direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C23⋊C4, C23⋊2C28, (C2×C4)⋊C28, (C2×C28)⋊2C4, C22⋊C4⋊1C14, (C22×C14)⋊1C4, (C2×D4).1C14, (D4×C14).7C2, (C2×C14).21D4, C22.2(C7×D4), C23.1(C2×C14), C22.2(C2×C28), C14.21(C22⋊C4), (C22×C14).1C22, (C7×C22⋊C4)⋊2C2, C2.3(C7×C22⋊C4), (C2×C14).19(C2×C4), SmallGroup(224,48)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23⋊C4
G = < a,b,c,d,e | a7=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >
Smallest permutation representation of C7×C23⋊C4
►On 56 points
Generators in S56
(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 19)(2 20)(3 21)(4 15)(5 16)(6 17)(7 18)(8 48)(9 49)(10 43)(11 44)(12 45)(13 46)(14 47)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(28 35)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 50)
(1 42)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)(15 53)(16 54)(17 55)(18 56)(19 50)(20 51)(21 52)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 43)(7 44)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 55)(23 56)(24 50)(25 51)(26 52)(27 53)(28 54)(29 40)(30 41)(31 42)(32 36)(33 37)(34 38)(35 39)
(8 27 15 53)(9 28 16 54)(10 22 17 55)(11 23 18 56)(12 24 19 50)(13 25 20 51)(14 26 21 52)(29 40)(30 41)(31 42)(32 36)(33 37)(34 38)(35 39)
G:=sub<Sym(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), (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,48)(9,49)(10,43)(11,44)(12,45)(13,46)(14,47)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(28,35)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,50), (1,42)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,53)(16,54)(17,55)(18,56)(19,50)(20,51)(21,52)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49), (1,45)(2,46)(3,47)(4,48)(5,49)(6,43)(7,44)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,55)(23,56)(24,50)(25,51)(26,52)(27,53)(28,54)(29,40)(30,41)(31,42)(32,36)(33,37)(34,38)(35,39), (8,27,15,53)(9,28,16,54)(10,22,17,55)(11,23,18,56)(12,24,19,50)(13,25,20,51)(14,26,21,52)(29,40)(30,41)(31,42)(32,36)(33,37)(34,38)(35,39)>;
G:=Group( (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), (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,48)(9,49)(10,43)(11,44)(12,45)(13,46)(14,47)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(28,35)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,50), (1,42)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,53)(16,54)(17,55)(18,56)(19,50)(20,51)(21,52)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49), (1,45)(2,46)(3,47)(4,48)(5,49)(6,43)(7,44)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,55)(23,56)(24,50)(25,51)(26,52)(27,53)(28,54)(29,40)(30,41)(31,42)(32,36)(33,37)(34,38)(35,39), (8,27,15,53)(9,28,16,54)(10,22,17,55)(11,23,18,56)(12,24,19,50)(13,25,20,51)(14,26,21,52)(29,40)(30,41)(31,42)(32,36)(33,37)(34,38)(35,39) );
G=PermutationGroup([[(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)], [(1,19),(2,20),(3,21),(4,15),(5,16),(6,17),(7,18),(8,48),(9,49),(10,43),(11,44),(12,45),(13,46),(14,47),(22,29),(23,30),(24,31),(25,32),(26,33),(27,34),(28,35),(36,51),(37,52),(38,53),(39,54),(40,55),(41,56),(42,50)], [(1,42),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,27),(9,28),(10,22),(11,23),(12,24),(13,25),(14,26),(15,53),(16,54),(17,55),(18,56),(19,50),(20,51),(21,52),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,43),(7,44),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(22,55),(23,56),(24,50),(25,51),(26,52),(27,53),(28,54),(29,40),(30,41),(31,42),(32,36),(33,37),(34,38),(35,39)], [(8,27,15,53),(9,28,16,54),(10,22,17,55),(11,23,18,56),(12,24,19,50),(13,25,20,51),(14,26,21,52),(29,40),(30,41),(31,42),(32,36),(33,37),(34,38),(35,39)]])
C7×C23⋊C4 is a maximal subgroup of
C7⋊C2≀C4 (C2×C28).D4 C23.D28 C23.2D28 C23⋊C4⋊5D7 C23⋊D28 C23.5D28
77 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4E | 7A | ··· | 7F | 14A | ··· | 14F | 14G | ··· | 14X | 14Y | ··· | 14AD | 28A | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C28 | C28 | D4 | C7×D4 | C23⋊C4 | C7×C23⋊C4 |
| kernel | C7×C23⋊C4 | C7×C22⋊C4 | D4×C14 | C2×C28 | C22×C14 | C23⋊C4 | C22⋊C4 | C2×D4 | C2×C4 | C23 | C2×C14 | C22 | C7 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 6 | 12 | 6 | 12 | 12 | 2 | 12 | 1 | 6 |
Matrix representation of C7×C23⋊C4 ►in GL4(𝔽29) generated by
| 25 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(29))| [25,0,0,0,0,25,0,0,0,0,25,0,0,0,0,25],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,28,0,0,0,0,0,1,0,0,28,0] >;
C7×C23⋊C4 in GAP, Magma, Sage, TeX
C_7\times C_2^3\rtimes C_4
% in TeX
G:=Group("C7xC2^3:C4"); // GroupNames label
G:=SmallGroup(224,48);
// by ID
G=gap.SmallGroup(224,48);
# by ID
G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,3363,2530]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^2=e^4=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,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations
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