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## G = C11×M4(2)  order 176 = 24·11

### Direct product of C11 and M4(2)

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C11×M4(2), C4.C44, C83C22, C887C2, C44.4C4, C22.C44, C44.22C22, (C2×C22).1C4, (C2×C44).8C2, C2.3(C2×C44), (C2×C4).2C22, C4.6(C2×C22), C22.12(C2×C4), SmallGroup(176,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C11×M4(2)
 Chief series C1 — C2 — C4 — C44 — C88 — C11×M4(2)
 Lower central C1 — C2 — C11×M4(2)
 Upper central C1 — C44 — C11×M4(2)

Generators and relations for C11×M4(2)
G = < a,b,c | a11=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Smallest permutation representation of C11×M4(2)
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 63 15 46 40 75 30)(2 80 64 16 47 41 76 31)(3 81 65 17 48 42 77 32)(4 82 66 18 49 43 67 33)(5 83 56 19 50 44 68 23)(6 84 57 20 51 34 69 24)(7 85 58 21 52 35 70 25)(8 86 59 22 53 36 71 26)(9 87 60 12 54 37 72 27)(10 88 61 13 55 38 73 28)(11 78 62 14 45 39 74 29)
(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), (1,79,63,15,46,40,75,30)(2,80,64,16,47,41,76,31)(3,81,65,17,48,42,77,32)(4,82,66,18,49,43,67,33)(5,83,56,19,50,44,68,23)(6,84,57,20,51,34,69,24)(7,85,58,21,52,35,70,25)(8,86,59,22,53,36,71,26)(9,87,60,12,54,37,72,27)(10,88,61,13,55,38,73,28)(11,78,62,14,45,39,74,29), (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), (1,79,63,15,46,40,75,30)(2,80,64,16,47,41,76,31)(3,81,65,17,48,42,77,32)(4,82,66,18,49,43,67,33)(5,83,56,19,50,44,68,23)(6,84,57,20,51,34,69,24)(7,85,58,21,52,35,70,25)(8,86,59,22,53,36,71,26)(9,87,60,12,54,37,72,27)(10,88,61,13,55,38,73,28)(11,78,62,14,45,39,74,29), (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)], [(1,79,63,15,46,40,75,30),(2,80,64,16,47,41,76,31),(3,81,65,17,48,42,77,32),(4,82,66,18,49,43,67,33),(5,83,56,19,50,44,68,23),(6,84,57,20,51,34,69,24),(7,85,58,21,52,35,70,25),(8,86,59,22,53,36,71,26),(9,87,60,12,54,37,72,27),(10,88,61,13,55,38,73,28),(11,78,62,14,45,39,74,29)], [(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)])

C11×M4(2) is a maximal subgroup of   C44.53D4  C44.46D4  C44.47D4  D444C4  D44.C4  C8⋊D22  C8.D22

110 conjugacy classes

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

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + image C1 C2 C2 C4 C4 C11 C22 C22 C44 C44 M4(2) C11×M4(2) kernel C11×M4(2) C88 C2×C44 C44 C2×C22 M4(2) C8 C2×C4 C4 C22 C11 C1 # reps 1 2 1 2 2 10 20 10 20 20 2 20

Matrix representation of C11×M4(2) in GL3(𝔽89) generated by

 45 0 0 0 1 0 0 0 1
,
 1 0 0 0 0 1 0 34 0
,
 88 0 0 0 1 0 0 0 88
G:=sub<GL(3,GF(89))| [45,0,0,0,1,0,0,0,1],[1,0,0,0,0,34,0,1,0],[88,0,0,0,1,0,0,0,88] >;

C11×M4(2) in GAP, Magma, Sage, TeX

C_{11}\times M_4(2)
% in TeX

G:=Group("C11xM4(2)");
// GroupNames label

G:=SmallGroup(176,23);
// by ID

G=gap.SmallGroup(176,23);
# by ID

G:=PCGroup([5,-2,-2,-11,-2,-2,220,901,58]);
// Polycyclic

G:=Group<a,b,c|a^11=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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