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G = C42.174D4order 128 = 27

156th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.174D4, C23.465C24, C22.1892- 1+4, (C2×Q8).23Q8, C4.57(C22⋊Q8), C429C4.30C2, C428C4.33C2, C2.18(Q83Q8), (C2×C42).566C22, (C22×C4).841C23, C22.316(C22×D4), C22.106(C22×Q8), (C22×Q8).436C22, C23.81C23.16C2, C2.C42.201C22, C23.63C23.27C2, C23.67C23.40C2, C2.42(C22.26C24), C2.25(C23.38C23), C2.65(C22.46C24), (C4×C4⋊C4).66C2, (C2×C4×Q8).34C2, (C2×C4).54(C2×Q8), (C2×C4).359(C2×D4), C2.33(C2×C22⋊Q8), (C2×C4).150(C4○D4), (C2×C4⋊C4).313C22, C22.341(C2×C4○D4), (C2×C42.C2).21C2, SmallGroup(128,1297)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.174D4
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C42.174D4
C1C23 — C42.174D4
C1C23 — C42.174D4
C1C23 — C42.174D4

Generators and relations for C42.174D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 356 in 226 conjugacy classes, 112 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×20], C22 [×3], C22 [×4], C2×C4 [×18], C2×C4 [×36], Q8 [×8], C23, C42 [×4], C42 [×8], C4⋊C4 [×30], C22×C4 [×3], C22×C4 [×12], C2×Q8 [×4], C2×Q8 [×4], C2.C42 [×14], C2×C42 [×3], C2×C42 [×2], C2×C4⋊C4 [×3], C2×C4⋊C4 [×12], C4×Q8 [×4], C42.C2 [×4], C22×Q8, C4×C4⋊C4, C428C4, C429C4, C23.63C23 [×4], C23.67C23 [×2], C23.81C23 [×4], C2×C4×Q8, C2×C42.C2, C42.174D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×6], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4 [×3], 2- 1+4 [×2], C2×C22⋊Q8, C22.26C24, C23.38C23, C22.46C24 [×2], Q83Q8 [×2], C42.174D4

Smallest permutation representation of C42.174D4
Regular action on 128 points
Generators in S128
(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)(65 66 67 68)(69 70 71 72)(73 74 75 76)(77 78 79 80)(81 82 83 84)(85 86 87 88)(89 90 91 92)(93 94 95 96)(97 98 99 100)(101 102 103 104)(105 106 107 108)(109 110 111 112)(113 114 115 116)(117 118 119 120)(121 122 123 124)(125 126 127 128)
(1 13 9 41)(2 14 10 42)(3 15 11 43)(4 16 12 44)(5 48 38 20)(6 45 39 17)(7 46 40 18)(8 47 37 19)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 69)(30 62 58 70)(31 63 59 71)(32 64 60 72)(33 123 127 91)(34 124 128 92)(35 121 125 89)(36 122 126 90)(65 111 100 87)(66 112 97 88)(67 109 98 85)(68 110 99 86)(73 101 105 77)(74 102 106 78)(75 103 107 79)(76 104 108 80)(81 93 113 117)(82 94 114 118)(83 95 115 119)(84 96 116 120)
(1 59 51 45)(2 58 52 48)(3 57 49 47)(4 60 50 46)(5 42 62 56)(6 41 63 55)(7 44 64 54)(8 43 61 53)(9 31 23 17)(10 30 24 20)(11 29 21 19)(12 32 22 18)(13 71 27 39)(14 70 28 38)(15 69 25 37)(16 72 26 40)(33 105 98 95)(34 108 99 94)(35 107 100 93)(36 106 97 96)(65 117 125 75)(66 120 126 74)(67 119 127 73)(68 118 128 76)(77 85 115 123)(78 88 116 122)(79 87 113 121)(80 86 114 124)(81 89 103 111)(82 92 104 110)(83 91 101 109)(84 90 102 112)
(1 89 3 91)(2 124 4 122)(5 76 7 74)(6 107 8 105)(9 121 11 123)(10 92 12 90)(13 125 15 127)(14 34 16 36)(17 79 19 77)(18 102 20 104)(21 85 23 87)(22 112 24 110)(25 67 27 65)(26 97 28 99)(29 115 31 113)(30 82 32 84)(33 41 35 43)(37 73 39 75)(38 108 40 106)(42 128 44 126)(45 103 47 101)(46 78 48 80)(49 109 51 111)(50 88 52 86)(53 98 55 100)(54 66 56 68)(57 83 59 81)(58 114 60 116)(61 95 63 93)(62 118 64 120)(69 119 71 117)(70 94 72 96)

G:=sub<Sym(128)| (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)(65,66,67,68)(69,70,71,72)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96)(97,98,99,100)(101,102,103,104)(105,106,107,108)(109,110,111,112)(113,114,115,116)(117,118,119,120)(121,122,123,124)(125,126,127,128), (1,13,9,41)(2,14,10,42)(3,15,11,43)(4,16,12,44)(5,48,38,20)(6,45,39,17)(7,46,40,18)(8,47,37,19)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,69)(30,62,58,70)(31,63,59,71)(32,64,60,72)(33,123,127,91)(34,124,128,92)(35,121,125,89)(36,122,126,90)(65,111,100,87)(66,112,97,88)(67,109,98,85)(68,110,99,86)(73,101,105,77)(74,102,106,78)(75,103,107,79)(76,104,108,80)(81,93,113,117)(82,94,114,118)(83,95,115,119)(84,96,116,120), (1,59,51,45)(2,58,52,48)(3,57,49,47)(4,60,50,46)(5,42,62,56)(6,41,63,55)(7,44,64,54)(8,43,61,53)(9,31,23,17)(10,30,24,20)(11,29,21,19)(12,32,22,18)(13,71,27,39)(14,70,28,38)(15,69,25,37)(16,72,26,40)(33,105,98,95)(34,108,99,94)(35,107,100,93)(36,106,97,96)(65,117,125,75)(66,120,126,74)(67,119,127,73)(68,118,128,76)(77,85,115,123)(78,88,116,122)(79,87,113,121)(80,86,114,124)(81,89,103,111)(82,92,104,110)(83,91,101,109)(84,90,102,112), (1,89,3,91)(2,124,4,122)(5,76,7,74)(6,107,8,105)(9,121,11,123)(10,92,12,90)(13,125,15,127)(14,34,16,36)(17,79,19,77)(18,102,20,104)(21,85,23,87)(22,112,24,110)(25,67,27,65)(26,97,28,99)(29,115,31,113)(30,82,32,84)(33,41,35,43)(37,73,39,75)(38,108,40,106)(42,128,44,126)(45,103,47,101)(46,78,48,80)(49,109,51,111)(50,88,52,86)(53,98,55,100)(54,66,56,68)(57,83,59,81)(58,114,60,116)(61,95,63,93)(62,118,64,120)(69,119,71,117)(70,94,72,96)>;

G:=Group( (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)(65,66,67,68)(69,70,71,72)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96)(97,98,99,100)(101,102,103,104)(105,106,107,108)(109,110,111,112)(113,114,115,116)(117,118,119,120)(121,122,123,124)(125,126,127,128), (1,13,9,41)(2,14,10,42)(3,15,11,43)(4,16,12,44)(5,48,38,20)(6,45,39,17)(7,46,40,18)(8,47,37,19)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,61,57,69)(30,62,58,70)(31,63,59,71)(32,64,60,72)(33,123,127,91)(34,124,128,92)(35,121,125,89)(36,122,126,90)(65,111,100,87)(66,112,97,88)(67,109,98,85)(68,110,99,86)(73,101,105,77)(74,102,106,78)(75,103,107,79)(76,104,108,80)(81,93,113,117)(82,94,114,118)(83,95,115,119)(84,96,116,120), (1,59,51,45)(2,58,52,48)(3,57,49,47)(4,60,50,46)(5,42,62,56)(6,41,63,55)(7,44,64,54)(8,43,61,53)(9,31,23,17)(10,30,24,20)(11,29,21,19)(12,32,22,18)(13,71,27,39)(14,70,28,38)(15,69,25,37)(16,72,26,40)(33,105,98,95)(34,108,99,94)(35,107,100,93)(36,106,97,96)(65,117,125,75)(66,120,126,74)(67,119,127,73)(68,118,128,76)(77,85,115,123)(78,88,116,122)(79,87,113,121)(80,86,114,124)(81,89,103,111)(82,92,104,110)(83,91,101,109)(84,90,102,112), (1,89,3,91)(2,124,4,122)(5,76,7,74)(6,107,8,105)(9,121,11,123)(10,92,12,90)(13,125,15,127)(14,34,16,36)(17,79,19,77)(18,102,20,104)(21,85,23,87)(22,112,24,110)(25,67,27,65)(26,97,28,99)(29,115,31,113)(30,82,32,84)(33,41,35,43)(37,73,39,75)(38,108,40,106)(42,128,44,126)(45,103,47,101)(46,78,48,80)(49,109,51,111)(50,88,52,86)(53,98,55,100)(54,66,56,68)(57,83,59,81)(58,114,60,116)(61,95,63,93)(62,118,64,120)(69,119,71,117)(70,94,72,96) );

G=PermutationGroup([(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),(65,66,67,68),(69,70,71,72),(73,74,75,76),(77,78,79,80),(81,82,83,84),(85,86,87,88),(89,90,91,92),(93,94,95,96),(97,98,99,100),(101,102,103,104),(105,106,107,108),(109,110,111,112),(113,114,115,116),(117,118,119,120),(121,122,123,124),(125,126,127,128)], [(1,13,9,41),(2,14,10,42),(3,15,11,43),(4,16,12,44),(5,48,38,20),(6,45,39,17),(7,46,40,18),(8,47,37,19),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,61,57,69),(30,62,58,70),(31,63,59,71),(32,64,60,72),(33,123,127,91),(34,124,128,92),(35,121,125,89),(36,122,126,90),(65,111,100,87),(66,112,97,88),(67,109,98,85),(68,110,99,86),(73,101,105,77),(74,102,106,78),(75,103,107,79),(76,104,108,80),(81,93,113,117),(82,94,114,118),(83,95,115,119),(84,96,116,120)], [(1,59,51,45),(2,58,52,48),(3,57,49,47),(4,60,50,46),(5,42,62,56),(6,41,63,55),(7,44,64,54),(8,43,61,53),(9,31,23,17),(10,30,24,20),(11,29,21,19),(12,32,22,18),(13,71,27,39),(14,70,28,38),(15,69,25,37),(16,72,26,40),(33,105,98,95),(34,108,99,94),(35,107,100,93),(36,106,97,96),(65,117,125,75),(66,120,126,74),(67,119,127,73),(68,118,128,76),(77,85,115,123),(78,88,116,122),(79,87,113,121),(80,86,114,124),(81,89,103,111),(82,92,104,110),(83,91,101,109),(84,90,102,112)], [(1,89,3,91),(2,124,4,122),(5,76,7,74),(6,107,8,105),(9,121,11,123),(10,92,12,90),(13,125,15,127),(14,34,16,36),(17,79,19,77),(18,102,20,104),(21,85,23,87),(22,112,24,110),(25,67,27,65),(26,97,28,99),(29,115,31,113),(30,82,32,84),(33,41,35,43),(37,73,39,75),(38,108,40,106),(42,128,44,126),(45,103,47,101),(46,78,48,80),(49,109,51,111),(50,88,52,86),(53,98,55,100),(54,66,56,68),(57,83,59,81),(58,114,60,116),(61,95,63,93),(62,118,64,120),(69,119,71,117),(70,94,72,96)])

38 conjugacy classes

class 1 2A···2G4A···4H4I···4Z4AA4AB4AC4AD
order12···24···44···44444
size11···12···24···48888

38 irreducible representations

dim1111111112224
type++++++++++--
imageC1C2C2C2C2C2C2C2C2D4Q8C4○D42- 1+4
kernelC42.174D4C4×C4⋊C4C428C4C429C4C23.63C23C23.67C23C23.81C23C2×C4×Q8C2×C42.C2C42C2×Q8C2×C4C22
# reps11114241144122

Matrix representation of C42.174D4 in GL6(𝔽5)

200000
030000
004000
000400
000040
000021
,
400000
040000
001000
000100
000030
000042
,
010000
400000
000100
004000
000020
000013
,
020000
200000
000100
001000
000044
000001

G:=sub<GL(6,GF(5))| [2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,2,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,4,0,0,0,0,0,2],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,1,0,0,0,0,0,3],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,4,1] >;

C42.174D4 in GAP, Magma, Sage, TeX

C_4^2._{174}D_4
% in TeX

G:=Group("C4^2.174D4");
// GroupNames label

G:=SmallGroup(128,1297);
// by ID

G=gap.SmallGroup(128,1297);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,723,184,675,80]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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