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G = C22×Dic7order 112 = 24·7

Direct product of C22 and Dic7

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

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

C1C7 — C22×Dic7
C1C7C14Dic7C2×Dic7 — C22×Dic7
C7 — C22×Dic7
C1C23

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 [×6], C4 [×4], C22 [×7], C7, C2×C4 [×6], C23, C14, C14 [×6], C22×C4, Dic7 [×4], C2×C14 [×7], C2×Dic7 [×6], C22×C14, C22×Dic7
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D7, C22×C4, Dic7 [×4], D14 [×3], C2×Dic7 [×6], C22×D7, C22×Dic7

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

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

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

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

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···2G4A···4H7A7B7C14A···14U
order12···24···477714···14
size11···17···72222···2

40 irreducible representations

dim1111222
type++++-+
imageC1C2C2C4D7Dic7D14
kernelC22×Dic7C2×Dic7C22×C14C2×C14C23C22C22
# reps16183129

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

28000
0100
00280
00028
,
28000
02800
0010
0001
,
1000
0100
00128
00921
,
1000
02800
0028
00327
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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