direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×C22⋊C4, D14.10D4, C23.13D14, (C2×C4)⋊5D14, C2.1(D4×D7), D14⋊C4⋊9C2, D14⋊5(C2×C4), C22⋊3(C4×D7), (C2×C28)⋊6C22, C14.17(C2×D4), (C22×D7)⋊2C4, C23.D7⋊3C2, C14.6(C22×C4), (C23×D7).1C2, (C2×C14).21C23, (C2×Dic7)⋊5C22, C22.13(C22×D7), (C22×C14).10C22, (C22×D7).33C22, (C2×C4×D7)⋊8C2, C2.8(C2×C4×D7), C7⋊1(C2×C22⋊C4), (C2×C14)⋊1(C2×C4), (C7×C22⋊C4)⋊8C2, SmallGroup(224,75)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×C22⋊C4
G = < a,b,c,d,e | a7=b2=c2=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >
Subgroups: 614 in 132 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, D7, D7, C14, C14, C14, C22⋊C4, C22⋊C4, C22×C4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C4×D7, C2×Dic7, C2×C28, C22×D7, C22×D7, C22×D7, C22×C14, D14⋊C4, C23.D7, C7×C22⋊C4, C2×C4×D7, C23×D7, D7×C22⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C4×D7, C22×D7, C2×C4×D7, D4×D7, D7×C22⋊C4
►On 56 points
(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 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(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 48 20 34)(2 49 21 35)(3 43 15 29)(4 44 16 30)(5 45 17 31)(6 46 18 32)(7 47 19 33)(8 50 22 36)(9 51 23 37)(10 52 24 38)(11 53 25 39)(12 54 26 40)(13 55 27 41)(14 56 28 42)
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(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), (1,48,20,34)(2,49,21,35)(3,43,15,29)(4,44,16,30)(5,45,17,31)(6,46,18,32)(7,47,19,33)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)>;
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(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), (1,48,20,34)(2,49,21,35)(3,43,15,29)(4,44,16,30)(5,45,17,31)(6,46,18,32)(7,47,19,33)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42) );
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,14),(7,13),(15,24),(16,23),(17,22),(18,28),(19,27),(20,26),(21,25),(29,38),(30,37),(31,36),(32,42),(33,41),(34,40),(35,39),(43,52),(44,51),(45,50),(46,56),(47,55),(48,54),(49,53)], [(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(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)], [(1,48,20,34),(2,49,21,35),(3,43,15,29),(4,44,16,30),(5,45,17,31),(6,46,18,32),(7,47,19,33),(8,50,22,36),(9,51,23,37),(10,52,24,38),(11,53,25,39),(12,54,26,40),(13,55,27,41),(14,56,28,42)]])
D7×C22⋊C4 is a maximal subgroup of
C24.24D14 C24.27D14 C42⋊7D14 C42⋊10D14 C4×D4×D7 C42⋊11D14 D28⋊23D4 C42⋊16D14 C24⋊2D14 C24.33D14 C14.402+ 1+4 D28⋊20D4 C14.422+ 1+4 D28⋊21D4 C14.512+ 1+4 C14.532+ 1+4 C14.1202+ 1+4 C14.1212+ 1+4 C14.612+ 1+4 C14.1222+ 1+4 C14.622+ 1+4 D28⋊10D4 C42⋊20D14 C42⋊21D14 C42⋊23D14 C42⋊24D14
D7×C22⋊C4 is a maximal quotient of
(C2×C28)⋊Q8 C22.58(D4×D7) (C2×C4)⋊9D28 D14⋊C42 D14⋊(C4⋊C4) D14⋊M4(2) D14⋊C8⋊C2 C23⋊C4⋊5D7 M4(2).19D14 M4(2).21D14 (D4×D7)⋊C4 D4⋊(C4×D7) D4⋊2D7⋊C4 (Q8×D7)⋊C4 Q8⋊(C4×D7) Q8⋊2D7⋊C4 C42⋊D14 C24.44D14 C23.44D28 C24.12D14 C24.13D14
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 7 | 7 | 7 | 7 | 14 | 14 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | D14 | D14 | C4×D7 | D4×D7 |
| kernel | D7×C22⋊C4 | D14⋊C4 | C23.D7 | C7×C22⋊C4 | C2×C4×D7 | C23×D7 | C22×D7 | D14 | C22⋊C4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 4 | 3 | 6 | 3 | 12 | 6 |
Matrix representation of D7×C22⋊C4 ►in GL4(𝔽29) generated by
| 22 | 1 | 0 | 0 |
| 16 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 10 | 28 | 0 | 0 |
| 12 | 19 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 |
G:=sub<GL(4,GF(29))| [22,16,0,0,1,10,0,0,0,0,1,0,0,0,0,1],[10,12,0,0,28,19,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[12,0,0,0,0,12,0,0,0,0,0,28,0,0,1,0] >;
D7×C22⋊C4 in GAP, Magma, Sage, TeX
D_7\times C_2^2\rtimes C_4
% in TeX
G:=Group("D7xC2^2:C4"); // GroupNames label
G:=SmallGroup(224,75);
// by ID
G=gap.SmallGroup(224,75);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,188,50,6917]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^2=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations