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## G = C22×C3⋊C8order 96 = 25·3

### Direct product of C22 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C22×C3⋊C8
 Chief series C1 — C3 — C6 — C12 — C3⋊C8 — C2×C3⋊C8 — C22×C3⋊C8
 Lower central C3 — C22×C3⋊C8
 Upper central C1 — C22×C4

Generators and relations for C22×C3⋊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 [×6], C3, C4, C4 [×3], C22 [×7], C6, C6 [×6], C8 [×4], C2×C4 [×6], C23, C12, C12 [×3], C2×C6 [×7], C2×C8 [×6], C22×C4, C3⋊C8 [×4], C2×C12 [×6], C22×C6, C22×C8, C2×C3⋊C8 [×6], C22×C12, C22×C3⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C8 [×4], C2×C4 [×6], C23, Dic3 [×4], D6 [×3], C2×C8 [×6], C22×C4, C3⋊C8 [×4], C2×Dic3 [×6], C22×S3, C22×C8, C2×C3⋊C8 [×6], C22×Dic3, C22×C3⋊C8

Smallest permutation representation of C22×C3⋊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)])

C22×C3⋊C8 is a maximal subgroup of
(C2×C12)⋊3C8  C12.C42  (C2×C24)⋊5C4  C12.4C42  C3⋊D4⋊C8  C3⋊C826D4  C12.5C42  C4⋊C4.234D6  C42.43D6  C4.(C2×D12)  C42.47D6  C3⋊C822D4  C3⋊C823D4  C3⋊C824D4  C3⋊C8.29D4  Dic3×C2×C8  C12.88(C2×Q8)  C12.7C42  D6⋊C840C2  C4○D44Dic3  (C6×D4).11C4  S3×C22×C8
C22×C3⋊C8 is a maximal quotient of
C42.285D6  C24.78C23

48 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 6A ··· 6G 8A ··· 8P 12A ··· 12H order 1 2 ··· 2 3 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 ··· 1 2 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 S3 Dic3 D6 Dic3 C3⋊C8 kernel C22×C3⋊C8 C2×C3⋊C8 C22×C12 C2×C12 C22×C6 C2×C6 C22×C4 C2×C4 C2×C4 C23 C22 # reps 1 6 1 6 2 16 1 3 3 1 8

Matrix representation of C22×C3⋊C8 in GL4(𝔽73) generated by

 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
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] >;

C22×C3⋊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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