Copied to
clipboard

G = C24.372C23  order 128 = 27

212nd non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.372C23, C23.535C24, C22.3112+ (1+4), (C22×C4)⋊37D4, C23⋊2D4⋊27C2, C23.198(C2×D4), C23.67(C4○D4), C23.4Q8⋊31C2, C23.10D4⋊61C2, C23.34D4⋊43C2, C23.23D4⋊71C2, C2.28(C23⋊3D4), (C23×C4).433C22, (C2×C42).612C22, (C22×C4).145C23, C22.360(C22×D4), (C22×D4).543C22, C23.83C23⋊61C2, C24.C22⋊106C2, C2.41(C22.32C24), C2.86(C22.19C24), C2.43(C22.29C24), C2.C42.260C22, C2.28(C22.34C24), (C2×C4×D4)⋊54C2, (C2×C4⋊D4)⋊24C2, (C2×C4).394(C2×D4), (C2×C4).170(C4○D4), (C2×C4⋊C4).362C22, C22.407(C2×C4○D4), (C2×C22⋊C4).223C22, SmallGroup(128,1367)

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

Derived series
C1 — C23 — C24.372C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C23.23D4 — C24.372C23
Lower central
C1 — C23 — C24.372C23
Upper central
C1 — C23 — C24.372C23
Jennings
C1 — C23 — C24.372C23

Subgroups: 708 in 310 conjugacy classes, 96 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×14], C22 [×3], C22 [×4], C22 [×34], C2×C4 [×6], C2×C4 [×38], D4 [×28], C23, C23 [×4], C23 [×26], C42 [×2], C22⋊C4 [×20], C4⋊C4 [×6], C22×C4 [×5], C22×C4 [×10], C22×C4 [×6], C2×D4 [×26], C24 [×2], C24 [×2], C2.C42 [×2], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C4×D4 [×4], C4⋊D4 [×4], C23×C4 [×2], C22×D4 [×2], C22×D4 [×4], C23.34D4, C23.23D4 [×2], C24.C22 [×2], C23⋊2D4, C23⋊2D4 [×2], C23.10D4, C23.10D4 [×2], C23.4Q8, C23.83C23, C2×C4×D4, C2×C4⋊D4, C24.372C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ (1+4) [×4], C22.19C24, C23⋊3D4, C22.29C24, C22.32C24 [×2], C22.34C24 [×2], C24.372C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=cb=bc, eae=ab=ba, faf=ac=ca, ad=da, ag=ga, bd=db, be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef=de=ed, df=fd, dg=gd, eg=ge >

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

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

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

31000000
22000000
00100000
00010000
00003232
00000030
00000200
00001402
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00130000
00040000
00000010
00001414
00001000
00000001
,
10000000
44000000
00100000
00140000
00001000
00000400
00000010
00002024
,
30000000
03000000
00400000
00040000
00000100
00001000
00001414
00000004

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4L4M···4R
order12···222222244444···44···4
size11···144448822224···48···8

32 irreducible representations

dim11111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ (1+4)
kernelC24.372C23C23.34D4C23.23D4C24.C22C23⋊2D4C23.10D4C23.4Q8C23.83C23C2×C4×D4C2×C4⋊D4C22×C4C2×C4C23C22
# reps11223311114444

In GAP, Magma, Sage, TeX 

C_2^4._{372}C_2^3
 
% in TeX
 
G:=Group("C2^4.372C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1367);
 
// by ID
 
G=gap.SmallGroup(128,1367);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,185,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^2=c*b=b*c,e*a*e=a*b=b*a,f*a*f=a*c=c*a,a*d=d*a,a*g=g*a,b*d=d*b,b*e=e*b,g*f*g^-1=b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀