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## G = C22×Dic7order 112 = 24·7

### Direct product of C22 and Dic7

Aliases: C22×Dic7, C23.2D7, C14.9C23, C22.11D14, C142(C2×C4), (C2×C14)⋊3C4, C72(C22×C4), C2.2(C22×D7), (C22×C14).3C2, (C2×C14).12C22, SmallGroup(112,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C22×Dic7
 Chief series C1 — C7 — C14 — Dic7 — C2×Dic7 — C22×Dic7
 Lower central C7 — C22×Dic7
 Upper central C1 — C23

Generators and relations for C22×Dic7
G = < a,b,c,d | a2=b2=c14=1, d2=c7, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 120 in 54 conjugacy classes, 43 normal (7 characteristic)
C1, C2, C2, C4, C22, C7, C2×C4, C23, C14, C14, C22×C4, Dic7, C2×C14, C2×Dic7, C22×C14, C22×Dic7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, Dic7, D14, C2×Dic7, C22×D7, C22×Dic7

Smallest permutation representation of C22×Dic7
Regular action on 112 points
Generators in S112
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 29)(15 54)(16 55)(17 56)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 15)(29 54)(30 55)(31 56)(32 43)(33 44)(34 45)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)(41 52)(42 53)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(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)
(1 71 8 78)(2 84 9 77)(3 83 10 76)(4 82 11 75)(5 81 12 74)(6 80 13 73)(7 79 14 72)(15 58 22 65)(16 57 23 64)(17 70 24 63)(18 69 25 62)(19 68 26 61)(20 67 27 60)(21 66 28 59)(29 100 36 107)(30 99 37 106)(31 112 38 105)(32 111 39 104)(33 110 40 103)(34 109 41 102)(35 108 42 101)(43 97 50 90)(44 96 51 89)(45 95 52 88)(46 94 53 87)(47 93 54 86)(48 92 55 85)(49 91 56 98)

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

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

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

C22×Dic7 is a maximal subgroup of
C14.C42  C23.11D14  C22⋊Dic14  Dic74D4  C22.D28  C23.18D14  Dic7⋊D4  D7×C22×C4  Dic7⋊A4
C22×Dic7 is a maximal quotient of
C23.21D14  Q8.Dic7

40 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 7A 7B 7C 14A ··· 14U order 1 2 ··· 2 4 ··· 4 7 7 7 14 ··· 14 size 1 1 ··· 1 7 ··· 7 2 2 2 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 D7 Dic7 D14 kernel C22×Dic7 C2×Dic7 C22×C14 C2×C14 C23 C22 C22 # reps 1 6 1 8 3 12 9

Matrix representation of C22×Dic7 in GL4(𝔽29) generated by

 28 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 28 0 0 0 0 28 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 28 0 0 9 21
,
 1 0 0 0 0 28 0 0 0 0 2 8 0 0 3 27
G:=sub<GL(4,GF(29))| [28,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,9,0,0,28,21],[1,0,0,0,0,28,0,0,0,0,2,3,0,0,8,27] >;

C22×Dic7 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_7
% in TeX

G:=Group("C2^2xDic7");
// GroupNames label

G:=SmallGroup(112,35);
// by ID

G=gap.SmallGroup(112,35);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,40,2404]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^14=1,d^2=c^7,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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