direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C7⋊D4, C28⋊8D4, C23.22D14, C7⋊4(C4×D4), D14⋊4(C2×C4), C22⋊2(C4×D7), (C22×C4)⋊2D7, D14⋊C4⋊18C2, (C22×C28)⋊9C2, Dic7⋊2(C2×C4), C14.41(C2×D4), Dic7⋊C4⋊18C2, (C4×Dic7)⋊16C2, C2.5(C4○D28), (C2×C4).103D14, C23.D7⋊14C2, C14.17(C4○D4), (C2×C28).77C22, C14.19(C22×C4), (C2×C14).46C23, C22.24(C22×D7), (C22×C14).38C22, (C2×Dic7).36C22, (C22×D7).24C22, (C2×C4×D7)⋊14C2, C2.20(C2×C4×D7), (C2×C14)⋊5(C2×C4), C2.3(C2×C7⋊D4), (C2×C7⋊D4).7C2, SmallGroup(224,123)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C7⋊D4
G = < a,b,c,d | a4=b7=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 334 in 94 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C4×D4, C4×D7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C2×C4×D7, C2×C7⋊D4, C22×C28, C4×C7⋊D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, C7⋊D4, C22×D7, C2×C4×D7, C4○D28, C2×C7⋊D4, C4×C7⋊D4
►On 112 points
(1 43 15 29)(2 44 16 30)(3 45 17 31)(4 46 18 32)(5 47 19 33)(6 48 20 34)(7 49 21 35)(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)(57 99 71 85)(58 100 72 86)(59 101 73 87)(60 102 74 88)(61 103 75 89)(62 104 76 90)(63 105 77 91)(64 106 78 92)(65 107 79 93)(66 108 80 94)(67 109 81 95)(68 110 82 96)(69 111 83 97)(70 112 84 98)
(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 57 8 64)(2 63 9 70)(3 62 10 69)(4 61 11 68)(5 60 12 67)(6 59 13 66)(7 58 14 65)(15 71 22 78)(16 77 23 84)(17 76 24 83)(18 75 25 82)(19 74 26 81)(20 73 27 80)(21 72 28 79)(29 85 36 92)(30 91 37 98)(31 90 38 97)(32 89 39 96)(33 88 40 95)(34 87 41 94)(35 86 42 93)(43 99 50 106)(44 105 51 112)(45 104 52 111)(46 103 53 110)(47 102 54 109)(48 101 55 108)(49 100 56 107)
(1 15)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 22)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(29 43)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 50)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(57 78)(58 84)(59 83)(60 82)(61 81)(62 80)(63 79)(64 71)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(85 106)(86 112)(87 111)(88 110)(89 109)(90 108)(91 107)(92 99)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)
G:=sub<Sym(112)| (1,43,15,29)(2,44,16,30)(3,45,17,31)(4,46,18,32)(5,47,19,33)(6,48,20,34)(7,49,21,35)(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)(57,99,71,85)(58,100,72,86)(59,101,73,87)(60,102,74,88)(61,103,75,89)(62,104,76,90)(63,105,77,91)(64,106,78,92)(65,107,79,93)(66,108,80,94)(67,109,81,95)(68,110,82,96)(69,111,83,97)(70,112,84,98), (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), (1,57,8,64)(2,63,9,70)(3,62,10,69)(4,61,11,68)(5,60,12,67)(6,59,13,66)(7,58,14,65)(15,71,22,78)(16,77,23,84)(17,76,24,83)(18,75,25,82)(19,74,26,81)(20,73,27,80)(21,72,28,79)(29,85,36,92)(30,91,37,98)(31,90,38,97)(32,89,39,96)(33,88,40,95)(34,87,41,94)(35,86,42,93)(43,99,50,106)(44,105,51,112)(45,104,52,111)(46,103,53,110)(47,102,54,109)(48,101,55,108)(49,100,56,107), (1,15)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,22)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(29,43)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,50)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(57,78)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)(64,71)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(85,106)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,99)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)>;
G:=Group( (1,43,15,29)(2,44,16,30)(3,45,17,31)(4,46,18,32)(5,47,19,33)(6,48,20,34)(7,49,21,35)(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)(57,99,71,85)(58,100,72,86)(59,101,73,87)(60,102,74,88)(61,103,75,89)(62,104,76,90)(63,105,77,91)(64,106,78,92)(65,107,79,93)(66,108,80,94)(67,109,81,95)(68,110,82,96)(69,111,83,97)(70,112,84,98), (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), (1,57,8,64)(2,63,9,70)(3,62,10,69)(4,61,11,68)(5,60,12,67)(6,59,13,66)(7,58,14,65)(15,71,22,78)(16,77,23,84)(17,76,24,83)(18,75,25,82)(19,74,26,81)(20,73,27,80)(21,72,28,79)(29,85,36,92)(30,91,37,98)(31,90,38,97)(32,89,39,96)(33,88,40,95)(34,87,41,94)(35,86,42,93)(43,99,50,106)(44,105,51,112)(45,104,52,111)(46,103,53,110)(47,102,54,109)(48,101,55,108)(49,100,56,107), (1,15)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,22)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(29,43)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,50)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(57,78)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)(64,71)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(85,106)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,99)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100) );
G=PermutationGroup([[(1,43,15,29),(2,44,16,30),(3,45,17,31),(4,46,18,32),(5,47,19,33),(6,48,20,34),(7,49,21,35),(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),(57,99,71,85),(58,100,72,86),(59,101,73,87),(60,102,74,88),(61,103,75,89),(62,104,76,90),(63,105,77,91),(64,106,78,92),(65,107,79,93),(66,108,80,94),(67,109,81,95),(68,110,82,96),(69,111,83,97),(70,112,84,98)], [(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)], [(1,57,8,64),(2,63,9,70),(3,62,10,69),(4,61,11,68),(5,60,12,67),(6,59,13,66),(7,58,14,65),(15,71,22,78),(16,77,23,84),(17,76,24,83),(18,75,25,82),(19,74,26,81),(20,73,27,80),(21,72,28,79),(29,85,36,92),(30,91,37,98),(31,90,38,97),(32,89,39,96),(33,88,40,95),(34,87,41,94),(35,86,42,93),(43,99,50,106),(44,105,51,112),(45,104,52,111),(46,103,53,110),(47,102,54,109),(48,101,55,108),(49,100,56,107)], [(1,15),(2,21),(3,20),(4,19),(5,18),(6,17),(7,16),(8,22),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(29,43),(30,49),(31,48),(32,47),(33,46),(34,45),(35,44),(36,50),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(57,78),(58,84),(59,83),(60,82),(61,81),(62,80),(63,79),(64,71),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(85,106),(86,112),(87,111),(88,110),(89,109),(90,108),(91,107),(92,99),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100)]])
C4×C7⋊D4 is a maximal subgroup of
C7⋊D4⋊C8 D14⋊2M4(2) Dic7⋊M4(2) C7⋊C8⋊26D4 C56⋊32D4 C56⋊D4 C56⋊18D4 C42.277D14 C24.24D14 C24.27D14 C24.30D14 C24.31D14 C14.82+ 1+4 C14.2- 1+4 C14.102+ 1+4 C14.52- 1+4 C14.112+ 1+4 C14.62- 1+4 C42.93D14 C42.94D14 C42.95D14 C42.97D14 C42.98D14 C42.102D14 C42.104D14 C4×D4×D7 C42⋊11D14 C42.108D14 C42⋊12D14 C42.228D14 C42⋊16D14 C42.229D14 C42.113D14 C42.114D14 C42⋊17D14 C42.118D14 Dic14⋊19D4 Dic14⋊20D4 C14.342+ 1+4 D28⋊19D4 C14.402+ 1+4 C14.732- 1+4 D28⋊20D4 C14.422+ 1+4 C14.432+ 1+4 C14.442+ 1+4 C14.452+ 1+4 C14.1152+ 1+4 D28⋊21D4 D28⋊22D4 Dic14⋊21D4 Dic14⋊22D4 C14.1182+ 1+4 C14.522+ 1+4 C14.532+ 1+4 C14.202- 1+4 C14.212- 1+4 C14.222- 1+4 C14.232- 1+4 C14.772- 1+4 C14.612+ 1+4 C14.622+ 1+4 C14.832- 1+4 C14.642+ 1+4 C14.842- 1+4 C14.662+ 1+4 C14.672+ 1+4 C24.72D14 C24.42D14 C14.452- 1+4 C14.1042- 1+4 (C2×C28)⋊15D4 C14.1452+ 1+4 C14.1072- 1+4 (C2×C28)⋊17D4 C14.1482+ 1+4
C4×C7⋊D4 is a maximal quotient of
(C2×C42).D7 (C2×C42)⋊D7 C24.3D14 C24.4D14 C24.12D14 C24.13D14 Dic7⋊(C4⋊C4) C22.23(Q8×D7) D14⋊C4⋊6C4 D14⋊C4⋊7C4 C42.48D14 C42.51D14 C42.56D14 C42.59D14 C56⋊32D4 C56⋊D4 C56⋊18D4 C56.93D4 C56.50D4 C24.62D14 C23.28D28
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | C4×D7 | C4○D28 |
| kernel | C4×C7⋊D4 | C4×Dic7 | Dic7⋊C4 | D14⋊C4 | C23.D7 | C2×C4×D7 | C2×C7⋊D4 | C22×C28 | C7⋊D4 | C28 | C22×C4 | C14 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 3 | 2 | 6 | 3 | 12 | 12 | 12 |
Matrix representation of C4×C7⋊D4 ►in GL4(𝔽29) generated by
| 17 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 28 | 28 | 0 | 0 |
| 5 | 4 | 0 | 0 |
| 0 | 0 | 18 | 28 |
| 0 | 0 | 1 | 0 |
| 0 | 8 | 0 | 0 |
| 11 | 0 | 0 | 0 |
| 0 | 0 | 11 | 2 |
| 0 | 0 | 26 | 18 |
| 0 | 21 | 0 | 0 |
| 18 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 11 | 1 |
G:=sub<GL(4,GF(29))| [17,0,0,0,0,17,0,0,0,0,12,0,0,0,0,12],[28,5,0,0,28,4,0,0,0,0,18,1,0,0,28,0],[0,11,0,0,8,0,0,0,0,0,11,26,0,0,2,18],[0,18,0,0,21,0,0,0,0,0,28,11,0,0,0,1] >;
C4×C7⋊D4 in GAP, Magma, Sage, TeX
C_4\times C_7\rtimes D_4
% in TeX
G:=Group("C4xC7:D4"); // GroupNames label
G:=SmallGroup(224,123);
// by ID
G=gap.SmallGroup(224,123);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,50,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^7=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations