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G = C22xC3:C8order 96 = 25·3

Direct product of C22 and C3:C8

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

Aliases: C22xC3:C8, C12.40C23, C23.5Dic3, C6:2(C2xC8), (C2xC6):3C8, C3:2(C22xC8), (C2xC4).99D6, (C2xC12).12C4, C12.42(C2xC4), (C22xC6).6C4, (C2xC4).9Dic3, C6.20(C22xC4), (C22xC4).11S3, C4.40(C22xS3), C4.14(C2xDic3), (C22xC12).12C2, C2.1(C22xDic3), (C2xC12).112C22, C22.11(C2xDic3), C4o(C2xC3:C8), (C2xC4)o(C3:C8), (C2xC6).31(C2xC4), (C2xC4)o(C2xC3:C8), SmallGroup(96,127)

Series: Derived Chief Lower central Upper central

C1C3 — C22xC3:C8
C1C3C6C12C3:C8C2xC3:C8 — C22xC3:C8
C3 — C22xC3:C8
C1C22xC4

Generators and relations for C22xC3:C8
 G = < a,b,c,d | a2=b2=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 98 in 76 conjugacy classes, 65 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2xC4, C23, C12, C12, C2xC6, C2xC8, C22xC4, C3:C8, C2xC12, C22xC6, C22xC8, C2xC3:C8, C22xC12, C22xC3:C8
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, C23, Dic3, D6, C2xC8, C22xC4, C3:C8, C2xDic3, C22xS3, C22xC8, C2xC3:C8, C22xDic3, C22xC3:C8

Smallest permutation representation of C22xC3:C8
Regular action on 96 points
Generators in S96
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)(33 96)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 70)(42 71)(43 72)(44 65)(45 66)(46 67)(47 68)(48 69)(49 77)(50 78)(51 79)(52 80)(53 73)(54 74)(55 75)(56 76)(57 87)(58 88)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 78)(26 79)(27 80)(28 73)(29 74)(30 75)(31 76)(32 77)(41 86)(42 87)(43 88)(44 81)(45 82)(46 83)(47 84)(48 85)(57 71)(58 72)(59 65)(60 66)(61 67)(62 68)(63 69)(64 70)
(1 42 93)(2 94 43)(3 44 95)(4 96 45)(5 46 89)(6 90 47)(7 48 91)(8 92 41)(9 51 83)(10 84 52)(11 53 85)(12 86 54)(13 55 87)(14 88 56)(15 49 81)(16 82 50)(17 77 59)(18 60 78)(19 79 61)(20 62 80)(21 73 63)(22 64 74)(23 75 57)(24 58 76)(25 33 66)(26 67 34)(27 35 68)(28 69 36)(29 37 70)(30 71 38)(31 39 72)(32 65 40)
(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)

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

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

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

C22xC3:C8 is a maximal subgroup of
(C2xC12):3C8  C12.C42  (C2xC24):5C4  C12.4C42  C3:D4:C8  C3:C8:26D4  C12.5C42  C4:C4.234D6  C42.43D6  C4.(C2xD12)  C42.47D6  C3:C8:22D4  C3:C8:23D4  C3:C8:24D4  C3:C8.29D4  Dic3xC2xC8  C12.88(C2xQ8)  C12.7C42  D6:C8:40C2  C4oD4:4Dic3  (C6xD4).11C4  S3xC22xC8
C22xC3:C8 is a maximal quotient of
C42.285D6  C24.78C23

48 conjugacy classes

class 1 2A···2G 3 4A···4H6A···6G8A···8P12A···12H
order12···234···46···68···812···12
size11···121···12···23···32···2

48 irreducible representations

dim11111122222
type++++-+-
imageC1C2C2C4C4C8S3Dic3D6Dic3C3:C8
kernelC22xC3:C8C2xC3:C8C22xC12C2xC12C22xC6C2xC6C22xC4C2xC4C2xC4C23C22
# reps161621613318

Matrix representation of C22xC3:C8 in GL4(F73) generated by

72000
07200
00720
00072
,
72000
0100
00720
00072
,
1000
0100
0080
00064
,
22000
07200
00072
00720
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,64],[22,0,0,0,0,72,0,0,0,0,0,72,0,0,72,0] >;

C22xC3:C8 in GAP, Magma, Sage, TeX

C_2^2\times C_3\rtimes C_8
% in TeX

G:=Group("C2^2xC3:C8");
// GroupNames label

G:=SmallGroup(96,127);
// by ID

G=gap.SmallGroup(96,127);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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