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G = C22.142C25order 128 = 27

123rd central stem extension by C22 of C25

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

Aliases: C23.83C24, C42.125C23, C22.142C25, C22.52- 1+4, D43Q839C2, C4⋊C4.327C23, (C2×C4).132C24, C4⋊Q8.227C22, (C2×D4).333C23, (C4×D4).252C22, C22⋊C4.56C23, (C2×Q8).311C23, (C4×Q8).238C22, C4⋊D4.236C22, (C2×C42).970C22, (C22×C4).402C23, C22⋊Q8.126C22, C2.48(C2×2- 1+4), C2.54(C2.C25), C422C2.24C22, C22.58C242C2, C4.4D4.182C22, C42.C2.166C22, C42⋊C2.246C22, C23.36C2353C2, C22.35C2420C2, C22.47C2436C2, C22.56C2411C2, C22.46C2436C2, C22.57C2413C2, C22.33C2418C2, C22.D4.18C22, (C2×C42.C2)⋊50C2, (C2×C4⋊C4).723C22, SmallGroup(128,2285)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.142C25
C1C2C22C23C22×C4C2×C42C2×C42.C2 — C22.142C25
C1C22 — C22.142C25
C1C22 — C22.142C25
C1C22 — C22.142C25

Generators and relations for C22.142C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=b, e2=ba=ab, f2=a, dcd-1=gcg=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf-1=bc=cb, ede-1=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 620 in 466 conjugacy classes, 380 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×26], C22, C22 [×2], C22 [×14], C2×C4 [×2], C2×C4 [×24], C2×C4 [×20], D4 [×14], Q8 [×6], C23, C23 [×4], C42 [×2], C42 [×14], C22⋊C4 [×36], C4⋊C4 [×68], C22×C4, C22×C4 [×18], C2×D4 [×10], C2×Q8 [×6], C2×C42, C2×C4⋊C4 [×10], C42⋊C2 [×8], C4×D4 [×18], C4×Q8 [×6], C4⋊D4 [×10], C22⋊Q8 [×26], C22.D4 [×20], C4.4D4 [×2], C42.C2 [×2], C42.C2 [×28], C422C2 [×20], C4⋊Q8 [×4], C2×C42.C2, C23.36C23 [×2], C22.33C24 [×8], C22.35C24 [×4], C22.46C24 [×4], C22.47C24 [×4], D43Q8 [×4], C22.56C24, C22.57C24 [×2], C22.58C24, C22.142C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2- 1+4 [×2], C25, C2×2- 1+4, C2.C25 [×2], C22.142C25

Smallest permutation representation of C22.142C25
On 64 points
Generators in S64
(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(1 19)(2 32)(3 17)(4 30)(5 56)(6 41)(7 54)(8 43)(9 57)(10 46)(11 59)(12 48)(13 63)(14 40)(15 61)(16 38)(18 50)(20 52)(21 45)(22 58)(23 47)(24 60)(25 39)(26 64)(27 37)(28 62)(29 49)(31 51)(33 53)(34 42)(35 55)(36 44)
(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 57 49 47)(2 60 50 46)(3 59 51 45)(4 58 52 48)(5 16 34 26)(6 15 35 25)(7 14 36 28)(8 13 33 27)(9 31 23 17)(10 30 24 20)(11 29 21 19)(12 32 22 18)(37 41 63 55)(38 44 64 54)(39 43 61 53)(40 42 62 56)
(1 55 51 43)(2 44 52 56)(3 53 49 41)(4 42 50 54)(5 30 36 18)(6 19 33 31)(7 32 34 20)(8 17 35 29)(9 25 21 13)(10 14 22 26)(11 27 23 15)(12 16 24 28)(37 45 61 57)(38 58 62 46)(39 47 63 59)(40 60 64 48)
(1 11)(2 12)(3 9)(4 10)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4AB
order122222222244444···4
size111122444422224···4

38 irreducible representations

dim1111111111144
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C22- 1+4C2.C25
kernelC22.142C25C2×C42.C2C23.36C23C22.33C24C22.35C24C22.46C24C22.47C24D43Q8C22.56C24C22.57C24C22.58C24C22C2
# reps1128444412124

Matrix representation of C22.142C25 in GL8(𝔽5)

40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
02000000
30000000
02240000
14330000
00000010
00000004
00001000
00000400
,
00100000
04430000
10000000
20310000
00000020
00000002
00002000
00000200
,
20000000
02000000
00200000
00020000
00000010
00000001
00004000
00000400
,
30000000
03000000
00200000
02020000
00000100
00001000
00000001
00000010
,
01000000
10000000
04430000
23010000
00001000
00000100
00000010
00000001

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,3,0,1,0,0,0,0,2,0,2,4,0,0,0,0,0,0,2,3,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0],[0,0,1,2,0,0,0,0,0,4,0,0,0,0,0,0,1,4,0,3,0,0,0,0,0,3,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[3,0,0,0,0,0,0,0,0,3,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,1,0,2,0,0,0,0,1,0,4,3,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C22.142C25 in GAP, Magma, Sage, TeX

C_2^2._{142}C_2^5
% in TeX

G:=Group("C2^2.142C2^5");
// GroupNames label

G:=SmallGroup(128,2285);
// by ID

G=gap.SmallGroup(128,2285);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,723,520,2019,570,136,1684]);
// Polycyclic

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

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