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G = C24.52D4order 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(C422C2), 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

C1C23 — C24.52D4
C1C2C22C23C24C23×C4C22×C22⋊C4 — C24.52D4
C1C23 — C24.52D4
C1C24 — C24.52D4
C1C24 — 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 [×14], C2 [×4], C4 [×12], C22, C22 [×34], C22 [×36], C2×C4 [×60], C23, C23 [×18], C23 [×52], C22⋊C4 [×24], C22×C4 [×12], C22×C4 [×36], C24, C24 [×6], C24 [×12], C2.C42 [×8], C2×C22⋊C4 [×4], C2×C22⋊C4 [×16], C23×C4 [×6], C25, C2×C2.C42 [×4], C22×C22⋊C4, C22×C22⋊C4 [×2], C24.52D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×6], C4○D4 [×8], C2×C22⋊C4, C42⋊C2 [×2], C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×8], C22.D4 [×10], C4.4D4 [×4], C422C2 [×4], C23.34D4, C23.23D4 [×2], C24.C22 [×4], C23.10D4 [×4], C23.11D4 [×4], C24.52D4

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

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

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

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