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G = C24.36D14  order 448 = 26·7

36th non-split extension by C24 of D14 acting via D14/C7=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.36D14, C14.312+ 1+4, C22≀C2⋊8D7, C28⋊D4⋊13C2, C28⋊2D4⋊15C2, (C2×D4).88D14, C22⋊C4.3D14, D14⋊D4⋊15C2, D14⋊C4⋊16C22, Dic7⋊D4⋊6C2, (C2×D28)⋊21C22, C24⋊D7⋊10C2, (C2×C28).33C23, C4⋊Dic7⋊28C22, D14.D4⋊15C2, (C2×C14).139C24, Dic7⋊C4⋊13C22, (C4×Dic7)⋊19C22, C23.D7⋊19C22, C2.33(D4⋊6D14), C7⋊1(C22.54C24), (D4×C14).113C22, C23.18D14⋊6C2, C23.D14⋊13C2, (C23×C14).71C22, (C2×Dic7).64C23, (C22×D7).58C23, C22.160(C23×D7), C23.111(C22×D7), (C22×C14).184C23, (C22×Dic7)⋊17C22, (C2×C4×D7)⋊11C22, (C7×C22≀C2)⋊10C2, (C2×C7⋊D4)⋊11C22, (C2×C4).33(C22×D7), (C7×C22⋊C4).4C22, SmallGroup(448,1048)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C24.36D14
Chief series
C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — D14.D4 — C24.36D14
Lower central
C7 — C2×C14 — C24.36D14
Upper central
C1 — C22 — C22≀C2

Generators and relations for C24.36D14
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=f2=d, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >

Subgroups: 1260 in 252 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, Dic7, C28, D14, C2×C14, C2×C14, C22≀C2, C22≀C2, C4⋊D4, C22.D4, C42⋊2C2, C4⋊1D4, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22×C14, C22×C14, C22.54C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C23.D7, C7×C22⋊C4, C7×C22⋊C4, C2×C4×D7, C2×D28, C22×Dic7, C2×C7⋊D4, D4×C14, D4×C14, C23×C14, C23.D14, D14.D4, D14⋊D4, C23.18D14, C28⋊2D4, Dic7⋊D4, C28⋊D4, C24⋊D7, C7×C22≀C2, C24.36D14
Quotients: C1, C2, C22, C23, D7, C24, D14, 2+ 1+4, C22×D7, C22.54C24, C23×D7, D4⋊6D14, C24.36D14

Smallest permutation representation of C24.36D14
►On 112 points
Generators in S112
(2 86)(4 88)(6 90)(8 92)(10 94)(12 96)(14 98)(16 100)(18 102)(20 104)(22 106)(24 108)(26 110)(28 112)(29 61)(30 44)(31 63)(32 46)(33 65)(34 48)(35 67)(36 50)(37 69)(38 52)(39 71)(40 54)(41 73)(42 56)(43 75)(45 77)(47 79)(49 81)(51 83)(53 57)(55 59)(58 72)(60 74)(62 76)(64 78)(66 80)(68 82)(70 84)

(2 16)(4 18)(6 20)(8 22)(10 24)(12 26)(14 28)(29 75)(30 62)(31 77)(32 64)(33 79)(34 66)(35 81)(36 68)(37 83)(38 70)(39 57)(40 72)(41 59)(42 74)(43 61)(44 76)(45 63)(46 78)(47 65)(48 80)(49 67)(50 82)(51 69)(52 84)(53 71)(54 58)(55 73)(56 60)(86 100)(88 102)(90 104)(92 106)(94 108)(96 110)(98 112)

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

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

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

61 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D···4I7A7B7C14A···14I14J···14AA14AB14AC14AD28A···28I
order12222222224444···477714···1414···1414141428···28
size11114444282844428···282222···24···48888···8

61 irreducible representations

dim1111111111222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D7D14D14D142+ 1+4D4⋊6D14
kernelC24.36D14C23.D14D14.D4D14⋊D4C23.18D14C28⋊2D4Dic7⋊D4C28⋊D4C24⋊D7C7×C22≀C2C22≀C2C22⋊C4C2×D4C24C14C2
# reps12221221213993318

Matrix representation of C24.36D14 ►in GL8(𝔽29)

10000000
128000000
00100000
100280000
00001000
00000100
000000280
000000028
,
10000000
01000000
102800000
100280000
00001000
000002800
000000280
00000001
,
280000000
028000000
002800000
000280000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000028000
000002800
000000280
000000028
,
127000000
028000000
028010000
028100000
00000700
000022000
000000025
00000040
,
100270000
001280000
010280000
000280000
000000025
00000040
00000700
000022000

G:=sub<GL(8,GF(29))| [1,1,0,1,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,27,28,28,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,25,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,28,28,28,0,0,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,7,0,0,0,0,0,0,4,0,0,0,0,0,0,25,0,0,0] >;
 

C24.36D14 in GAP, Magma, Sage, TeX 

C_2^4._{36}D_{14}
 
% in TeX
 
G:=Group("C2^4.36D14");
 
// GroupNames label
 
G:=SmallGroup(448,1048);
 
// by ID
 
G=gap.SmallGroup(448,1048);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,1571,570,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=f^2=d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^13>;
 
// generators/relations
 

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