C₁₁ and C₈:C₂²

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₁₁xC₈:C₂², D₈:₂C₂₂, C₈₈:₇C₂², C₄₄.₆₃D₄, SD₁₆:₁C₂₂, M₄(2):₁C₂₂, C₄₄.₄₈C₂³, C₈:(C₂xC₂₂), C₄oD₄:₂C₂₂, D₄:₂(C₂xC₂₂), (C₁₁xD₈):₆C₂, (C₂xD₄):₅C₂₂, Q₈:₂(C₂xC₂₂), (D₄xC₂₂):₁₄C₂, (C₂xC₂₂).₂₄D₄, C₂₂.₇₈(C₂xD₄), C₄.₁₄(D₄xC₁₁), C₂.₁₅(D₄xC₂₂), (C₁₁xSD₁₆):₅C₂, C₄.₅(C₂²xC₂₂), C₂².₅(D₄xC₁₁), (D₄xC₁₁):₁₁C₂², (C₁₁xM₄(2)):₅C₂, (C₂xC₄₄).₆₉C₂², (Q₈xC₁₁):₁₀C₂², (C₁₁xC₄oD₄):₇C₂, (C₂xC₄).₁₀(C₂xC₂₂), SmallGroup(352,171)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄ — C₁₁xC₈:C₂²

Chief series

C₁ — C₂ — C₄ — C₄₄ — D₄xC₁₁ — C₁₁xD₈ — C₁₁xC₈:C₂²

Lower central

C₁ — C₂ — C₄ — C₁₁xC₈:C₂²

Upper central

C₁ — C₂₂ — C₂xC₄₄ — C₁₁xC₈:C₂²

Generators and relations for C₁₁xC₈:C₂²
G = < a,b,c,d | a¹¹=b⁸=c²=d²=1, ab=ba, ac=ca, ad=da, cbc=b³, dbd=b⁵, cd=dc >

Subgroups: 116 in 68 conjugacy classes, 40 normal (24 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂xC₄, C₂xC₄, D₄, D₄, D₄, Q₈, C₂³, C₁₁, M₄(2), D₈, SD₁₆, C₂xD₄, C₄oD₄, C₂₂, C₂₂, C₈:C₂², C₄₄, C₄₄, C₂xC₂₂, C₂xC₂₂, C₈₈, C₂xC₄₄, C₂xC₄₄, D₄xC₁₁, D₄xC₁₁, D₄xC₁₁, Q₈xC₁₁, C₂²xC₂₂, C₁₁xM₄(2), C₁₁xD₈, C₁₁xSD₁₆, D₄xC₂₂, C₁₁xC₄oD₄, C₁₁xC₈:C₂²
Quotients: C₁, C₂, C₂², D₄, C₂³, C₁₁, C₂xD₄, C₂₂, C₈:C₂², C₂xC₂₂, D₄xC₁₁, C₂²xC₂₂, D₄xC₂₂, C₁₁xC₈:C₂²

Smallest permutation representation of C₁₁xC₈:C₂²
►On 88 points

Generators in S₈₈

(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)

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

(12 87)(13 88)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 44)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(45 77)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)

(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 23)(20 24)(21 25)(22 26)(34 84)(35 85)(36 86)(37 87)(38 88)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)

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

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

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

121 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	8A	8B	11A	···	11J	22A	···	22J	22K	···	22T	22U	···	22AX	44A	···	44T	44U	···	44AD	88A	···	88T
order	1	2	2	2	2	2	4	4	4	8	8	11	···	11	22	···	22	22	···	22	22	···	22	44	···	44	44	···	44	88	···	88
size	1	1	2	4	4	4	2	2	4	4	4	1	···	1	1	···	1	2	···	2	4	···	4	2	···	2	4	···	4	4	···	4

121 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+	+	+							+	+			+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₁₁	C₂₂	C₂₂	C₂₂	C₂₂	C₂₂	D₄	D₄	D₄xC₁₁	D₄xC₁₁	C₈:C₂²	C₁₁xC₈:C₂²
kernel	C₁₁xC₈:C₂²	C₁₁xM₄(2)	C₁₁xD₈	C₁₁xSD₁₆	D₄xC₂₂	C₁₁xC₄oD₄	C₈:C₂²	M₄(2)	D₈	SD₁₆	C₂xD₄	C₄oD₄	C₄₄	C₂xC₂₂	C₄	C₂²	C₁₁	C₁
# reps	1	1	2	2	1	1	10	10	20	20	10	10	1	1	10	10	1	10

Matrix representation of C₁₁xC₈:C₂² ►in GL₄(F₈₉) generated by

39	0	0	0
0	39	0	0
0	0	39	0
0	0	0	39

0	0	1	0
0	0	0	88
0	1	0	0
1	0	0	0

1	0	0	0
0	88	0	0
0	0	0	88
0	0	88	0

1	0	0	0
0	1	0	0
0	0	88	0
0	0	0	88

G:=sub<GL(4,GF(89))| [39,0,0,0,0,39,0,0,0,0,39,0,0,0,0,39],[0,0,0,1,0,0,1,0,1,0,0,0,0,88,0,0],[1,0,0,0,0,88,0,0,0,0,0,88,0,0,88,0],[1,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88] >;

C₁₁xC₈:C₂² in GAP, Magma, Sage, TeX

C_{11}\times C_8\rtimes C_2^2

% in TeX

G:=Group("C11xC8:C2^2");

// GroupNames label

G:=SmallGroup(352,171);

// by ID

G=gap.SmallGroup(352,171);

# by ID

G:=PCGroup([6,-2,-2,-2,-11,-2,-2,1081,3242,7924,3970,88]);

// Polycyclic

G:=Group<a,b,c,d|a^11=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;

// generators/relations

G = C11xC8:C22 order 352 = 25·11

Direct product of C11 and C8:C22

G = C₁₁xC₈:C₂² order 352 = 2⁵·11

Direct product of C₁₁ and C₈:C₂²