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## G = C42⋊23D14order 448 = 26·7

### 23rd semidirect product of C42 and D14 acting via D14/C7=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C42⋊23D14
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22⋊C4 — C42⋊23D14
 Lower central C7 — C2×C14 — C42⋊23D14
 Upper central C1 — C22 — C42⋊2C2

Generators and relations for C4223D14
G = < a,b,c,d | a4=b4=c14=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=b-1, dcd=c-1 >

Subgroups: 1484 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C422C2, C422C2, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×D7, C22×C14, C22.32C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C2×C7⋊D4, C23×D7, C4×D28, C4.D28, D7×C22⋊C4, C22⋊D28, D14.D4, D14⋊D4, Dic7.D4, D28⋊C4, D14.5D4, C4⋊D28, D14⋊Q8, C4⋊C4⋊D7, C7×C422C2, C4223D14
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.32C24, C23×D7, D7×C4○D4, D48D14, C4223D14

Smallest permutation representation of C4223D14
On 112 points
Generators in S112
(1 74 13 94)(2 82 14 88)(3 76 8 96)(4 84 9 90)(5 78 10 98)(6 72 11 92)(7 80 12 86)(15 81 22 87)(16 75 23 95)(17 83 24 89)(18 77 25 97)(19 71 26 91)(20 79 27 85)(21 73 28 93)(29 109 36 47)(30 55 37 103)(31 111 38 49)(32 43 39 105)(33 99 40 51)(34 45 41 107)(35 101 42 53)(44 59 106 66)(46 61 108 68)(48 63 110 70)(50 65 112 58)(52 67 100 60)(54 69 102 62)(56 57 104 64)
(1 64 22 38)(2 58 23 32)(3 66 24 40)(4 60 25 34)(5 68 26 42)(6 62 27 36)(7 70 28 30)(8 59 17 33)(9 67 18 41)(10 61 19 35)(11 69 20 29)(12 63 21 37)(13 57 15 31)(14 65 16 39)(43 82 50 95)(44 89 51 76)(45 84 52 97)(46 91 53 78)(47 72 54 85)(48 93 55 80)(49 74 56 87)(71 101 98 108)(73 103 86 110)(75 105 88 112)(77 107 90 100)(79 109 92 102)(81 111 94 104)(83 99 96 106)
(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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 21)(16 20)(17 19)(22 28)(23 27)(24 26)(29 65)(30 64)(31 63)(32 62)(33 61)(34 60)(35 59)(36 58)(37 57)(38 70)(39 69)(40 68)(41 67)(42 66)(43 109)(44 108)(45 107)(46 106)(47 105)(48 104)(49 103)(50 102)(51 101)(52 100)(53 99)(54 112)(55 111)(56 110)(71 76)(72 75)(73 74)(77 84)(78 83)(79 82)(80 81)(85 88)(86 87)(89 98)(90 97)(91 96)(92 95)(93 94)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C ··· 4G 4H 4I 4J 4K 4L 7A 7B 7C 14A ··· 14I 14J 14K 14L 28A ··· 28R 28S ··· 28AA order 1 2 2 2 2 2 2 2 2 2 4 4 4 ··· 4 4 4 4 4 4 7 7 7 14 ··· 14 14 14 14 28 ··· 28 28 ··· 28 size 1 1 1 1 4 14 14 28 28 28 2 2 4 ··· 4 14 14 28 28 28 2 2 2 2 ··· 2 8 8 8 4 ··· 4 8 ··· 8

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D7 C4○D4 D14 D14 D14 2+ 1+4 D7×C4○D4 D4⋊8D14 kernel C42⋊23D14 C4×D28 C4.D28 D7×C22⋊C4 C22⋊D28 D14.D4 D14⋊D4 Dic7.D4 D28⋊C4 D14.5D4 C4⋊D28 D14⋊Q8 C4⋊C4⋊D7 C7×C42⋊2C2 C42⋊2C2 D14 C42 C22⋊C4 C4⋊C4 C14 C2 C2 # reps 1 1 1 1 2 1 1 1 1 1 2 1 1 1 3 4 3 9 9 2 6 12

Matrix representation of C4223D14 in GL6(𝔽29)

 17 0 0 0 0 0 0 17 0 0 0 0 0 0 28 0 27 0 0 0 0 28 0 27 0 0 0 0 1 0 0 0 0 0 0 1
,
 28 28 0 0 0 0 2 1 0 0 0 0 0 0 27 11 0 0 0 0 18 2 0 0 0 0 0 0 27 11 0 0 0 0 18 2
,
 1 0 0 0 0 0 27 28 0 0 0 0 0 0 21 21 0 0 0 0 8 26 0 0 0 0 8 8 8 8 0 0 21 3 21 3
,
 28 0 0 0 0 0 2 1 0 0 0 0 0 0 21 21 0 0 0 0 26 8 0 0 0 0 8 8 8 8 0 0 3 21 3 21

G:=sub<GL(6,GF(29))| [17,0,0,0,0,0,0,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,27,0,1,0,0,0,0,27,0,1],[28,2,0,0,0,0,28,1,0,0,0,0,0,0,27,18,0,0,0,0,11,2,0,0,0,0,0,0,27,18,0,0,0,0,11,2],[1,27,0,0,0,0,0,28,0,0,0,0,0,0,21,8,8,21,0,0,21,26,8,3,0,0,0,0,8,21,0,0,0,0,8,3],[28,2,0,0,0,0,0,1,0,0,0,0,0,0,21,26,8,3,0,0,21,8,8,21,0,0,0,0,8,3,0,0,0,0,8,21] >;

C4223D14 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{23}D_{14}
% in TeX

G:=Group("C4^2:23D14");
// GroupNames label

G:=SmallGroup(448,1157);
// by ID

G=gap.SmallGroup(448,1157);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,184,675,570,192,18822]);
// Polycyclic

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

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