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G = C23×D7order 112 = 24·7

Direct product of C23 and D7

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

Aliases: C23×D7, C7⋊C24, C14⋊C23, (C22×C14)⋊3C2, (C2×C14)⋊4C22, SmallGroup(112,42)

Series: Derived Chief Lower central Upper central

C1C7 — C23×D7
C1C7D7D14C22×D7 — C23×D7
C7 — C23×D7
C1C23

Generators and relations for C23×D7
 G = < a,b,c,d,e | a2=b2=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 440 in 134 conjugacy classes, 83 normal (5 characteristic)
C1, C2 [×7], C2 [×8], C22 [×7], C22 [×28], C7, C23, C23 [×14], D7 [×8], C14 [×7], C24, D14 [×28], C2×C14 [×7], C22×D7 [×14], C22×C14, C23×D7
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], C22×D7 [×7], C23×D7

Smallest permutation representation of C23×D7
On 56 points
Generators in S56
(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)
(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)
(1 33)(2 32)(3 31)(4 30)(5 29)(6 35)(7 34)(8 38)(9 37)(10 36)(11 42)(12 41)(13 40)(14 39)(15 45)(16 44)(17 43)(18 49)(19 48)(20 47)(21 46)(22 52)(23 51)(24 50)(25 56)(26 55)(27 54)(28 53)

G:=sub<Sym(56)| (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56), (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,35)(7,34)(8,38)(9,37)(10,36)(11,42)(12,41)(13,40)(14,39)(15,45)(16,44)(17,43)(18,49)(19,48)(20,47)(21,46)(22,52)(23,51)(24,50)(25,56)(26,55)(27,54)(28,53)>;

G:=Group( (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56), (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,35)(7,34)(8,38)(9,37)(10,36)(11,42)(12,41)(13,40)(14,39)(15,45)(16,44)(17,43)(18,49)(19,48)(20,47)(21,46)(22,52)(23,51)(24,50)(25,56)(26,55)(27,54)(28,53) );

G=PermutationGroup([(1,34),(2,35),(3,29),(4,30),(5,31),(6,32),(7,33),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56)], [(1,27),(2,28),(3,22),(4,23),(5,24),(6,25),(7,26),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49)], [(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56)], [(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,35),(7,34),(8,38),(9,37),(10,36),(11,42),(12,41),(13,40),(14,39),(15,45),(16,44),(17,43),(18,49),(19,48),(20,47),(21,46),(22,52),(23,51),(24,50),(25,56),(26,55),(27,54),(28,53)])

C23×D7 is a maximal subgroup of   C22⋊D28  C23⋊D14
C23×D7 is a maximal quotient of   D46D14  Q8.10D14  D48D14  D4.10D14

40 conjugacy classes

class 1 2A···2G2H···2O7A7B7C14A···14U
order12···22···277714···14
size11···17···72222···2

40 irreducible representations

dim11122
type+++++
imageC1C2C2D7D14
kernelC23×D7C22×D7C22×C14C23C22
# reps1141321

Matrix representation of C23×D7 in GL4(𝔽29) generated by

1000
02800
0010
0001
,
28000
0100
00280
00028
,
1000
0100
00280
00028
,
1000
0100
0001
002818
,
28000
0100
0001
0010
G:=sub<GL(4,GF(29))| [1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,0,28,0,0,1,18],[28,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C23×D7 in GAP, Magma, Sage, TeX

C_2^3\times D_7
% in TeX

G:=Group("C2^3xD7");
// GroupNames label

G:=SmallGroup(112,42);
// by ID

G=gap.SmallGroup(112,42);
# by ID

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

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

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