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G = C23.21M4(2)  order 128 = 27

3rd non-split extension by C23 of M4(2) acting via M4(2)/C8=C2

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

Aliases: C23.21M4(2), C22⋊C43C8, C2.10(C8×D4), C222(C4⋊C8), (C2×C8).319D4, (C23×C8).7C2, C2.4(C89D4), C24.95(C2×C4), C23.19(C2×C8), C22.98(C4×D4), C4.114C22≀C2, (C22×C4).79Q8, C23.70(C4⋊C4), (C22×C4).680D4, C22.29(C8○D4), C4.111(C22⋊Q8), C22.39(C22×C8), (C22×C8).25C22, C2.C42.19C4, (C23×C4).635C22, (C2×C42).261C22, C22.7C426C2, C23.268(C22×C4), C22.50(C2×M4(2)), C2.3(C23.8Q8), (C22×C4).1627C23, C4.131(C22.D4), C2.3(C42.6C22), (C2×C4)⋊2(C2×C8), (C2×C4⋊C8)⋊13C2, C2.11(C2×C4⋊C8), (C2×C4).90(C4⋊C4), C22.66(C2×C4⋊C4), (C2×C4).341(C2×Q8), (C2×C4).1525(C2×D4), (C2×C22⋊C8).20C2, (C4×C22⋊C4).13C2, (C2×C22⋊C4).28C4, (C2×C4).933(C4○D4), (C22×C4).116(C2×C4), SmallGroup(128,582)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.21M4(2)
C1C2C4C2×C4C22×C4C23×C4C23×C8 — C23.21M4(2)
C1C22 — C23.21M4(2)
C1C22×C4 — C23.21M4(2)
C1C2C2C22×C4 — C23.21M4(2)

Generators and relations for C23.21M4(2)
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=b, ab=ba, dad-1=eae-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd5 >

Subgroups: 276 in 174 conjugacy classes, 80 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×8], C22 [×3], C22 [×8], C22 [×12], C8 [×6], C2×C4 [×2], C2×C4 [×10], C2×C4 [×20], C23, C23 [×6], C23 [×4], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×2], C2×C8 [×4], C2×C8 [×18], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C2.C42 [×2], C22⋊C8 [×2], C4⋊C8 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C22×C8 [×2], C22×C8 [×2], C22×C8 [×6], C23×C4, C22.7C42 [×2], C4×C22⋊C4, C2×C22⋊C8, C2×C4⋊C8 [×2], C23×C8, C23.21M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C4⋊C8 [×4], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C22×C8, C2×M4(2), C8○D4 [×2], C23.8Q8, C2×C4⋊C8, C42.6C22, C8×D4 [×2], C89D4 [×2], C23.21M4(2)

Smallest permutation representation of C23.21M4(2)
On 64 points
Generators in S64
(1 43)(2 31)(3 45)(4 25)(5 47)(6 27)(7 41)(8 29)(9 58)(10 18)(11 60)(12 20)(13 62)(14 22)(15 64)(16 24)(17 35)(19 37)(21 39)(23 33)(26 53)(28 55)(30 49)(32 51)(34 57)(36 59)(38 61)(40 63)(42 56)(44 50)(46 52)(48 54)
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 56)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 33)(16 34)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 57)(25 46)(26 47)(27 48)(28 41)(29 42)(30 43)(31 44)(32 45)
(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 17 47 13)(2 10 48 22)(3 19 41 15)(4 12 42 24)(5 21 43 9)(6 14 44 18)(7 23 45 11)(8 16 46 20)(25 61 56 34)(26 39 49 58)(27 63 50 36)(28 33 51 60)(29 57 52 38)(30 35 53 62)(31 59 54 40)(32 37 55 64)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I4J4K4L4M···4T8A···8P8Q···8X
order12···222224···444444···48···88···8
size11···122221···122224···42···24···4

56 irreducible representations

dim111111111222222
type++++++++-
imageC1C2C2C2C2C2C4C4C8D4D4Q8C4○D4M4(2)C8○D4
kernelC23.21M4(2)C22.7C42C4×C22⋊C4C2×C22⋊C8C2×C4⋊C8C23×C8C2.C42C2×C22⋊C4C22⋊C4C2×C8C22×C4C22×C4C2×C4C23C22
# reps1211214416422448

Matrix representation of C23.21M4(2) in GL5(𝔽17)

160000
016000
00100
000160
000016
,
160000
01000
00100
00010
00001
,
10000
016000
001600
00010
00001
,
20000
00400
01000
000013
00010
,
40000
00800
015000
00002
00090

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[2,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,13,0],[4,0,0,0,0,0,0,15,0,0,0,8,0,0,0,0,0,0,0,9,0,0,0,2,0] >;

C23.21M4(2) in GAP, Magma, Sage, TeX

C_2^3._{21}M_4(2)
% in TeX

G:=Group("C2^3.21M4(2)");
// GroupNames label

G:=SmallGroup(128,582);
// by ID

G=gap.SmallGroup(128,582);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^5>;
// generators/relations

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