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G = C2xC11:D4order 176 = 24·11

Direct product of C2 and C11:D4

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

Aliases: C2xC11:D4, C22:2D4, C23:D11, C22:2D22, D22:3C22, C22.10C23, Dic11:2C22, C11:3(C2xD4), (C2xC22):3C22, (C22xC22):2C2, (C2xDic11):4C2, (C22xD11):3C2, C2.10(C22xD11), SmallGroup(176,36)

Series: Derived Chief Lower central Upper central

C1C22 — C2xC11:D4
C1C11C22D22C22xD11 — C2xC11:D4
C11C22 — C2xC11:D4
C1C22C23

Generators and relations for C2xC11:D4
 G = < a,b,c,d | a2=b11=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 260 in 54 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2xC4, D4, C23, C23, C11, C2xD4, D11, C22, C22, C22, Dic11, D22, D22, C2xC22, C2xC22, C2xC22, C2xDic11, C11:D4, C22xD11, C22xC22, C2xC11:D4
Quotients: C1, C2, C22, D4, C23, C2xD4, D11, D22, C11:D4, C22xD11, C2xC11:D4

Smallest permutation representation of C2xC11:D4
On 88 points
Generators in S88
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)
(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 23 12 34)(2 33 13 44)(3 32 14 43)(4 31 15 42)(5 30 16 41)(6 29 17 40)(7 28 18 39)(8 27 19 38)(9 26 20 37)(10 25 21 36)(11 24 22 35)(45 67 56 78)(46 77 57 88)(47 76 58 87)(48 75 59 86)(49 74 60 85)(50 73 61 84)(51 72 62 83)(52 71 63 82)(53 70 64 81)(54 69 65 80)(55 68 66 79)
(2 11)(3 10)(4 9)(5 8)(6 7)(13 22)(14 21)(15 20)(16 19)(17 18)(23 34)(24 44)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(46 55)(47 54)(48 53)(49 52)(50 51)(57 66)(58 65)(59 64)(60 63)(61 62)(67 78)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)

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

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

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

C2xC11:D4 is a maximal subgroup of
C22.2D44  Dic11:4D4  C22:D44  D22.D4  D22:D4  Dic11.D4  C22.D44  C23.23D22  C44:7D4  C23:D22  C44:2D4  Dic11:D4  C44:D4  C24:D11  C2xD4xD11  D4:6D22
C2xC11:D4 is a maximal quotient of
C44.48D4  C23.23D22  C44:7D4  D44:6C22  C23.18D22  C44.17D4  C23:D22  C44:2D4  Dic11:D4  C44:D4  C44.C23  Dic11:Q8  D22:3Q8  C44.23D4  Q8:D22  D4.8D22  D4.9D22  C24:D11

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B11A···11E22A···22AI
order122222224411···1122···22
size111122222222222···22···2

50 irreducible representations

dim111112222
type++++++++
imageC1C2C2C2C2D4D11D22C11:D4
kernelC2xC11:D4C2xDic11C11:D4C22xD11C22xC22C22C23C22C2
# reps11411251520

Matrix representation of C2xC11:D4 in GL4(F89) generated by

88000
08800
00880
00088
,
598800
102700
00088
00171
,
16800
08800
007974
007210
,
882100
0100
003318
007856
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[59,10,0,0,88,27,0,0,0,0,0,1,0,0,88,71],[1,0,0,0,68,88,0,0,0,0,79,72,0,0,74,10],[88,0,0,0,21,1,0,0,0,0,33,78,0,0,18,56] >;

C2xC11:D4 in GAP, Magma, Sage, TeX

C_2\times C_{11}\rtimes D_4
% in TeX

G:=Group("C2xC11:D4");
// GroupNames label

G:=SmallGroup(176,36);
// by ID

G=gap.SmallGroup(176,36);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,182,4004]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^11=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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