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G = C24.52D4  order 128 = 27

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

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C24.52D4, C25.5C22, C24.630C23, C24.43(C2×C4), (C22×C4).122D4, C22.123(C4×D4), C23.720(C2×D4), (C23×C4).8C22, C22.67C22≀C2, C23.343(C4○D4), C22.99(C4⋊D4), C23.39(C22⋊C4), C23.299(C22×C4), C2.3(C23.10D4), C2.4(C23.34D4), C2.2(C23.11D4), C2.7(C23.23D4), C22.49(C4.4D4), C22.73(C42⋊C2), C22.24(C42⋊2C2), C2.6(C24.C22), C22.72(C22.D4), (C2×C22⋊C4)⋊12C4, (C22×C4).98(C2×C4), (C2×C2.C42)⋊4C2, (C22×C22⋊C4).4C2, C22.140(C2×C22⋊C4), SmallGroup(128,172)

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

Derived series
C1 — C23 — C24.52D4
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C22⋊C4 — C24.52D4
Lower central
C1 — C23 — C24.52D4
Upper central
C1 — C24 — C24.52D4
Jennings
C1 — C24 — C24.52D4

Generators and relations for C24.52D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=d, faf-1=ab=ba, ac=ca, ad=da, eae-1=abc, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bde-1 >

Subgroups: 788 in 368 conjugacy classes, 108 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C23, C23, C23, C22⋊C4, C22×C4, C22×C4, C24, C24, C24, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C25, C2×C2.C42, C22×C22⋊C4, C22×C22⋊C4, C24.52D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C23.34D4, C23.23D4, C24.C22, C23.10D4, C23.11D4, C24.52D4

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

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

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

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

44 conjugacy classes

class 1 2A···2O2P2Q2R2S4A···4X
order12···222224···4
size11···144444···4

44 irreducible representations

dim1111222
type+++++
imageC1C2C2C4D4D4C4○D4
kernelC24.52D4C2×C2.C42C22×C22⋊C4C2×C22⋊C4C22×C4C24C23
# reps14388416

Matrix representation of C24.52D4 ►in GL7(𝔽5)

1000000
0400000
0310000
0001000
0004400
0000010
0000034
,
1000000
0400000
0040000
0004000
0000400
0000010
0000001
,
1000000
0400000
0040000
0004000
0000400
0000040
0000004
,
4000000
0400000
0040000
0001000
0000100
0000010
0000001
,
3000000
0400000
0040000
0004000
0001100
0000011
0000004
,
2000000
0230000
0030000
0003100
0002200
0000040
0000004

G:=sub<GL(7,GF(5))| [1,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4],[2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,3,2,0,0,0,0,0,1,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4] >;
 

C24.52D4 in GAP, Magma, Sage, TeX 

C_2^4._{52}D_4
 
% in TeX
 
G:=Group("C2^4.52D4");
 
// GroupNames label
 
G:=SmallGroup(128,172);
 
// by ID
 
G=gap.SmallGroup(128,172);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,2,448,141,422,387,58]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=d,f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c,b*c=c*b,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^-1>;
 
// generators/relations
 

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