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## G = C2×Q8×C7⋊C3order 336 = 24·3·7

### Direct product of C2, Q8 and C7⋊C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×Q8×C7⋊C3
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C22×C7⋊C3 — C2×C4×C7⋊C3 — C2×Q8×C7⋊C3
 Lower central C7 — C14 — C2×Q8×C7⋊C3
 Upper central C1 — C22 — C2×Q8

Generators and relations for C2×Q8×C7⋊C3
G = < a,b,c,d,e | a2=b4=d7=e3=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 190 in 76 conjugacy classes, 57 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C6, C7, C2×C4, Q8, C12, C2×C6, C14, C14, C2×Q8, C7⋊C3, C2×C12, C3×Q8, C28, C2×C14, C2×C7⋊C3, C2×C7⋊C3, C6×Q8, C2×C28, C7×Q8, C4×C7⋊C3, C22×C7⋊C3, Q8×C14, C2×C4×C7⋊C3, Q8×C7⋊C3, C2×Q8×C7⋊C3
Quotients: C1, C2, C3, C22, C6, Q8, C23, C2×C6, C2×Q8, C7⋊C3, C3×Q8, C22×C6, C2×C7⋊C3, C6×Q8, C22×C7⋊C3, Q8×C7⋊C3, C23×C7⋊C3, C2×Q8×C7⋊C3

Smallest permutation representation of C2×Q8×C7⋊C3
On 112 points
Generators in S112
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)
(1 50 8 43)(2 51 9 44)(3 52 10 45)(4 53 11 46)(5 54 12 47)(6 55 13 48)(7 56 14 49)(15 29 22 36)(16 30 23 37)(17 31 24 38)(18 32 25 39)(19 33 26 40)(20 34 27 41)(21 35 28 42)(57 106 64 99)(58 107 65 100)(59 108 66 101)(60 109 67 102)(61 110 68 103)(62 111 69 104)(63 112 70 105)(71 85 78 92)(72 86 79 93)(73 87 80 94)(74 88 81 95)(75 89 82 96)(76 90 83 97)(77 91 84 98)
(1 78 8 71)(2 79 9 72)(3 80 10 73)(4 81 11 74)(5 82 12 75)(6 83 13 76)(7 84 14 77)(15 57 22 64)(16 58 23 65)(17 59 24 66)(18 60 25 67)(19 61 26 68)(20 62 27 69)(21 63 28 70)(29 99 36 106)(30 100 37 107)(31 101 38 108)(32 102 39 109)(33 103 40 110)(34 104 41 111)(35 105 42 112)(43 92 50 85)(44 93 51 86)(45 94 52 87)(46 95 53 88)(47 96 54 89)(48 97 55 90)(49 98 56 91)
(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 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)(37 38 40)(39 42 41)(44 45 47)(46 49 48)(51 52 54)(53 56 55)(58 59 61)(60 63 62)(65 66 68)(67 70 69)(72 73 75)(74 77 76)(79 80 82)(81 84 83)(86 87 89)(88 91 90)(93 94 96)(95 98 97)(100 101 103)(102 105 104)(107 108 110)(109 112 111)

G:=sub<Sym(112)| (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105), (1,50,8,43)(2,51,9,44)(3,52,10,45)(4,53,11,46)(5,54,12,47)(6,55,13,48)(7,56,14,49)(15,29,22,36)(16,30,23,37)(17,31,24,38)(18,32,25,39)(19,33,26,40)(20,34,27,41)(21,35,28,42)(57,106,64,99)(58,107,65,100)(59,108,66,101)(60,109,67,102)(61,110,68,103)(62,111,69,104)(63,112,70,105)(71,85,78,92)(72,86,79,93)(73,87,80,94)(74,88,81,95)(75,89,82,96)(76,90,83,97)(77,91,84,98), (1,78,8,71)(2,79,9,72)(3,80,10,73)(4,81,11,74)(5,82,12,75)(6,83,13,76)(7,84,14,77)(15,57,22,64)(16,58,23,65)(17,59,24,66)(18,60,25,67)(19,61,26,68)(20,62,27,69)(21,63,28,70)(29,99,36,106)(30,100,37,107)(31,101,38,108)(32,102,39,109)(33,103,40,110)(34,104,41,111)(35,105,42,112)(43,92,50,85)(44,93,51,86)(45,94,52,87)(46,95,53,88)(47,96,54,89)(48,97,55,90)(49,98,56,91), (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)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34)(37,38,40)(39,42,41)(44,45,47)(46,49,48)(51,52,54)(53,56,55)(58,59,61)(60,63,62)(65,66,68)(67,70,69)(72,73,75)(74,77,76)(79,80,82)(81,84,83)(86,87,89)(88,91,90)(93,94,96)(95,98,97)(100,101,103)(102,105,104)(107,108,110)(109,112,111)>;

G:=Group( (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105), (1,50,8,43)(2,51,9,44)(3,52,10,45)(4,53,11,46)(5,54,12,47)(6,55,13,48)(7,56,14,49)(15,29,22,36)(16,30,23,37)(17,31,24,38)(18,32,25,39)(19,33,26,40)(20,34,27,41)(21,35,28,42)(57,106,64,99)(58,107,65,100)(59,108,66,101)(60,109,67,102)(61,110,68,103)(62,111,69,104)(63,112,70,105)(71,85,78,92)(72,86,79,93)(73,87,80,94)(74,88,81,95)(75,89,82,96)(76,90,83,97)(77,91,84,98), (1,78,8,71)(2,79,9,72)(3,80,10,73)(4,81,11,74)(5,82,12,75)(6,83,13,76)(7,84,14,77)(15,57,22,64)(16,58,23,65)(17,59,24,66)(18,60,25,67)(19,61,26,68)(20,62,27,69)(21,63,28,70)(29,99,36,106)(30,100,37,107)(31,101,38,108)(32,102,39,109)(33,103,40,110)(34,104,41,111)(35,105,42,112)(43,92,50,85)(44,93,51,86)(45,94,52,87)(46,95,53,88)(47,96,54,89)(48,97,55,90)(49,98,56,91), (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)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34)(37,38,40)(39,42,41)(44,45,47)(46,49,48)(51,52,54)(53,56,55)(58,59,61)(60,63,62)(65,66,68)(67,70,69)(72,73,75)(74,77,76)(79,80,82)(81,84,83)(86,87,89)(88,91,90)(93,94,96)(95,98,97)(100,101,103)(102,105,104)(107,108,110)(109,112,111) );

G=PermutationGroup([[(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,85),(37,86),(38,87),(39,88),(40,89),(41,90),(42,91),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,99),(51,100),(52,101),(53,102),(54,103),(55,104),(56,105)], [(1,50,8,43),(2,51,9,44),(3,52,10,45),(4,53,11,46),(5,54,12,47),(6,55,13,48),(7,56,14,49),(15,29,22,36),(16,30,23,37),(17,31,24,38),(18,32,25,39),(19,33,26,40),(20,34,27,41),(21,35,28,42),(57,106,64,99),(58,107,65,100),(59,108,66,101),(60,109,67,102),(61,110,68,103),(62,111,69,104),(63,112,70,105),(71,85,78,92),(72,86,79,93),(73,87,80,94),(74,88,81,95),(75,89,82,96),(76,90,83,97),(77,91,84,98)], [(1,78,8,71),(2,79,9,72),(3,80,10,73),(4,81,11,74),(5,82,12,75),(6,83,13,76),(7,84,14,77),(15,57,22,64),(16,58,23,65),(17,59,24,66),(18,60,25,67),(19,61,26,68),(20,62,27,69),(21,63,28,70),(29,99,36,106),(30,100,37,107),(31,101,38,108),(32,102,39,109),(33,103,40,110),(34,104,41,111),(35,105,42,112),(43,92,50,85),(44,93,51,86),(45,94,52,87),(46,95,53,88),(47,96,54,89),(48,97,55,90),(49,98,56,91)], [(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),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112)], [(2,3,5),(4,7,6),(9,10,12),(11,14,13),(16,17,19),(18,21,20),(23,24,26),(25,28,27),(30,31,33),(32,35,34),(37,38,40),(39,42,41),(44,45,47),(46,49,48),(51,52,54),(53,56,55),(58,59,61),(60,63,62),(65,66,68),(67,70,69),(72,73,75),(74,77,76),(79,80,82),(81,84,83),(86,87,89),(88,91,90),(93,94,96),(95,98,97),(100,101,103),(102,105,104),(107,108,110),(109,112,111)]])

50 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A ··· 4F 6A ··· 6F 7A 7B 12A ··· 12L 14A ··· 14F 28A ··· 28L order 1 2 2 2 3 3 4 ··· 4 6 ··· 6 7 7 12 ··· 12 14 ··· 14 28 ··· 28 size 1 1 1 1 7 7 2 ··· 2 7 ··· 7 3 3 14 ··· 14 3 ··· 3 6 ··· 6

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 3 3 3 6 type + + + - image C1 C2 C2 C3 C6 C6 Q8 C3×Q8 C7⋊C3 C2×C7⋊C3 C2×C7⋊C3 Q8×C7⋊C3 kernel C2×Q8×C7⋊C3 C2×C4×C7⋊C3 Q8×C7⋊C3 Q8×C14 C2×C28 C7×Q8 C2×C7⋊C3 C14 C2×Q8 C2×C4 Q8 C2 # reps 1 3 4 2 6 8 2 4 2 6 8 4

Matrix representation of C2×Q8×C7⋊C3 in GL7(𝔽337)

 336 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 74 297 0 0 0 0 0 297 263 0 0 0 0 0 0 0 263 40 0 0 0 0 0 40 74 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 336
,
 0 1 0 0 0 0 0 336 0 0 0 0 0 0 0 0 0 336 0 0 0 0 0 1 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 336
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 125 0 0 0 0 0 1 124
,
 128 0 0 0 0 0 0 0 128 0 0 0 0 0 0 0 208 0 0 0 0 0 0 0 208 0 0 0 0 0 0 0 1 0 124 0 0 0 0 0 0 336 0 0 0 0 0 1 336

G:=sub<GL(7,GF(337))| [336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[74,297,0,0,0,0,0,297,263,0,0,0,0,0,0,0,263,40,0,0,0,0,0,40,74,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336],[0,336,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,125,124],[128,0,0,0,0,0,0,0,128,0,0,0,0,0,0,0,208,0,0,0,0,0,0,0,208,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,124,336,336] >;

C2×Q8×C7⋊C3 in GAP, Magma, Sage, TeX

C_2\times Q_8\times C_7\rtimes C_3
% in TeX

G:=Group("C2xQ8xC7:C3");
// GroupNames label

G:=SmallGroup(336,166);
// by ID

G=gap.SmallGroup(336,166);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,151,260,122,455]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^7=e^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^4>;
// generators/relations

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