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G = C24.152D4  order 128 = 27

7th non-split extension by C24 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C24.152D4, C4.18(C2×C42), (C2×C4).20C42, C42⋊C2⋊16C4, (C22×C4).36Q8, C23.61(C4⋊C4), (C2×M4(2))⋊14C4, (C22×C4).255D4, C23.729(C2×D4), C22.4Q16⋊44C2, C22.53(C8⋊C22), (C23×C4).218C22, (C22×C8).372C22, C2.2(M4(2)⋊C4), C23.115(C22⋊C4), C4.18(C2.C42), (C22×C4).1302C23, C2.2(C23.37D4), C2.2(C23.38D4), C2.2(C23.36D4), C22.42(C8.C22), (C22×M4(2)).12C2, C22.11(C2.C42), (C2×C4⋊C4)⋊23C4, C4⋊C4⋊35(C2×C4), (C2×C8)⋊22(C2×C4), C4.28(C2×C4⋊C4), (C2×C4).39(C4⋊C4), C4.82(C2×C22⋊C4), C22.54(C2×C4⋊C4), (C2×C4).181(C2×Q8), (C2×C4).1293(C2×D4), (C22×C4⋊C4).10C2, (C2×C4⋊C4).741C22, (C2×C42⋊C2).8C2, (C22×C4).254(C2×C4), (C2×C4).346(C22×C4), (C2×C4).252(C22⋊C4), C22.107(C2×C22⋊C4), C2.13(C2×C2.C42), SmallGroup(128,468)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — C24.152D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C24.152D4
Lower central
C1 — C2 — C4 — C24.152D4
Upper central
C1 — C23 — C23×C4 — C24.152D4
Jennings
C1 — C2 — C2 — C22×C4 — C24.152D4

Generators and relations for C24.152D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=cb=bc, ab=ba, ac=ca, eae-1=faf-1=ad=da, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bde3 >

Subgroups: 388 in 224 conjugacy classes, 116 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C23×C4, C22.4Q16, C22×C4⋊C4, C2×C42⋊C2, C22×M4(2), C24.152D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C8⋊C22, C8.C22, C2×C2.C42, C23.36D4, C23.37D4, C23.38D4, M4(2)⋊C4, C24.152D4

Smallest permutation representation of C24.152D4
►On 64 points
Generators in S64
(1 12)(2 9)(3 14)(4 11)(5 16)(6 13)(7 10)(8 15)(17 59)(18 64)(19 61)(20 58)(21 63)(22 60)(23 57)(24 62)(25 46)(26 43)(27 48)(28 45)(29 42)(30 47)(31 44)(32 41)(33 55)(34 52)(35 49)(36 54)(37 51)(38 56)(39 53)(40 50)

(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)

(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)

(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)

(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)

(1 64 44 49)(2 21 45 38)(3 62 46 55)(4 19 47 36)(5 60 48 53)(6 17 41 34)(7 58 42 51)(8 23 43 40)(9 59 28 52)(10 24 29 33)(11 57 30 50)(12 22 31 39)(13 63 32 56)(14 20 25 37)(15 61 26 54)(16 18 27 35)
 
G:=sub<Sym(64)| (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15)(17,59)(18,64)(19,61)(20,58)(21,63)(22,60)(23,57)(24,62)(25,46)(26,43)(27,48)(28,45)(29,42)(30,47)(31,44)(32,41)(33,55)(34,52)(35,49)(36,54)(37,51)(38,56)(39,53)(40,50), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(25,46)(26,47)(27,48)(28,41)(29,42)(30,43)(31,44)(32,45)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (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), (1,64,44,49)(2,21,45,38)(3,62,46,55)(4,19,47,36)(5,60,48,53)(6,17,41,34)(7,58,42,51)(8,23,43,40)(9,59,28,52)(10,24,29,33)(11,57,30,50)(12,22,31,39)(13,63,32,56)(14,20,25,37)(15,61,26,54)(16,18,27,35)>;
 
G:=Group( (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15)(17,59)(18,64)(19,61)(20,58)(21,63)(22,60)(23,57)(24,62)(25,46)(26,43)(27,48)(28,45)(29,42)(30,47)(31,44)(32,41)(33,55)(34,52)(35,49)(36,54)(37,51)(38,56)(39,53)(40,50), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(25,46)(26,47)(27,48)(28,41)(29,42)(30,43)(31,44)(32,45)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,57), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (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), (1,64,44,49)(2,21,45,38)(3,62,46,55)(4,19,47,36)(5,60,48,53)(6,17,41,34)(7,58,42,51)(8,23,43,40)(9,59,28,52)(10,24,29,33)(11,57,30,50)(12,22,31,39)(13,63,32,56)(14,20,25,37)(15,61,26,54)(16,18,27,35) );
 
G=PermutationGroup([[(1,12),(2,9),(3,14),(4,11),(5,16),(6,13),(7,10),(8,15),(17,59),(18,64),(19,61),(20,58),(21,63),(22,60),(23,57),(24,62),(25,46),(26,43),(27,48),(28,45),(29,42),(30,47),(31,44),(32,41),(33,55),(34,52),(35,49),(36,54),(37,51),(38,56),(39,53),(40,50)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,57),(24,58),(25,46),(26,47),(27,48),(28,41),(29,42),(30,43),(31,44),(32,45),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,49),(40,50)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,52),(18,53),(19,54),(20,55),(21,56),(22,49),(23,50),(24,51),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,57)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(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)], [(1,64,44,49),(2,21,45,38),(3,62,46,55),(4,19,47,36),(5,60,48,53),(6,17,41,34),(7,58,42,51),(8,23,43,40),(9,59,28,52),(10,24,29,33),(11,57,30,50),(12,22,31,39),(13,63,32,56),(14,20,25,37),(15,61,26,54),(16,18,27,35)]])
 

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4X8A···8H
order12···222224···44···48···8
size11···122222···24···44···4

44 irreducible representations

dim1111111122244
type++++++-++-
imageC1C2C2C2C2C4C4C4D4Q8D4C8⋊C22C8.C22
kernelC24.152D4C22.4Q16C22×C4⋊C4C2×C42⋊C2C22×M4(2)C2×C4⋊C4C42⋊C2C2×M4(2)C22×C4C22×C4C24C22C22
# reps1411188852122

Matrix representation of C24.152D4 ►in GL8(𝔽17)

10000000
01000000
001600000
000160000
000016000
000001600
00001010
00001001
,
160000000
016000000
00100000
00010000
000016000
000001600
000000160
000000016
,
10000000
01000000
001600000
000160000
000016000
000001600
000000160
000000016
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
10000000
016000000
001300000
00040000
0000160150
000000161
00000110
00001010
,
04000000
40000000
000130000
001300000
000090153
000016069
000081017
00007117

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,15,16,1,1,0,0,0,0,0,1,0,0],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,9,16,8,7,0,0,0,0,0,0,10,1,0,0,0,0,15,6,1,1,0,0,0,0,3,9,7,7] >;
 

C24.152D4 in GAP, Magma, Sage, TeX 

C_2^4._{152}D_4
 
% in TeX
 
G:=Group("C2^4.152D4");
 
// GroupNames label
 
G:=SmallGroup(128,468);
 
// by ID
 
G=gap.SmallGroup(128,468);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,2019,248,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=c*b=b*c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*d=d*b,b*e=e*b,b*f=f*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=b*d*e^3>;
 
// generators/relations
 

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