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## G = (D4×C14)⋊10C4order 448 = 26·7

### 6th semidirect product of D4×C14 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — (D4×C14)⋊10C4
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C23.D7 — C23.21D14 — (D4×C14)⋊10C4
 Lower central C7 — C14 — C2×C14 — (D4×C14)⋊10C4
 Upper central C1 — C4 — C22×C4 — C2×C4○D4

Generators and relations for (D4×C14)⋊10C4
G = < a,b,c,d | a14=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, bd=db, dcd-1=a7b2c >

Subgroups: 532 in 158 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, C28, C2×C14, C2×C14, C2×C14, C23⋊C4, C42⋊C2, C2×C4○D4, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C23.C23, C4×Dic7, C4⋊Dic7, C23.D7, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C23⋊Dic7, C23.21D14, C14×C4○D4, (D4×C14)⋊10C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C2×C22⋊C4, C2×Dic7, C7⋊D4, C22×D7, C23.C23, C23.D7, C22×Dic7, C2×C7⋊D4, C2×C23.D7, (D4×C14)⋊10C4

Smallest permutation representation of (D4×C14)⋊10C4
On 112 points
Generators in S112
(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 60 36 23)(2 61 37 24)(3 62 38 25)(4 63 39 26)(5 64 40 27)(6 65 41 28)(7 66 42 15)(8 67 29 16)(9 68 30 17)(10 69 31 18)(11 70 32 19)(12 57 33 20)(13 58 34 21)(14 59 35 22)(43 107 73 86)(44 108 74 87)(45 109 75 88)(46 110 76 89)(47 111 77 90)(48 112 78 91)(49 99 79 92)(50 100 80 93)(51 101 81 94)(52 102 82 95)(53 103 83 96)(54 104 84 97)(55 105 71 98)(56 106 72 85)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 71)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 85)(68 86)(69 87)(70 88)
(2 35)(3 13)(4 33)(5 11)(6 31)(7 9)(8 29)(10 41)(12 39)(14 37)(15 17)(16 67)(18 65)(19 27)(20 63)(21 25)(22 61)(24 59)(26 57)(28 69)(30 42)(32 40)(34 38)(43 48 73 78)(44 77 74 47)(45 46 75 76)(49 56 79 72)(50 71 80 55)(51 54 81 84)(52 83 82 53)(58 62)(64 70)(66 68)(85 99 106 92)(86 91 107 112)(87 111 108 90)(88 89 109 110)(93 105 100 98)(94 97 101 104)(95 103 102 96)

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H ··· 4O 7A 7B 7C 14A ··· 14I 14J ··· 14AA 28A ··· 28L 28M ··· 28AD order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 ··· 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 2 2 4 4 1 1 2 2 2 4 4 28 ··· 28 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + - + - image C1 C2 C2 C2 C4 C4 C4 D4 D7 Dic7 D14 Dic7 D14 Dic7 C7⋊D4 C23.C23 (D4×C14)⋊10C4 kernel (D4×C14)⋊10C4 C23⋊Dic7 C23.21D14 C14×C4○D4 C22×C28 D4×C14 Q8×C14 C2×C28 C2×C4○D4 C22×C4 C22×C4 C2×D4 C2×D4 C2×Q8 C2×C4 C7 C1 # reps 1 4 2 1 4 2 2 4 3 6 3 3 6 3 24 2 12

Matrix representation of (D4×C14)⋊10C4 in GL4(𝔽29) generated by

 8 21 0 0 8 10 0 0 0 0 8 21 0 0 8 10
,
 12 0 0 0 0 12 0 0 0 0 17 0 0 0 0 17
,
 0 0 21 23 0 0 6 8 8 6 0 0 23 21 0 0
,
 1 0 0 0 7 28 0 0 0 0 9 14 0 0 19 20
G:=sub<GL(4,GF(29))| [8,8,0,0,21,10,0,0,0,0,8,8,0,0,21,10],[12,0,0,0,0,12,0,0,0,0,17,0,0,0,0,17],[0,0,8,23,0,0,6,21,21,6,0,0,23,8,0,0],[1,7,0,0,0,28,0,0,0,0,9,19,0,0,14,20] >;

(D4×C14)⋊10C4 in GAP, Magma, Sage, TeX

(D_4\times C_{14})\rtimes_{10}C_4
% in TeX

G:=Group("(D4xC14):10C4");
// GroupNames label

G:=SmallGroup(448,774);
// by ID

G=gap.SmallGroup(448,774);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,422,297,1684,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^7*b^2*c>;
// generators/relations

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